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0.999...= 1

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Re: 0.999...= 1

#181

Post by andrewla »

CogitoCogito wrote: post #94
knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. I've never considered them right at all. By saying something like 0.9... x 10 = 9.9... and then saying that 9.9... - 0.9... = 9 you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you'r…

Limits and completeness are convenient shorthands here. But let's start from a more basic perspective.

You say

> you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 0.

Okay, well what is an alternate definition that makes more sense intuitively?

0.333...., for example, is one that seems pretty intuitive. We can get to .333... by iterated long division of 1 by 3.

    3 | 1
        0    (3 * 0 = 0) => 0.
        10   (add zero)
         9   (3 * 3 = 9) => 0.3
         10  (add zero)
          9  (3 * 3 = 9) => 0.33
          ...
And we can verify the reverse by doing the same trick above; 0.333... = 10 * 0.333... - 3 => 3 = 9 * 0.333... => 0.333... = 3 / 9 = 1 / 3.

So does this trick always work? If we have a repeated decimal, can we always multiply by 10 ^ (length of repeated sequence), subtract off, and get the value of that repeated decimal? If so, then it is reasonable to say that 0.999.... is equal to 1.

We can't really go in the forward direction without cheating (that is, going from 1 -> .999...); the best we can do is to modify long division to allow us to do it:

    3 | 3
        0    (3 * 0 = 0)  => 0.
        30   (add zero)
        27   (3 * 9 = 27) => 0.9
         30  (add zero)
         27  (3 * 9 = 27) => 0.99
         ...
And so on.

Obviously this isn't in Peano arithmetic exactly, but I think it holds together. If we allow repeated decimals in general to be valid representations of rational numbers, then we have to accept 0.999.... is equal to 1.

Re: 0.999...= 1

#182

Post by Okx »

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.

Re: 0.999...= 1

#183

Post by smlckz »

frogpelt wrote: post #141
smlckz wrote: post #113

Reminds me of the paradoxes of Zeno [1], especially the paradox of Achilles and the tortoise. At least one can simply prove that 0.999... = 1 without much hard work. Maybe less controversial than the following: 1 + 2 + 3 + ... [somehow] = -1/12 {{Riemann's zeta(-1)?}} 1 + 2 + 4 + 8 + 16 + ... [somehow] = -1 As well as the weird prime product (Product of 1/(1-(p^-2)) for p prime) and the sum of x^-2 from x=1 to [ sigh…

Your two examples can be debunked though. See here: https://www.youtube.com/watch?v=YuIIjLr6vUA

ah, Mathologer video.

Have seen that.

Another one by 3b1b on that topic: https://youtube.com/watch?v=sD0NjbwqlYw

Re: 0.999...= 1

#184

Post by OskarS »

amelius wrote: post #163
OskarS wrote: post #159

Earlier quoted context omitted.

It's still not true. No no sense is it true. It is true that 0.999... = lim[eps->0] (1 - eps), but it is ALSO true that lim[eps->0] (1 - eps) = 1. That's how limits work. If you accept both of those two (which you should, because they're correct), then since equality is transitive, 0.999... = 1. Therefore it is not less than 1, it is equal to 1.

I think you should look more closely at what I did with the order of operations, and the fact that "<" now is part of the expression acted over by the limit.

Limits are defined for functions and "1 - eps 
    lim[eps->0]( 1 - eps
is equal to

   1 - 0
which is obviously false. Again though, limits are defined for functions, not inequalities. If you have the limit

    lim[eps->0] (x - eps)
Then it is equal to

    x - 0
which is equal to x.

You're simply wrong here. You can read that wikipedia article if you don't trust me, or you can watch any of the thousands of youtube videos of mathematicians explaining this to you.

This is one of those things that is hard to do. Once you have an idea in your head that you're sure is right, it is incredibly difficult to dislodge it. It takes an enormous amount of humility and intellectual flexibility. But it is healthy and good to do it, every time you do it, you become a better person for it.

Re: 0.999...= 1

#185

Post by ColinWright »

moonchild wrote: post #50
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

It's much more intuitive that 10 0.9999...=9.9999... than that 2 0.9999...=1.9999...

You've met the formatting limitations. You mean:

"It's much more intuitive that 10 * 0.9999... = 9.9999... than that 2 * 0.9999...= 1.9999..."

Re: 0.999...= 1

#186

Post by CogitoCogito »

marcosdumay wrote: post #166
CogitoCogito wrote: post #133

Earlier quoted context omitted.

> You only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude that it has to be equivalent to 1. Sure you can provide a hand-wavy argument and try to give some intuition if you'd like. That doesn't make it any sort of logical proof though. I guess it depends on what you're after.

The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means. You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.

> The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means.

> You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.

I _know_ that.

I think I maybe should just bow out of this conversation. I'm apparently incapable of explaining myself in a way that is understandable to people here. I'll consider this my fault.

I'll just summarize: I don't think any "proof" that 0.9... = 1 that is only expressed in terms of arithmetic operations and does not make use of limits is legitimate. In other words I claim that a proof like "0.9... = x" means "9.9... = 10x" means "9 = 9x" is illegitimate. Instead of taking "0.9... = 1" on faith it takes "10 x 0.9... = 9.9..." and "9.9... - 0.9... = 1.0... = 1" on faith. There's no proof here. It's just shifting around symbols. Of course there are logical proofs, but they make use of limits/completeness/properties of real numbers explicitly.

Feel free to disagree...

Re: 0.999...= 1

#187

Post by Sharlin »

lmkg wrote: post #137
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…

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.

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.

Re: 0.999...= 1

#188

Post by Falling3 »

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.

That's a great approach

Re: 0.999...= 1

#189

Post by sorokod »

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…

People mistakenly believe that decimal numbers don't have multiple representations - which, in a way is correct.

It is correct if you take the limit, people usually do not.

Re: 0.999...= 1

#190

Post by tom-thistime »

lmkg wrote: post #137
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…

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 identity of a group is unique ... so long as you define "unique" to mean "if element e1 and element e2 are equal, then we say they are the same element."

You could count things differently and say the identity is "not unique", it would just lead to a lot of stupid and un-illuminating consequences.

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