## Is 1 = 0.999... ? Really?

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## Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
13
41%
No, 1 ≠ 0.999...
16
50%
Other
3
9%

### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:1/9 = 0.111...

Your word “are” also means “equals” in my case above, a finite expression “is” (are) equals an infinite expression.

The way I understand it, what you're saying is:

1) $$0.111\dotso$$ is an infinite sequence.
(Disagree.)

2) $$\frac{1}{9}$$ is a finite sequence.
(Disagree.)

3) $$0.111\dotso = \frac{1}{9}$$ is true.
(Disagree.)

4) If #1, #2 and #3 are true, it follows that there is at least one infinite sequence that is a finite sequence.
(I'm inclined to agree with this.)

The problem is that $$\frac{1}{9}$$ and $$0.111\dotso$$ are not sequences. They are numbers. (And it's also not true that $$0.111\dotso = \frac{1}{9}$$ but that's a peripheral issue.)

Ok, we’re going to dig at each other every so often, so I’ll just ignore your post after this.

So...

I found it interesting that you omitted the sequence:

1/10+1/100+1/1000+1/10000 etc...

But just decided to write 0.111...

The first part IMPLIES the sequence!

1/9 is finite in that it is a rational in its fraction form!!

0.111... is a repeating rational in its decimal form.

All of this shit:

1/9

0.111...

1/10+1/1000*1/1000 etc...

All equal each other!

I have no clue why you are playing such subtle word games that don’t change the content of what I wrote whatsoever, but here you are, doing just that!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:I found it interesting that you omitted the sequence:

1/10+1/100+1/1000+1/10000 etc...

That's not a sequence, that's a sum. You are confusing the two.

$$0.111\dots$$, which is the same as $$\frac{1}{10} + \frac{1}{10^2} + \frac{1}{10^3} + \cdots$$, is an infinite sum. It is not an infinite sequence. There's a huge difference between the two.

1/9 is finite in that it is a rational in its fraction form!!

It's not a finite sequence. It's not a sequence. It's a NUMBER.

I have no clue why you are playing such subtle word games

That would be you.

You need to learn language.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:I found it interesting that you omitted the sequence:

1/10+1/100+1/1000+1/10000 etc...

That's not a sequence, that's a sum. You are confusing the two.

$$0.111\dots$$, which is the same as $$\frac{1}{10} + \frac{1}{10^2} + \frac{1}{10^3} + \cdots$$, is an infinite sum. It is not an infinite sequence. There's a huge difference between the two.

1/9 is finite in that it is a rational in its fraction form!!

It's not a finite sequence. It's not a sequence. It's a NUMBER.

I have no clue why you are playing such subtle word games

That would be you.

You need to learn language.

It’s only an infinite sum if it converges, an infinite sequence is just an infinitely expanding discernible pattern.

I’m aware that 1/9 is a number. I stated that it’s a rational in fractional form. Because there’s a divisor, it’s also an operation.

Ecmandu
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### Re: Is 1 = 0.999... ? Really?

It’s only an infinite sum if it converges

Not really.

An infinite sum is simply a sum cosisting of an infinite number of terms. Whether it converges or not has nothing to do with it.

I’m aware that 1/9 is a number. I stated that it’s a rational in fractional form. Because there’s a divisor, it’s also an operation.

But you don't seem to be aware that it is not a finite sequence.

Actually, it is you who are 1) playing word games, and 2) avoiding addressing other people's claims.

As for me, I think I responded to almost every claim you made. Can you show me a claim I did not respond to?

Perhaps you simply don't like how I responded to your claims? If this is the case, can you explain why? What exactly are your expectations?
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
It’s only an infinite sum if it converges

Not really.

An infinite sum is simply a sum cosisting of an infinite number of terms. Whether it converges or not has nothing to do with it.

I’m aware that 1/9 is a number. I stated that it’s a rational in fractional form. Because there’s a divisor, it’s also an operation.

But you don't seem to be aware that it is not a finite sequence.

Actually, it is you who are 1) playing word games, and 2) avoiding addressing other people's claims.

As for me, I think I responded to almost every claim you made. Can you show me a claim I did not respond to?

Perhaps you simply don't like how I responded to your claims? If this is the case, can you explain why? What exactly are your expectations?

Dude, Magnus! Honestly!

“It’s a SUM with an infinite number of terms!! That’s what convergence fucking is! A fucking SUM!!

Not a sequence, not a series! It’s a fucking SUM! A SOLUTION to the fucking additive infinite series!

You never addressed the argument that proves infinite and finite behave differently in anything resembling a rational manner.

It is a mathematical FACT that when you remove something (and notice when I pointed out that when you “add” to an infinite set, it’s so absurd that not even YOU are arguing that! ) so the only argument you think you have is removal!

This has been explained to you!

If you remove the first one:

Boy —>
Boy —> clone
Boy —> clone

Etc...

All that NEED fucking occur is that all the boys take ONE step forward, and EVERYONE is holding hands again. This is IMPOSSIBLE!! With finite sets!!

Impossible!!! It’s a fucking PROOF that the infinite works differently than the finite!!

You figured that out. That it disproved you.

So what did you do? You ignored it and then posted this:

Boy
Boy —> clone
Boy
Boy —> clone
Boy
Boy —> clone

Etc...

And I jumped in and said “if you move the first boy up one step and then the bottom two (now) up one step and the (now) bottom three up one step Etc... all at once, everyone will still be holding hands again! But only in infinity is this a FACT!! If this is finite, it’s impossible to do this! Thus: infinite and finite WORK differently!

And you know what ?

You blew me off!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:It’s a SUM with an infinite number of terms!! That’s what convergence fucking is! A fucking SUM!

That's not what convergence is.

https://en.wikipedia.org/wiki/Convergent_series

Wikipedia wrote:A series is convergent if the sequence of its partial sums $$(S_{1},S_{2},S_{3},\dots)$$ tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number $$\ell$$ such that for every arbitrarily small positive number $$\varepsilon$$, there is a (sufficiently large) integer $$N$$ such that for all $$n\geq N$$,

$$\left|S_{n}-\ell \right\vert <\varepsilon .$$

This means that the infinite sum $$0.9 + 0.09 + 0.009 + \cdots$$ converges to $$1$$. (Which does not mean that it is equal to $$1$$.)
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:It’s a SUM with an infinite number of terms!! That’s what convergence fucking is! A fucking SUM!

That's not what convergence is.

https://en.wikipedia.org/wiki/Convergent_series

Wikipedia wrote:A series is convergent if the sequence of its partial sums $$(S_{1},S_{2},S_{3},\dots)$$ tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number $$\ell$$ such that for every arbitrarily small positive number $$\varepsilon$$, there is a (sufficiently large) integer $$N$$ such that for all $$n\geq N$$,

$$\left|S_{n}-\ell \right\vert <\varepsilon .$$

This means that the infinite sum $$0.9 + 0.09 + 0.009 + \cdots$$ converges to $$1$$. (Which does not mean that it is equal to $$1$$.)

It does mean that it’s a SUM!!!

YOU’RE the one who used the word “sum” incorrectly, not me!

But here you are AGAIN nit-picking over stupid shit and avoiding arguments that have to do with either:

1.) 0.999... 1 (or not)
2.) orders of infinity exist (or not)
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:
Magnus Anderson wrote:
Ecmandu wrote:It’s a SUM with an infinite number of terms!! That’s what convergence fucking is! A fucking SUM!

That's not what convergence is.

https://en.wikipedia.org/wiki/Convergent_series

Wikipedia wrote:A series is convergent if the sequence of its partial sums $$(S_{1},S_{2},S_{3},\dots)$$ tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number $$\ell$$ such that for every arbitrarily small positive number $$\varepsilon$$, there is a (sufficiently large) integer $$N$$ such that for all $$n\geq N$$,

$$\left|S_{n}-\ell \right\vert <\varepsilon .$$

This means that the infinite sum $$0.9 + 0.09 + 0.009 + \cdots$$ converges to $$1$$. (Which does not mean that it is equal to $$1$$.)

It does mean that it’s a SUM!!!

YOU’RE the one who used the word “sum” incorrectly, not me!

But here you are AGAIN nit-picking over stupid shit and avoiding arguments that have to do with either:

1.) 0.999... 1 (or not)
2.) orders of infinity exist (or not)

Converges to means the same exact thing as “equals”

You have arguments to look at!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:If you remove the first one:

Boy —>
Boy —> clone
Boy —> clone

Etc...

All that NEED fucking occur is that all the boys take ONE step forward, and EVERYONE is holding hands again.

[..]

You figured that out. That it disproved you.

So what did you do? You ignored it.

I did not ignore it. I responded to it by stating that it's not something that you can do because it is strictly forbidden by your previous claims.

Let's go back to page 98 where I stated:

Magnus wrote:We started with the following situation:

Boy1 -> Clone1
Boy2 -> Clone2
Boy3 -> Clone3
etc

We put the two sets in one-to-one correspondence. We paired every boy with exactly one clone and every clone with exactly one boy. This means that every boy is paired (which means there are no unpaired boys) and that every clone is paired (which means there are no unpaired clones.)

Once you remove Clone1 from the set of clones, you get the following situation:

Boy1
Boy2 -> Clone2
Boy3 -> Clone3
etc

Boy1 is now unpaired because we removed the clone he was paired with. At this point, there is no one-to-one correspondence between the two sets. In order to restore it, there must be a clone in the set of clones that is not paired -- an unpaired clone. But there are NO unpaired clones. We STATED it earlier. And if there were unpaired clones, that would mean there was no one-to-one correspondence in the first place. But we did put the two sets in one-to-one correspondence, didn't we?

A possible way out is to say that by removing Clone1 a new clone is generated. But the problem with this is . . . that's not what the word "remove" means. To remove a clone does not mean to remove a clone and add a new one.

Another possible way out is to say that there is no need for an unpaired clone to exist. You can just pair Boy1 with one of the paired clones. But the result of that wouldn't be a one-to-one correspondence. You'd have a clone paired with TWO boys. One-to-one correspondence requires that every clone is paired with EXACTLY ONE boy.

Note the bolded part.

In order to restore one-to-one correspondence between the two sets, there must be an unpaired clone to pair with an unpaired boy. But there is no such a clone. All of the clones are already paired. Thus, regardless of how you move your clones, you cannot restore one-to-one correspondence.

You responded to this by saying that the word "infinity" refers to a never-ending process of increase which means that new clones are added continually. So when we remove a clone, a new one is added automatically.

And my response to this was that the word "infinity" does not refer to a never-ending process of increase (that it does not refer to a process at all.)
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Converges to means the same exact thing as “equals”

That's not true.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:Converges to means the same exact thing as “equals”

That's not true.

You’re still doing it! You’re ego is invested in nit-picking and not arguments!!!

There is a difference between ‘converges to’ (which is convergence) and ‘converges towards’ which is not convergence. But!! Even that’s a contradiction because the word convergence IN AND OF ITSELF is defined as the finite conclusion of a sequence or series. Infinite or not.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

It's part of your argument that infinite sequences are both finite and infinite sequences.

That in turn is part of your argument that infinite sequences are algorithms.

That in turn is part of your argument that the word "infinity" refers to a never-ending processes of increase.

That in turn is part of your argument that infinities do not come in sizes.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:There is a difference between ‘converges to’ (which is convergence) and ‘converges towards’ which is not convergence. But!! Even that’s a contradiction because the word convergence IN AND OF ITSELF is defined as the finite conclusion of a sequence or series. Infinite or not.

Not true.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:

It's part of your argument that infinite sequences are both finite and infinite sequences.

That in turn is part of your argument that infinite sequences are algorithms.

That in turn is part of your argument that the word "infinity" refers to a never-ending processes of increase.

That in turn is part of your argument that infinities do not come in sizes.

This part is transitive:

1/9 implies 0.111...

0.111... implies 1/9

If they both imply each other, they are equalities.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

I have no idea what that means.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:I have no idea what that means.

And that’s why this debate is over. Because you don’t understand, really, much of anything said here!

But let me be kind to you for a moment!

2+3=5
3+2=5

That means 2 and 3 are transitive: they mean the same thing!

I’ve seen you write a bunch of fancy symbols, but you don’t even understand kindergarten math!

That’s why we are butting heads here!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

This isn't supposed to be a contest of beliefs but a cooperative effort to resolve disagreements. (But then again, this is a forum, so pretty much everything anyone does here is some sort of competition where people try to prove themselves to be the smartest guy in the room.)

Ecmandu wrote:2+3=5
3+2=5

That means 2 and 3 are transitive: they mean the same thing!

What do you mean by "transitive"?

https://en.wikipedia.org/wiki/Transitive_relation

Wikipedia wrote:In mathematics, a homogeneous relation R over a set X is transitive if for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.

Either way, it's definitely not true that $$2$$ and $$3$$ mean the same thing.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:This isn't supposed to be a contest of beliefs but a cooperative effort to resolve disagreements. (But then again, this is a forum, so pretty much everything anyone does here is some sort of competition where people try to prove themselves to be the smartest guy in the room.)

Ecmandu wrote:2+3=5
3+2=5

That means 2 and 3 are transitive: they mean the same thing!

What do you mean by "transitive"?

https://en.wikipedia.org/wiki/Transitive_relation

Wikipedia wrote:In mathematics, a homogeneous relation R over a set X is transitive if for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.

Either way, it's definitely not true that $$2$$ and $$3$$ mean the same thing.

Magnus,

I have to admit, at this point, I enjoy teaching you because you don’t quit!

Transitive (strictly speaking) (as an example)

Is:

a*b = b*a

I gave you a more advanced version in the last post; what I should have said is that:

2+3 = 3+2

3+2 = 2+3

Etc...

When you introduce a new variable (such as “5”) (c) it becomes a different term than purely transitive, Wikipedia is wrong.
Last edited by Ecmandu on Wed Jun 17, 2020 8:32 pm, edited 1 time in total.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Transitive (strictly speaking) (as an example)

Is:

a*b = b*a

That looks like commutativity.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:You never addressed the argument that proves infinite and finite behave differently in anything resembling a rational manner.

Here it is:
viewtopic.php?f=4&t=190558&p=2768316#p2768299

And you are ignoring it (:
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:Transitive (strictly speaking) (as an example)

Is:

a*b = b*a

That looks like commutativity.

Oh man, that’s embarrassing for me.

You have to understand that I had brain damage (was in a coma for four hours) because of a head injury.

I went from being a super-genius to just your run of the mill genius.

Yes, your neurons were not misfiring on this!

It’s communicative!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:It’s communicative!

You mean commutative (:
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:It’s communicative!

You mean commutative (:

*chuckles*
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

You know Magnus,

Brain damage did not impair my logic, just my memory.

The link you just sent me implies that I’m not allowed to make ANY argument that shows FOR A FACT that infinite and finite behave differently (supposedly (according to you) by my own reasoning).

Your argument about me contradicting myself by having every boy step forward and still all be holding hands is a fantasy of yours! It violates YOUR reasoning! Not what I’ve presented in this thread.

You know why I know I’ll win this debate?

Because I know god doesn’t exist.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:You know Magnus,

Brain damage did not impair my logic, just my memory.

The link you just sent me implies that I’m not allowed to make ANY argument that shows FOR A FACT that infinite and finite behave differently (supposedly (according to you) by my own reasoning).

Your argument about me contradicting myself by having every boy step forward and still all be holding hands is a fantasy of yours! It violates YOUR reasoning! Not what I’ve presented in this thread.

You know why I know I’ll win this debate?

Because I know god doesn’t exist.

Let me put this to you a different way.

There is a highest order of cardinality that in laypersons terms means “the infinite cardinal”

This is a proof of god.

Cantor knew it to.

Our every sentence in this thread is also about whether god exists or not!

Very high stakes for lots of people.
Ecmandu
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