Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

0.999...= 1

en.wikipedia.org

Post reply on HN
Print view View original

401–410 of 647 posts

Re: 0.999...= 1

#401

Post by dahart »

CogitoCogito wrote: post #286
dahart wrote: post #257

Earlier quoted context omitted.

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…

I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste.

> does it matter if the proof of a fact is incorrect if the fact itself is correct?

Your language isn't allowing for a notion of precision, or for multiple forms of correctness, and it's not considering audience, communication or level of expertise either. I don't think it's a question of correct vs incorrect, I think you're asking for more precision, and/or for a form that meets your own higher standard.

It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition.

BTW, reading the blog post you linked to on surreals, the "proof" looks to me to be more hand-wavy than Euler's proof that .9bar = 1. The proof begins by stating there are a finite number of 9s, in direct contradiction to the hypothesis. 10^-inf = 0, so from this blog post I don't yet see any reason why surreals clarify anything here, it feels like the opposite, it feels like obfuscation.

This could be an argument over representation and not the values of numbers. If you start by defining 0.9bar to be a different number than 1 for the specific reason that it's written down a different way, then fine. That's what the surreal "proof" tells me. Euler's proof is talking about the value of 0.9bar in the limit, not the representation. (Even if that's stated without rigorous definitions of limits.) The proof is saying the values of .9bar and 1 are the same. If the surreal number .9bar were strictly less than 1, that must mean there's another surreal number closer to 1, but there isn't, so I don't accept the surreal argument as valid logic, other than playing a semantic trick by saying 'look I defined they way we write a number to be meaningful, therefore 0.999... is by definition different than 1.'

By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."

Re: 0.999...= 1

#402

Post by jl2718 »

The proof relies on the assertion that the supremum of an increasing sequence is equal to the limit. This is mathematical dogma, and should be introduced as such. Once that is accepted, it becomes obvious.

This is illustrative of what I see as a fundamental problem in mathematics education: nobody ever teaches the rules. In this case, the rules of simple arithmetic hit a dead end for mathematicians, so they invented a new rule that allowed them to go further without breaking any old rules. This is generally acceptable in proofs, although it can have significant implications, such as two mutually exclusive but otherwise acceptable rules causing a divergence in fields of study.

When I was taught this, it was like, “Look how smart I am for applying this obtusely-stated limit rule that you were never told about.” This is how you keep people out of math. The point of teaching it is to make it easy, not hard.

Re: 0.999...= 1

#403

Post by fourseventy »

The best way to think about it is that 1 - 0.999... = 0.000...

The result of 1 minus 0.999... is 0.000 with zeroes that go to infinity. And I think its easier to reason that 0.000 with repeating zeroes forever is in fact equal to zero.

Re: 0.999...= 1

#404

Post by hinkley »

champagneben wrote: post #51
hinkley wrote: post #40

Earlier quoted context omitted.

Yeah that’s way more complicated than it needs to be and I’m tempted to replace that whole section with: x = 0.9999... 2x = 1.9999... 2x - x = 1 x = 1

How do you jump from x = 0.999... to 2x = 1.999..

Picture an infinite ticker tape in your mind.

Write

   0.9
 + 0.9
 —————-
   1.8
Now keep extending the 9’s. You have to carry a one, so fix the carry and then add more nines.

Where people keep getting tripped up is thinking you can stop when you get tired, or die, or when the universe ends in heat death. You don’t get to stop. You never get to stop. It’s nines all the way down.

Re: 0.999...= 1

#405

Post by oh_sigh »

brutt wrote: post #322
brlewis wrote: post #317

Earlier quoted context omitted.

No, there's no 1. 2OEH8eoCRo0 is exactly right. Subtract 0.999... from 1 and you get 0.000...

0.999... + 0.000...1 = 1 0.000...1 = 1/∞ 0.999... = 1 - 1/∞

What exactly is the value of the number that ends in a 1 but has an infinite number of 9s before it?

Re: 0.999...= 1

#406

Post by throwaway894345 »

bluquark wrote: post #379
throwaway894345 wrote: post #329

Earlier quoted context omitted.

I think the problem is that people are often only taught base-10 so they confuse numerals with numbers. If you learn base 2 and then base 16 and then base pi, you start to realize that numbers are something more abstract than whichever numeral system we use to represent them. Rightly or wrongly, the way I imagine integers now is an infinite set of different numerals (base infinity?) such that there is only ever 1 dig…

Interesting. I find it funny to imagine what if you did have pictorial representations. They'd have more and more complex strokes and knots, and you would also need an infinitely large paper or infinitely precise pen in order to not end up repeating a number sooner or later as you enumerate them. Come to think of it, this is one way of thinking about the relationship between symbols and geometry.

Who says pictorials have to be 2-dimensional? ;)

Re: 0.999...= 1

#407

Post by happythomist »

analog31 wrote: post #245
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've had the same experience, even debating this topic with engineers. I think there are actually two hang-ups. 1. People have had it drilled into their heads that humans can't comprehend infinity. It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas), and that tricks such as infinity and the infinitesimal were not rigorous. Mathematical…

> It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas)...

Aquinas specifically objected to the notion of an essentially ordered infinite causal series. He had no objection to an accidentally ordered infinite causal series or other kinds of infinite series.

This distinction is extremely important for the purposes of understanding his proofs of God's existence, and people often unfairly reject his arguments because they conflate the two.

More reading here: http://edwardfeser.blogspot.com/2010/08/edwards-on-infinite-...

Re: 0.999...= 1

#408

Post by brlewis »

theincredulousk wrote: post #308
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.

great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3. I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in…

Whatever base you use, it will only be useful for rationals. You can't use any base to significantly improve the representation of pi. In base 7, 3.1 would be a nice approximation, but you can't go beyond approximations.

Re: 0.999...= 1

#409

Post by [deleted] »

RHSeeger wrote: post #335
lukifer wrote: post #228

Earlier quoted context omitted.

I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly p…

> it could just as easily be reframed as 0.000...0001 = 0 But it can't be, because there's nothing after "0.000..."; that ... goes on infinitely. It's literally "0s forever, never stopping". It's not a process of "keep adding 0s", it's the end result of never adding 0s. It's not a process, it is a noun.

[deleted]

Re: 0.999...= 1

#410

Post by andrewprock »

jl2718 wrote: post #402

The proof relies on the assertion that the supremum of an increasing sequence is equal to the limit. This is mathematical dogma, and should be introduced as such. Once that is accepted, it becomes obvious. This is illustrative of what I see as a fundamental problem in mathematics education: nobody ever teaches the rules. In this case, the rules of simple arithmetic hit a dead end for mathematicians, so they invented…

This is in large part due to the difficulty with reasoning about infinite representations. You do have to add axioms to your system to be able to reason about 0.9999...

Stating that 0.9999... = 1 without exposing these new tools meant to grapple with concepts that physically cannot be grappled with is a huge mistake.

Post reply on HN