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0.999...= 1

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Re: 0.999...= 1

#201

Post by kangnkodos »

rrmm wrote: post #48
traderjane wrote: post #38

Professor N.J. Wildberger is probably among the most well known "ultrafinitist" on YouTube. https://www.youtube.com/watch?v=WabHm1QWVCA I mention him because I would think he sympathizes with those who have concern over the meaning of this kind of notation.

Wildberger is great. His lectures that he teaches at UNSW (i think) are interesting, and he usually keeps a clear dividing line between std math and his own predilections. It threads the line between being a kook and legitimate published mathematician very finely. I actually have some sympathies with his contention that real numbers (limit points of infinite series) are somehow a different animal than rational number…

But on computers, you get things such as

console.log(0.1 + 0.2)

// 0.30000000000000004

A mathematician might say that this shows that you do not really have accurate floating point values and arithmetic in your computer, but instead something close to it.

Re: 0.999...= 1

#202

Post by solotronics »

Okx wrote: post #182
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 usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.

is there any difference between a black hole and nothing? (somewhat joking but I was thinking of a physical analogy of the limit approaching zero)

Re: 0.999...= 1

#203

Post by 0xff00ffee »

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…

Yep. Agree 100%. It is like the blue dress.

I think the problem is the repeating function. Infinite things are non-intuitive and should be presented differently.

Even here on HN you still see people confused about "convergence" and "identity". 0.999... doesn't CONVERGE, it literally is 1.

I suspect this persists even with students that have had second year college calculus that discusses convergent series and sums.

Re: 0.999...= 1

#204

Post by _bxg1 »

Okx wrote: post #182
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 usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.

Why not 0.00...1?

Re: 0.999...= 1

#205

Post by tom-thistime »

_bxg1 wrote: post #204
Okx wrote: post #182

Earlier quoted context omitted.

I usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.

Why not 0.00...1?

Then the question is, is 0.00....1 equal to zero? We use the definition of equality above and we say yes.

EDIT: The number above seems well defined. It's lim n->inf (10^-n). That's zero.

Re: 0.999...= 1

#206

Post by Sharlin »

tomtomtom1 wrote: post #180
johnhattan wrote: post #174

Earlier quoted context omitted.

I remember a conversation I had with my daughter in the car when she was starting out with algebra... Me: Is 9.999... the same as 10, or is it just really close to 10? Kid: Really close. It never gets all the way there. Me: Well then how close? What do you get when you subtract 9.999... from 10? Kid: (pause) An infinite number of zeroes. . .and then a one. . .wait, you can't do that. Me: Right. You just have an infin…

> An infinite number of zeroes. . .and then a one. . .wait, you can't do that. why not? why can't an infinitely small number exist?

It can, and infinitesimals are a part of so-called nonstandard analysis, but you cannot write an infinitesimal using decimal notation. "0.999…1" is simply meaningless, a contradiction. If you have a "…" it means there's no place where you could put a "last" digit. If "0.999…1" doesn't feel impossible enough, then what would "0.999…9" mean?

Re: 0.999...= 1

#207

Post by dacohenii »

hilbertseries wrote: post #177
bognition wrote: post #168

Earlier quoted context omitted.

No .666666 is not equal to .6700000 0.666... is equal to 0.666...7

Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?

I'm pretty sure it's a real number in that it falls within ℝ (the set of all real numbers).

Re: 0.999...= 1

#208

Post by Sharlin »

tom-thistime wrote: post #199
Sharlin wrote: post #187

Earlier quoted context omitted.

Yes, but at the same time it is common for people to insist that 0.999… only "approaches" unity as if it were a series approaching a limit, rather than an unique number. Intuition is a funny thing.

Saying "a number isn't a limit" is true, but it's only really relevant if you're talking to someone who genuinely has no idea what limits are. In actual math the number 0.99... can be defined as the limit of a series.

Every number is a limit, yes, but people think of 0.999… as a series. Not rigorously, of course, but that's a common argument even by people (perhaps especially by people) who have a highschool or even math-minor level understanding of series and limits.

Re: 0.999...= 1

#209

Post by amelius »

gspr wrote: post #176
amelius wrote: post #149

Earlier quoted context omitted.

Yes, because someone defined it that way. It is because the "limit" in 0.999... = lim[eps->0] 1-eps is implicit and defined as being applied before anything else. But you might as well define that implicit limit as applying over the entire expression. UPDATE: So instead of interpreting the expression as: (lim[eps->0] 1-eps) which is indeed false, you can also interpret the expression as: lim[eps->0] ((1-eps) which is…

> which is indeed false, you can also interpret the expression as: > lim[eps->0] ((1-eps) Assuming you don't mean some special notion of limit, I would guess that by `(1-eps) If so, `f` does indeed have a limit from above that is `false` and a limit from below that is `true`. Where do you wanna go from here? Edit: Corrected stupid wrong assertion about limit from below, d'oh.

> If so, `f` does indeed have a limit from above that is `false` and a limit from below that is `true`. Where do you wanna go from here?

That's the other way around. From above you get true, and from below you get false.

Note that 0.999... represents the limit from above for 1-eps. Hence the result is true.

Re: 0.999...= 1

#210

Post by dahart »

CogitoCogito wrote: post #193
dahart wrote: post #172

Earlier quoted context omitted.

> why can’t you have an infinite amount of 9s and then just put a 7 after that? You can. The proof still works if you do that. What you’re refusing to accept here is the definition of infinity.

> You can. The proof still works if you do that. > What you’re refusing to accept here is the definition of infinity. I'm refusing to accept the definition of infinity? I have no idea what you mean by that. Would you make the same statement were you aware that I do in fact have a PhD in mathematics in the field of analysis? That I have in fact studied logic? Just as a hypothetical scenario.

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 automatically right, lots of PhDs in math have failed the Monty Hall problem, lots of people with advanced degrees are wrong all the time. I myself am a walking example on occasion.

I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work?

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