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0.999...= 1

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Re: 0.999...= 1

#241

Post by mrighele »

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…

What set in stone the equality for me was learning about limits and series, because 0.999... is essentially a funny way to represent a serie.

Before that, despite accepting the proofs that were given to me, there was always something in the back of the brain telling me "mmmm there is something wrong in that". The only thing close to that was a reasoning like the following:

1 divided by 3 = 1/3 = 0.333..., but then 3 * 0.333... = 0.999... so 1 = 0.999...

This comment in the wikipedia page nails it down:

"The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart. Nonsense."

Re: 0.999...= 1

#242

Post by jdoliner »

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…

When I was a child I was convinced by a pretty simple conversation with my father:

Me: 0.9999... is not the same as 1

Him: Well if it's not the same is it more than 1 or less than 1?

Me: Less

Him: Okay then how much less is it?

At this point I started trying to do 1 - 0.999..., using the methods I'd been taught, and after a few iterations of "borrowing" the 1 I realized the answer was 0.000... which I was pretty convinced was equal to 0.

Re: 0.999...= 1

#243

Post by ramshorns »

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.

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
  .
  .
  .

Re: 0.999...= 1

#244

Post by oconnor663 »

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.

Though you could treat it as an infinite series in a similar way:

0.9 -> 0.99 -> 0.999 -> ... -> ?

0.1 -> 0.01 -> 0.001 -> ... -> ?

Re: 0.999...= 1

#245

Post by analog31 »

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 infinity has been a settled matter for all practical purposes since the early 20th century AFAIK.

2. Related to the above, most people also believe that there is always a gap in any knowledge, and something hiding in that gap. Thus it's perfectly natural to believe that there's something hiding between 0.999... and 1, that we just haven't found yet. Knowing for certain that there is nothing between 0.999... and 1 is regarded as a kind of arrogance.

I think the way to approach this with children is to teach math as an abstract topic, that's not necessarily rooted in the objects of everyday life. For instance there's no physics experiment that can test the necessity of any math being carried beyond roughly the 15th decimal place. Yet we enjoy exploring it anyway.

Re: 0.999...= 1

#246

Post by zests »

Is 1 a prime number? No, because we define it not to be. Why do we define it not to be a prime number? That's the real question.

Is 0.999... = 1? Yes, because we define decimal numbers to behave that way. Why do we define them to behave that way? That's the real question.

Re: 0.999...= 1

#247

Post by hota_mazi »

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…

Not only can numbers have more than one representation, but they can also have zero!

Looking at you, irrational numbers.

Re: 0.999...= 1

#248

Post by tom-thistime »

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.

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.

Re: 0.999...= 1

#249

Post by hota_mazi »

jdoliner wrote: post #242
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…

When I was a child I was convinced by a pretty simple conversation with my father: Me: 0.9999... is not the same as 1 Him: Well if it's not the same is it more than 1 or less than 1? Me: Less Him: Okay then how much less is it? At this point I started trying to do 1 - 0.999..., using the methods I'd been taught, and after a few iterations of "borrowing" the 1 I realized the answer was 0.000... which I was pretty conv…

Hehe... smart man.

Another one is that

1 / 3 * 3 = 1

0.333... * 3 = 1

0.999... = 1

Re: 0.999...= 1

#250

Post by ubercow13 »

jefftk wrote: post #151

What if you have 0.9̅4? Can we say 0.9̅5 > 0.9̅4 > 0.9̅3? More on what happens if you allow this: https://mathwithbaddrawings.com/2013/08/13/the-kaufman-decim...

That's fun but it's not clear how interesting those numbers are compared to real numbers, which have turned out to be pretty interesting over the years
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