## 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...
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42%
No, 1 ≠ 0.999...
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48%
Other
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### Re: Is 1 = 0.999... ? Really?

You assert that 3 is (1+1+1)

By that logic,

1+1+1 = 1/2+1/2+1/2+1/2+1/2+1/2!!!!!!!

And 1/2 = 1/4+1/4 etc...

And 1/4 = 1/8+1/8. Etc....

That means that ALL whole numbers equal zero!!!

Do you want to hear my logic now?
You're going to keep subdividing the terms into smaller and smaller fractions until you end up at 1/infinity and then you are going to say that all the terms are equal to zero and so the total is equal to zero.

Do I win a prize?
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### Re: Is 1 = 0.999... ? Really?

Back to page 16:

viewtopic.php?p=2617441#p2617441

Carleas wrote:James, are you just articulating Zeno's paradox?

Suppose I want to go 1 mile. To get there, I have to go 90% of the way. Then I have to [go] 90% of the remaining distance, and 90% of that distance, and 90% of that distance. If I go the whole mile, how many times did I go 90% of the remaining distance? An infinite number of times.

It seems like you want to say that, once I've gone 90% of the remaining distance an infinite number of times, there's still a remaining infinitesimal distance to go. Are you positing some indivisible quantum of number-ness that isn't part of the infinite number of times we go 90% of the remaining distance? Why can't we go 90% of that distance?

I'm not talking about time or speed, I'm talking about distance. There is 1 mile. Between 0 and 1 miles are an infinite series of distances 90% of the way from 0 to 1 mile. To go 1 mile, we must go an infinite number of these 90% segments. We don't care how long or how fast we traverse these 90% segments, just that they all must be traversed. So we get an infinite sum .9 + .09 + .009 + .0009 + ...

We know we've gone 1 mile, and that we've traversed an infinite number of these segments. But you seem to be saying that after traversing infinitely many segments, we still have some distance to cover. Moreover, you must be claiming that we can't go 90% of the remaining distance (else, it would just be another 90% segment in the infinite sum); so there was a distance that could only be traversed all at once (i.e., a distance that was traversed without ever having completed 90% of that distance).

If you're trying to move from point $$A$$ to point $$B$$ by moving in steps that are $$90\%$$ of the remaining distance in size, you will never ever arrive at point $$B$$. Indeed, even if you make an infinite number of such steps, you will never arrive at $$B$$. (Not even infinity raised to infinity is enough. Indeed, no matter how large the number is, it's not enough.)

That's the only thing that Zeno's Paradox shows: if you're trying to cross an infinitely divisible distance between two points in the manner specified by Zeno, you will never ever cross that distance. Zeno makes a mistake by making a jump from this conclusion to "You cannot cross the distance regardless of how you move."

But you actually can. Let's see how.

If you crossed the distance, this means that you made a number of steps that are $$90\%$$ of the remaining distance in size PLUS a number of steps that are of different size. For example, if the distance between the two points is equal to $$1$$ mile, you might have crossed $$0.999$$ miles by making $$3$$ steps that are $$90\%$$ of the remaining distance in size and $$0.001$$ miles by making two steps that are $$0.0005$$ miles in size. Or perhaps you crossed $$0.\dot9$$ miles by making an infinite number of steps and $$0.\dot01$$ miles by making a single step. Either way, the distance between the two points cannot be expressed as a number of the form $$\sum_{i=1}^{n} \frac{9}{10^i}$$.

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

phyllo wrote:

You assert that 3 is (1+1+1)

By that logic,

1+1+1 = 1/2+1/2+1/2+1/2+1/2+1/2!!!!!!!

And 1/2 = 1/4+1/4 etc...

And 1/4 = 1/8+1/8. Etc....

That means that ALL whole numbers equal zero!!!

Do you want to hear my logic now?
You're going to keep subdividing the terms into smaller and smaller fractions until you end up at 1/infinity and then you are going to say that all the terms are equal to zero and so the total is equal to zero.

Do I win a prize?

Well... that’s what I explained using YOUR logic.

What’s my logic?

Let’s say you have a whole orange and cut it in half.

You have two things going on.

You no longer have a whole orange. You have two halves of an orange.

The two halves are no longer considered an orange and yet they are.

When you divide a quantity, you are no longer left with a quantity but rather pieces still equal to the whole quantity.

This is important as an articulation because in debates about infinity, people switch back and forth to different concepts and treat them the same as it suits their needs, like Magnus is doing for his “greater than infinity” arguments. Like you are doing by stating your that 3 = (1+1+1)

There’s another compelling argument. This is blasphemy to convergent series mathematicians ...

I made this argument 7 years ago.

When you are counting the rational numbers, they are slowly working their way to listing all the irrational numbers

1234

12345

123456

They are crawling ever slowly towards listing every irrational and transcendent number.

When the rationals CONVERGE with infinity, all the irrational numbers are expressed.

The reason that this is blasphemy to mathematicians is because they require non listabl numbers at the convergence of the rational numbers to prove orders of infinity.
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### Re: Is 1 = 0.999... ? Really?

Mathematics is the manipulation of symbols according to some rules. I followed the rules when I combined/separated the terms of the series.

All references to oranges and apples are unnecessary and irrelevant. They contribute nothing valuable to the discussion of the mathematics.
phyllo
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:Mathematics is the manipulation of symbols according to some rules. I followed the rules when I combined/separated the terms of the series.

All references to oranges and apples are unnecessary and irrelevant. They contribute nothing valuable to the discussion of the mathematics.

The rules in the instance you are using leads to absurdity.

The orange is an analogy for the number 1.
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### Re: Is 1 = 0.999... ? Really?

Number 6 : Who are you?

Number 2 : The new Number 2.

Number 6 : Who is Number 1?

Number 2 : You are Number 6.

Number 6 : I am not a number, I am a free man.
phyllo
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:Number 6 : Who are you?

Number 2 : The new Number 2.

Number 6 : Who is Number 1?

Number 2 : You are Number 6.

Number 6 : I am not a number, I am a free man.

I’ll reply to you a bit further.

When you’re dealing with infinities, you CANNOT remove the parentheses!!!!

Do you understand that?

Order of operations: parentheses come first.

The reason I gave the analogy of the orange is because let’s say an orange is 3* the size of a normal orange (because we were using your argument that 3 = 1+1+1 in an infinity), once you divide it, it is no longer an orange (no longer 3)
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:Number 6 : Who are you?

Number 2 : The new Number 2.

Number 6 : Who is Number 1?

Number 2 : You are Number 6.

Number 6 : I am not a number, I am a free man.

Lol

The Prisoner.. such a cult classic.
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ

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

Magnus Anderson wrote:
In any case, you seem to think that numbers that can only be described as approaching a limit without ever attaining the limit are a third kind of number after finite and infinite. Have a name for these kinds of numbers?

I don't.

Then answer the question. Is $$0.\dot9$$ a finite number, an infinite number, or something else. If it's something else, what do you call it?

Magnus Anderson wrote:Take $$B' = \{p_1, p_3, p_5, p_7, \dotso\}$$. You can relabel its elements by renaming $$p_3$$ to $$p_2$$, $$p_5$$ to $$p_3$$, $$p_7$$ to $$p_4$$ and so. That's what you're doing, right? By doing so, you can "prove" that $$B'$$ is equal in size to $$A$$. But by relabelling them in a different way, you can "prove" that $$B'$$ is actually bigger than $$A$$. Rename $$p_1$$ to $$p_0.5$$, $$p_3$$ to $$p_1$$, $$p_5$$ to $$p_1.5$$, $$p_7$$ to $$p_2$$, and so on. Voila! You have a set $$B' = \{p_{0.5}, p_1, p_{1.5}, p_2, \dotso\}$$ that is "clearly" bigger than $$A$$!

That would only prove it to you. You're still using your own logic.

And you're missing the point. The point is to pair up each element in the set with a natural number. If for every natural number, you can match it with an element in the set, then you've shown that the set is infinite.

Imagine pairing every odd point in B with every point in A before the points are removed from B. You would draw the correspondences the same way, right? You would draw a line straight across from a1 to b1. You would then draw a line at an angle from a3 to b2. You would then draw a line at a an even wider angle from a5 to b3. And so on. In this scenario, we agree that there are no differences between the length of the lines, and that they are both infinite. Therefore, you don't end up running out of points in line B before running out of points in line A, and visa-versa. You end up traversing line A at a faster rate than line B, but because neither line has an end, you never run out of points. The angles drawn between the lines just keep getting wider and wider and wider, without ever becoming 90 degrees. <-- This is exactly how I envision the lines being draw between points in the scenario after the points are removed from line B and the remaining points filling the gaps. Therefore, it doesn't prove anything about line B being shorter (not to me). I agree that after removing the points from line B and shifting the remaining points to fill the gaps, drawing the mapping between the points would have to be done with these angled lines. But why would I think this proves that line B must be shorter when I just envision this scenario being no different from the scenario I just described before removing the points and shifting the remaining points? You're barking up the wrong tree. You need to focus on proving that what applies to finite sets also applies to infinite sets, not on the fact that certain points were removed (I know they were removed).

Magnus Anderson wrote:Let's take a look at an example involving a line consisting of a finite number of football players.

No! No more finite lines.

I'm very well aware that certain points in B got remove. I'm the one who said to remove them! I'm not saying they magically come back. Stop belaboring the point!

Magnus Anderson wrote:
The reason why you can relabel is because you're using a method of counting whereby you pair natural numbers with elements in the set.

And that's not a valid method of counting because the set of natural numbers does not have a size on its own.

Let's map $$N = \{1, 2, 3, \dotso\}$$ to $$B' = \{p_1, p_3, p_5, \dotso\}$$.

You can use bijection to do so:

$$f(x) = p_{2x - 1}$$

$$1 \mapsto p_1$$
$$2 \mapsto p_3$$
$$3 \mapsto p_5$$
$$\cdots$$

This makes the two sets equal in size.

But you can also use any other kind of function e.g. injection:

$$f(x) = p_{4x - 1}$$

$$\hspace{0.83cm} p_1$$
$$1 \mapsto p_3$$
$$\hspace{0.83cm} p_5$$
$$2 \mapsto p_7$$
$$\hspace{0.83cm} p_9$$
$$3 \mapsto p_{11}$$
$$\hspace{0.83cm} p_{13}$$
$$\cdots$$

This makes $$A$$ smaller than $$B'$$.

With this kind of "logic", you can literally prove anything you want.

Read my words very carefully, Magnus. I did not say the rule is: map the natural numbers onto members of the set any which way you want... I said: if there is a way to map all the naturals onto members of the set such that all members get mapped, then you know there are just as many members in the set as there are natural numbers. There just has to be a way (really, you should watch the vsauce video). Incidentally, the scenarios you depicted also show there are an infinite amount of members in the set. Mapping the naturals onto every odd number, for example, will show that there are an infinite number of odd numbers. Imagine then extending that to include the even numbers as well... wouldn't that for sure show that there are an infinite number of natural numbers?

Look:

Suppose A = {p1, p2, p3, ... }

$$1 \mapsto p_1$$
$$2 \mapsto p_3$$
$$3 \mapsto p_5$$
$$\cdots$$

^ That's one way.

$$\hspace{0.83cm} p_1$$
$$1 \mapsto p_3$$
$$\hspace{0.83cm} p_5$$
$$2 \mapsto p_7$$
$$\hspace{0.83cm} p_9$$
$$3 \mapsto p_{11}$$
$$\hspace{0.83cm} p_{13}$$
$$\cdots$$

^ That's another way.

$$1 \mapsto p_1$$
$$2 \mapsto p_2$$
$$3 \mapsto p_3$$
$$\cdots$$

^ That's a third way.

Can you spot the one where all the naturals map onto all the points? That's right, it's behind door #3! The rule: so long as there is a way. And there is a way. There's also a lot of ways not to do it. But we're not picking any arbitrary method. We're asking: is there a way to map all the naturals onto all the member. And the answer in the above case is yes. Therefore, this shows there are just as many members in the set as there are natural numbers.

Magnus Anderson wrote:I understand very well how the rule works. I'm simply saying it's not a valid rule.

If people say that a rule is valid, does that mean it's valid?

No, you have to think it through and determine for yourself. For myself, the determination is trivially simple: there are an infinite number of natural numbers. If you can pair them up one-to-one with members of a set, with no natural numbers remaining and no members of the set remaining, then the set must also have an infinite number of members. Trivial! I mean, like, really trivial! It's how counting works.

If you don't think the rule is valid, then show me how it fails in the case of line B after removing the points. If it fails, then you must run out of points before you run out of natural numbers. What is the last number you use before running out of points.

Magnus Anderson wrote:I'm certainly not a machine that convinces people (:

I'll say.

If you don't want to prove to me that what applies to finite sets also applies to infinite sets, that's your call. But I am telling you what I need in order to be convinced. You can come up with scenario after scenario after scenario of ways to show that removing members of a set means that members were removed from the set--switching out points in a line for people in a queue or football players on a team or carts in a train or whatever--but since you have been informed that not only do I get the point (and am growing nauseous about hearing about it), but it's not what I need to be convinced, this is just an exercise in futility for you. So you go ahead and keep repeating the same argument over and over and over again; it's gonna get you nowhere.
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### Re: Is 1 = 0.999... ? Really?

gib wrote:Then answer the question. Is $$0.\dot9$$ a finite number, an infinite number, or something else. If it's something else, what do you call it?

I answered the question. I have no name for that kind of number. It's not a finite number (in the sense that it's not a number that can be expressed as a finite sum of rational numbers) and it's not an infinite number (in the sense that it's not a number greater than every integer.) Why is it so important to categorize it? Occassionally, I would call it an infinite number but only in the sense that it's a number that cannot be expressed as a finite sum of rational numbers. If this confuses you, perhaps what can be of help is to take into consideration the fact that one and the same term can have multiple meanings.

That would only prove it to you. You're still using your own logic.

It's not about my or your logic. It's about logic. And what I'm doing is either logical or it is not. If it is not, I'd have to see where's the flaw. If it is, you'd have to make an effort to understand it.

I did not say the rule is: map the natural numbers onto members of the set any which way you want...

Yes, you did not. I know very well what you said.

I said: if there is a way to map all the naturals onto members of the set such that all members get mapped, then you know there are just as many members in the set as there are natural numbers.

And what I'm saying is that you don't. I can use the same exact way of thinking that you're using (it goes by the name "confirmation bias") to prove that $$N$$ is smaller than $$B'$$. I can say: if there is a way to map all the naturals onto members of the set $$B'$$ such that every member of $$N$$ is associated with a distinct member from $$B'$$ but not every member of the set $$B'$$ is associated with a member from $$N$$, then "you know" that $$N$$ is smaller than $$B'$$.

$$N = \{1, 2, 3, \dotso\}$$
$$B' = \{p1, p3, p5, \dotso\}$$

$$f(x) = p_{4x - 1}$$

$$\hspace{0.83cm} p_1$$
$$1 \mapsto p_3$$
$$\hspace{0.83cm} p_5$$
$$2 \mapsto p_7$$
$$\hspace{0.83cm} p_9$$
$$3 \mapsto p_{11}$$
$$\hspace{0.83cm} p_{13}$$
$$\cdots$$

Incidentally, the scenarios you depicted also show there are an infinite amount of members in the set. Mapping the naturals onto every odd number, for example, will show that there are an infinite number of odd numbers. Imagine then extending that to include the even numbers as well... wouldn't that for sure show that there are an infinite number of natural numbers?

Yes, there is an infinite number of members in both sets. We know that. That's not what we're talking about (or what we're supposed to be talking about.) We're talking about size/cardinality. The question is: the two sets $$N$$ and $$B'$$, are they equal in size? The question is not: are they both infinite? We agree that they are both infinite.

Read my words very carefully, Magnus.

You did not ask me for advice but I did not ask you for instructions either. So I think I'm justified in giving you a small hint.

Never ever tell your interlocutors what to do unless they really want you to tell them what to do. It's a way to ruin the discussion.

Can you spot the one where all the naturals map onto all the points? That's right, it's behind door #3! The rule: so long as there is a way. And there is a way. There's also a lot of ways not to do it. But we're not picking any arbitrary method. We're asking: is there a way to map all the naturals onto all the member. And the answer in the above case is yes. Therefore, this shows there are just as many members in the set as there are natural numbers.

It does not. If you can use the same kind of thinking to arrive at the opposite conclusion, then that should tell you there is something wrong with it.

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

If you don't want to prove to me that what applies to finite sets also applies to infinite sets, that's your call. But I am telling you what I need in order to be convinced. You can come up with scenario after scenario after scenario of ways to show that removing members of a set means that members were removed from the set--switching out points in a line for people in a queue or football players on a team or carts in a train or whatever--but since you have been informed that not only do I get the point (and am growing nauseous about hearing about it), but it's not what I need to be convinced, this is just an exercise in futility for you. So you go ahead and keep repeating the same argument over and over and over again; it's gonna get you nowhere.

If this discussion is making you nauseous, then you shouldn't be participating in it.

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

In discussions revolving around the stuff that interest me philosophically, many come to expect that in any given exchange they'll find themselves convinced by the argument in one post only to read the next post and be convinced instead, that, no, this makes more sense.

But that's because in regard to value judgments and political prejudices and identity and free will and God, there never seems to be a way to actually pin the whole truth down.

With math though, some figure there surely must be a way to encompass it. But it turns out that the flaw here is that in discussing math without actually connecting the words to people and things out in the world, it still comes down to sets of assumptions about what you insist the words mean...and are telling us about other words.

Me, I don't have either the education or background to follow the exchange here with any degree of sophistication at all. Instead, I try to grapple with the implications of an exchange of this sort that can go on this long and still nobody is able to convince everyone else that they do indeed grasp the whole truth.

You know, going back to the whole truth about existence itself.
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Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382

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

gib wrote:Imagine pairing every odd point in B with every point in A before the points are removed from B. You would draw the correspondences the same way, right? You would draw a line straight across from a1 to b1. You would then draw a line at an angle from a3 to b2. You would then draw a line at a an even wider angle from a5 to b3. And so on. In this scenario, we agree that there are no differences between the length of the lines, and that they are both infinite. Therefore, you don't end up running out of points in line B before running out of points in line A, and visa-versa. You end up traversing line A at a faster rate than line B, but because neither line has an end, you never run out of points. The angles drawn between the lines just keep getting wider and wider and wider, without ever becoming 90 degrees. <-- This is exactly how I envision the lines being draw between points in the scenario after the points are removed from line B and the remaining points filling the gaps. Therefore, it doesn't prove anything about line B being shorter (not to me).

You don't run out of points, that's for sure. For every point on line $$B$$ there is an odd point on line $$A$$. That's where we agree. Where we disagree is that this means that $$B$$ is equal in length to $$A$$ with odd points taken out. I insist that it does not.

Do you think that $$A = \{1, 2, 3, \dotso\}$$ and $$B = \{1, 2, 3, \dots\}$$ are giving us enough information to conclude that the two sets are equal in size?

You obviously do. Like Silhouette, you think the two descriptions represent two infinite sets that are equal in size.

But I don't.

This is evident in the fact that you can specify any kind of relation between the two sets. You can specify bijection but you can also specify injection. It's an arbitrary decision.

You can say the two lines are equal. Fine. But if you remove one element from $$A$$, you can no longer say they are equal. Indeed, the size of these two sets is no longer an arbitrary decision. So the fact that you can still specify a bijective relation between the two sets proves nothing.

I can say that $$x = 3$$ and $$y = 2$$. These are arbitrary decisions. But the result of their addition, $$x + y$$, is not an arbitrary decision. It's something that must logically follow from previously accepted premises. If you accept that $$x = 3$$ and $$y = 2$$, and that the operation of addition means what it normally means, then the result of $$x + y$$ cannot be anything other than $$5$$. The fact that you can change your premises (e.g. change $$x$$ to $$5$$) to get a different result (e.g. $$7$$) does not mean that that different result is the result to this particular operation. That's the kind of mistake that you're making.

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

Why do you keep ignoring me Magnus ???

Let’s say you have sets:

1,2,3,4,5,6,7...

1,3,5,7,9,11....

In that latter set, that value is larger!!

For example:

0.333...

Is larger in value than

0.111...

My issue with you Magnus is that you consider this an ORDER of infinity, you actually consider one infinity to be larger than another infinity.

The only way you can prove that ! Is to prove non correspondence.

I’m meeting you halfway Magnus
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Why do you keep ignoring me Magnus ???

Because I can't make any sense out of your posts.

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

Magnus Anderson wrote:
Ecmandu wrote:Why do you keep ignoring me Magnus ???

Because I can't make any sense out of your posts.

So I need a reality check here:

Gib, Silhouette, Phyllo...

Am I incomprehensible???
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:
Magnus Anderson wrote:
Ecmandu wrote:Why do you keep ignoring me Magnus ???

Because I can't make any sense out of your posts.

So I need a reality check here:

Gib, Silhouette, Phyllo...

Am I incomprehensible???

Actually, Silhouette, since Magnus is done with that debate (since he doesn’t understand me). Shall we’ll move on to ours? Are you good with that?
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Max wrote:Certainly, the word "infinite" does not mean "not being able to be something else at some other point in time".

Mags wrote:..but then wouldn’t that simply mean that something is either infinite or not? which I ‘think’ Silhouette.. I don’t want to put words into his mouth, is also saying.

I am not exactly sure what you mean, so I'll have to make a guess.

I suppose what you mean is that something is either finite or infinite i.e. that it cannot be both at the same time.

I agree with that. The number of elements within a set is either finite or it is infinite. There is no third option here.

Unfortunately, I cannot understand how that relates to what I said in the above quote.

The following two statements certainly do not make a claim that a line of apples existing at some point in time at some point in space is both finite and infinite.

Statement 1: "At some point in time at some point in space, there exists an infinite line of apples."

Statement 2: "At some other point in time, no apples exist anywhere in space."

They merely state that at one point in time the line is made out of an infinite number of apples and that at some other point in time the line is made out of zero apples.

When put that way, sure..

As for Silhouette's claim that:

The issue is that adding apples to an infinite set of apples doesn't make "infinity bigger" because expanding the bounds of something to make it bigger only applies to the bounded and not the boundless.

It's not true that you cannot make an infinite quantity bigger. As for the rest, it's difficult to respond to because it's difficult to understand.

Is there such a notion as, more infinite? when a set is already in the state of being infinite.. can something that is already endless, be more-so? Where is the end, to add to it?
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ

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

MagsJ wrote:
Magnus Anderson wrote:
Max wrote:Certainly, the word "infinite" does not mean "not being able to be something else at some other point in time".

Mags wrote:..but then wouldn’t that simply mean that something is either infinite or not? which I ‘think’ Silhouette.. I don’t want to put words into his mouth, is also saying.

I am not exactly sure what you mean, so I'll have to make a guess.

I suppose what you mean is that something is either finite or infinite i.e. that it cannot be both at the same time.

I agree with that. The number of elements within a set is either finite or it is infinite. There is no third option here.

Unfortunately, I cannot understand how that relates to what I said in the above quote.

The following two statements certainly do not make a claim that a line of apples existing at some point in time at some point in space is both finite and infinite.

Statement 1: "At some point in time at some point in space, there exists an infinite line of apples."

Statement 2: "At some other point in time, no apples exist anywhere in space."

They merely state that at one point in time the line is made out of an infinite number of apples and that at some other point in time the line is made out of zero apples.

When put that way, sure..

As for Silhouette's claim that:

The issue is that adding apples to an infinite set of apples doesn't make "infinity bigger" because expanding the bounds of something to make it bigger only applies to the bounded and not the boundless.

It's not true that you cannot make an infinite quantity bigger. As for the rest, it's difficult to respond to because it's difficult to understand.

Is there such a notion as, more infinite? when a set is already in the state of being infinite.. can something that is already endless, be more-so? Where is the end, to add to it?

Oh, that who’s name should never be spoken, he already tried this by putting it at the beginning!!

To which I replied:

That means the end moves +1, which is the same as adding it to the end!

But Magnus doesn’t understand me, so that’s the end of our debate.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

So anyways,

No response yet from Silhouette, so I’ll just post it.

Silhouette and I agree that 1/10*1/10 (repeating) never expresses the 1 at the tail end!

Ok, so far so good.

That means the only way that you can shift the decimal is not from right to left, but from left to right!

That means that:

0.999... + 0.111... must equal 1.111... if 0.999... equals 1

There’s a problem with this!

0.999... + 0.111...

Equals: 1.1...0

And we know that 0.0...1 is the number that makes 0.999... equal one.

That means that there is a discrepancy of 0.0...2 which makes not the smallest possible number (equal to zero) that can possibly be made!

Thus, Silhouette’s argument thus far, has been falsified.

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

Magnus Anderson wrote:I answered the question. I have no name for that kind of number. It's not a finite number (in the sense that it's not a number that can be expressed as a finite sum of rational numbers) and it's not an infinite number (in the sense that it's not a number greater than every integer.) Why is it so important to categorize it? So that I can understand what the hell's going on in your mind. Occassionally, I would call it an infinite number but only in the sense that it's a number that cannot be expressed as a finite sum of rational numbers. If this confuses you, perhaps what can be of help is to take into consideration the fact that one and the same term can have multiple meanings.

Oh, obviously. I just question whether this is a commonly known definition (at least among specialists) or you're just going off on your own inventing your own customized definitions.

So for you, a finite number must be expressible in terms of a finite sum of rational numbers. So if it can't be expressed as a finite sum or if it can't be expressed with rational numbers, then it isn't finite. If we're gonna talk about this, we might as well give it a name. I'm just gonna throw one out there. How 'bout "semi-finite"?

^ By this definition, would $$0.\dot3$$ be a semi-finite number? I wouldn't say so since it can be expressed as 1/3 (a finite sum consisting of just one rational number). You could say the same of $$0.\dot9$$ since it can be expressed as 1/3 x 3, but this gives you 1 which you dispute (which is a whole other can of worms we could debate). It seems pretty obvious that you would say an infinite sum of irrational numbers isn't finite (although it wouldn't necessarily be infinite either). What about a finite sum of irrational numbers? For most irrational numbers, if you sum them, you *probably* still get an irrational number, which I'm guessing you'd say is semi-finite. But what about $$\pi$$ + (4 - $$\pi$$)? 4 - $$\pi$$ gives us 0.85840734641... I don't have a mathematical proof that this is also an irrational number but my gut tells me it is. So you'd have a finite sum of two irrational numbers which equals 1, obviously a finite number. But obviously, 1 can also be expressed as a finite sum of rational numbers; 1 is a finite sum of rational numbers consisting of just one term (1). But what about $$\pi$$ + 1. This equals 4.14159265358979... which is obviously also an irrational number. I don't know of any way to express this as a sum of rational numbers, but I derived it with a sum of one irrational number and one rational number. <-- Do you agree that it still counts as a semi-finite number?

Anyway, I always thought of finite numbers as just not infinite numbers. You seem to think of $$0.\dot9$$ or $$\pi$$ as not finite because, what, they don't have clear boundaries? I mean, a number like 4 clearly starts at exactly 0 and ends at exactly 4. But if you think there exists an infinitesimal between $$0.\dot9$$ and 1, but you can't say "the 9s end exactly here", then the boundaries of $$0.\dot9$$ are not clear, at least at the end closer to 1. <-- Is that why you don't think of it as finite? But then what do you say about $$0.\dot3$$ which can be expressed as 1/3? I would think you'd say it doesn't have a clear boundary either, at least not at the side furthest from 0. Does this make it semi-finite? Or does the fact that it can be expressed as 1/3 make it finite?

Magnus Anderson wrote:And what I'm saying is that you don't. I can use the same exact way of thinking that you're using (it goes by the name "confirmation bias") to prove that $$N$$ is smaller than $$B'$$. I can say: if there is a way to map all the naturals onto members of the set $$B'$$ such that every member of $$N$$ is associated with a distinct member from $$B'$$ but not every member of the set $$B'$$ is associated with a member from $$N$$, then "you know" that $$N$$ is smaller than $$B'$$.

$$N = \{1, 2, 3, \dotso\}$$
$$B' = \{p1, p3, p5, \dotso\}$$

$$f(x) = p_{4x - 1}$$

$$\hspace{0.83cm} p_1$$
$$1 \mapsto p_3$$
$$\hspace{0.83cm} p_5$$
$$2 \mapsto p_7$$
$$\hspace{0.83cm} p_9$$
$$3 \mapsto p_{11}$$
$$\hspace{0.83cm} p_{13}$$
$$\cdots$$

This wouldn't prove that B' has more members than N. Once again, the rule is: if you can map all the members of N onto all the members of B, then you know there are just as many members in B as there are in N (infinite). You're twisting it to say: if you can map all the members of N onto some members of B and find that there are some members of B left over, then you know there are more members in B than in N. <-- That's not the rule (though I understand why you think it can be restated as such). What your example proves is that there are just as many odd members in B as there are in N (infinite). But because the number of odd members is infinite, and because adding more members to an already infinite set is still infinite, then adding the even members of B to the mapping still gives you the same amount (infinite). So you can map all the natural numbers onto every odd member of B, and you can map all the natural numbers onto every member of B (odd and even), and yes, Magnus, I'm saying they will both be the same amount (infinite) (this is what I've been saying all along). You really, really, really should watch the vsauce video I posted. He goes exactly into this--how to map the naturals onto two infinite sets. Here it is again:

^ Go to 5:00

Magnus Anderson wrote:You did not ask me for advice but I did not ask you for instructions either. So I think I'm justified in giving you a small hint.

Never ever tell your interlocutors what to do unless they really want you to tell them what to do. It's a way to ruin the discussion.

Oh God.
My thoughts | My art | My music | My poetry

In fact, the idea that there's more differences between groups than there is between individuals is actually the fundamental racist idea.
- Jordan Peterson

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right outta high school i tried to get a job as a proctologist but i couldn't find an opening.
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### Re: Is 1 = 0.999... ? Really?

gib wrote:So that I can understand what the hell's going on in your mind.

You don't need to categorize it in order to understand what's going on inside my mind. I already told you that $$0.\dot9$$ is a number that is greater than every number of the form $$\sum_{i=1}^{n} \frac{9}{10^i}, n \in N$$ but less than $$1$$. Why is that not enough? Why do you need it to be categorized?

So for you, a finite number must be expressible in terms of a finite sum of rational numbers.

That sounds good.

By this definition, would $$0.\dot3$$ be a semi-finite number?

No. That's because $$0.\dot3 \neq \frac{1}{3}$$.

You could say the same of $$0.\dot9$$ since it can be expressed as 1/3 x 3, but this gives you 1 which you dispute (which is a whole other can of worms we could debate).

Do you agree that $$0.\dot9$$ represents the infinite sum $$\sum_{i=1}^{\infty} \frac{9}{10^i}$$? If so, you have to accept that $$0.\dot9$$ is less than $$1$$. Everything else is irrelevant. There is no $$n > 0$$ such that $$\sum_{i=1}^{n} \frac{9}{10^n} = 1$$.

This wouldn't prove that B' has more members than N. Once again, the rule is: if you can map all the members of N onto all the members of B, then you know there are just as many members in B as there are in N (infinite). You're twisting it to say: if you can map all the members of N onto some members of B and find that there are some members of B left over, then you know there are more members in B than in N. <-- That's not the rule (though I understand why you think it can be restated as such).

I understand very well that's not the rule (and what the rule is.) I am not saying that's the rule. I am not restating the rule. I am questioning the rule.

What you don't understand is that merely stating that's the right way of determining whether any two infinite sets are equal in size or not does not make it the right way.

You adopted a conclusion that someone else arrived at without understanding the steps they took to arrive at it. And you think it's correct because it's widely accepted. And that's all perfectly fine. What's problematic is that you don't seem to be able to comprehend that conclusions can be questioned and that the fact that they are widely accepted does not mean they are correct.

You really, really, really should watch the vsauce video I posted.

Oh God.

Signs of frustration are another thing that can ruin discussions. Sure, it's a fashionable thing, but that does not make it right.

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

Interesting video Gib.. still got a 1/3 or so of it to go.

Some infinities can be bigger than others? Does anybody know what infinity means..? I now see why most say that infinity is hard to imagine/fathom, because most obviously cannot.

Sure, the elements in the set can be numbered.. that was never my issue, it’s that infinity means infinite, not more infinite.
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ

I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Wait, What! - MagsJ

Nobilis Est Ira Leonis | Om Surya Devaay namah | Manus justa nardus

MagsJ
The Londonist: a chic geek

Posts: 19951
Joined: Wed Nov 01, 2006 2:59 pm
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
gib wrote:So that I can understand what the hell's going on in your mind.

You don't need to categorize it in order to understand what's going on inside my mind.

Too funny

..a classic ILP exchange.
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ

I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Wait, What! - MagsJ

Nobilis Est Ira Leonis | Om Surya Devaay namah | Manus justa nardus

MagsJ
The Londonist: a chic geek

Posts: 19951
Joined: Wed Nov 01, 2006 2:59 pm
Location: London, NC1 … … … … … … … … … … … … … … … … … …Suryaloka.. the sun

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

Ecmandu wrote:So anyways,

He’s finally got that right!
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ

I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Wait, What! - MagsJ

Nobilis Est Ira Leonis | Om Surya Devaay namah | Manus justa nardus

MagsJ
The Londonist: a chic geek

Posts: 19951
Joined: Wed Nov 01, 2006 2:59 pm
Location: London, NC1 … … … … … … … … … … … … … … … … … …Suryaloka.. the sun

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