Is 1 = 0.999... ? Really?

[Magnus Anderson]

but when I inform someone who isn't you that I am - that's a little too much for you...

It's unnecessary and irrelevant.

FYI, this "standard algorithm" of addition that you're trying to bring into doubt is known as "addition".

Not quite. And I'm not doubting it.

non-mathematician intuitions

It's called logic.

Has nobody actually bothered to check what all the professional mathematicians have concluded on this subject?
I'll assume not, because if you do a quick search you'll find they all side with me that \(1 = 0.\dot9\). It's almost as if I wasn't kidding when I said I knew what I was talking about, and I haven't just been posturing all along like the dissenters here.
But I guess all the amateurs here can reject that because they know better, huh?

Perhaps because some of us are interested in what is logical rather than what is popular?

You have the support of the majority, so you shouldn't worry.

This is just an appeal to authority after all, so it can be ignored

Exactly.

[Magnus Anderson]

From a utilitarian perspective, is 0.9... useful in a way that 1 is not?

Note that the utilitarian perspective is irrelevant. The question is whether or not the two symbols represent one and the same quantity. That's the subject of this thread, not what's useful in practice.

[Ecmandu]

From a utilitarian perspective, is 0.9... useful in a way that 1 is not?

Note that the utilitarian perspective is irrelevant. The question is whether or not the two symbols represent one and the same quantity. That's the subject of this thread, not what's useful in practice.

*cough* I call bullshit!

Everything about math is what works in practice!

Edit: If they have different functions, they are not equal

[Silhouette]

but when I inform someone who isn't you that I am - that's a little too much for you...

It's unnecessary and irrelevant.

Yes it was equally unnecessary for you to admit you're not a mathematician, but it's relevant since we're talking about mathematics and it explains a lot.
I was being as unnecessary as you in letting you know that I am, and doing so was just as relevant for the same reason: that we're talking about mathematics.

Even pointing out your double standards is lost on you, and even then you can only come back with something that's only half true.

FYI, this "standard algorithm" of addition that you're trying to bring into doubt is known as "addition".

Not quite. And I'm not doubting it.

But this here is a straight up lie:

Just earlier you were saying this where you're literally questioning why Ecmandu was using the "standard algorithm" to add numbers together. You admit that addition can't add \(0.\dot0{1}\) to \(9.\dot9\) and conclude that this proves nothing about there being no difference between two representations of the same quantity \(10\), throwing addition itself into doubt rather than face the (already proven) possibility that you've been wrong this whole time. It's not too late to do the right thing, by the way.

non-mathematician intuitions

It's called logic.

Insofar as A = ¬A is logic, sure: your non-mathematical intuitions are logical contradictions.
I've said many times that every single thing you're saying is logically valid for finites only - but it's not sound. Infinites are not finites, they're the literal opposite - you've not once acknowledged this overwhelmingly simple but crucial and undeniable fact.

Perhaps because some of us are interested in what is logical rather than what is popular?

You have the support of the majority, so you shouldn't worry.

Yes, some of us are interested in what is logical rather than popular - including me and the professional mathematicians who agree with me.

I have no doubt that you're interested in what's logical too, but that doesn't mean you're as capable at attaining and/or accepting it - at least in the field that you admit isn't your field. But over time even any such good intentions of yours have started to show signs of wavering since you're doubling down so hard that you don't seem to be able to even acknowledge the obvious truths that I keep correcting you with - lately you seem more concerned with pot-shots at me, fallacious argument and feeding your cognitive biases.

Professionals and I only hold the popular view because it is logical. Your fallacious argument here is to accuse me of "argumentum ad populam" and to apply the formal fallacy of "affirming the consequent" to it within the bounds of the informal fallacy of the "false dilemma", by treating "logic" and "popularity" as mutually exclusive, and concluding that since an argument isn't logical because it's popular, my argument being popular therefore entails that I'm not interested in the logical.

It's frustrating how you're either unable or unwilling to see the degree to which I'm absolutely steamrolling you here to the point where the ease of doing so is becoming frustrating.

And "the utilitarian perspective" is absolutely relevant - if you knew anything about mathematics you'd know that its revolutions have been in large part due to the treatment of the mathematically invalid as valid. This is why the proven fact that \(1 = 0.\dot9\) is less interesting than the utilitarian possibility of treating the two representations of the same quantity as actually different quantities. The fact that in truth they're the same is already a done deal: the jury has been out for a long time. JSS resurrecting it was just his contrarian ambition to make a difference in a world that apparently rejected him, and naturally a few amateurs were foolhardy enough to go along with it.

From a utilitarian perspective, is 0.9... useful in a way that 1 is not?

I know you’ll see the difference there.

So I have this to ask of you, “is your dogged defense of this rational given that “every mathematician” agrees with you.

What’s ultimately funny about this thread to me (now that I’ve got my bearings) is that we’re all debating each other about something!

May the best argument(s) win.

It's Cantor's construction of things like Epsilon numbers, somewhat implied by quantities like \(0.\dot9\) that seem to have made constructive ripples in the mathematical world, rather than \(0.\dot9\) itself being useful. \(0.\dot9 = 1\) is just a fun exercise for casual math enthusiasts, because it's another confirmed example of the superficially seemingly unintuitive turning out to be true. The relatively mainstream intrigue that it has the potential to attract is the same as for optical illusions and magic tricks - for the mass uninitiated to make a noise about and then forget about. Epsilon numbers et al. are more rigorous but less easily penetrable, so as is always the case, the actually significant stuff goes over most people's heads in favour of sophistry and the opportunity for amateurs to try to convince other amateurs that they're smart.

Unfortunately while the best argument(s) win, they're rarely accepted by most - this forum is a mass graveyard of examples of this.
As I explained to Magnus just above, professional mathematicians agreeing with me isn't why I'm "doggedly" defending the correct answer. The correct answer is only popular because it's logical, and the fact that it's popular doesn't make that the sole foundation of my defense - which reminds me of yet another fallacy to add to the list that others are trying use against me: "the fallacy of the single cause". The best arguments have already won - it's more a case of getting people to accept that they're the best arguments even if they "like" the idea that they had the unappreciated intelligence all along to see a mistake in the work of experts.

[obsrvr524]

Actually I've known all this math since over half my life ago, and I have been regularly keeping up with advanced level maths to this day for fun - I actually really enjoy it.

And still can't get it right. I'm impressed.

So you consider yourself a mathematician?
Mr Donald Trump considers himself a US President.
Your point?

What you appear to NOT be is an adequate logician for resolving questionable issues. Memorizing maths formulas does little in that regard and I'm sure actually diminishes the necessary talent.

[Silhouette]

What you appear to NOT be is an adequate logician for resolving questionable issues. Memorizing maths formulas does little in that regard and I'm sure actually diminishes the necessary talent.

And this is just why I don't consider your opinion to be valid on whether I got some maths right.

This wishful thinking that "knowing the math" and "being able to resolve questionable issues" is some kind of trade-off is simply often not the case, to which the countless mathematical innovators at the very top levels can attest by virtue of their breakthrough accomplishments. In many cases, memorisation alone isn't enough of course, but that isn't enough to let you off the hook for not knowing/memorising maths formulas. Consider Terry Tao, "regarded by many people as the world's greatest mathematician", who will openly talk about how learning tricks and formulae makes all the difference when it comes to resolving questionable issues. That's the reality, and it's the Trumps of the mathematical world who merely think they can innovate where far greater minds have failed before, like you/JSS and Magnus are/were doing. I'm just relaying what the greater minds say, and I'm not saying I'm up there with them, just that I'm further up than the Dunning-Kruger effect will allow some people to accept.

Look, if you both really can't accept that I'm right, forget I said anything at all and listen to all the many professional mathematicians who are saying the same thing. I'm not here, ok? I'm just a messenger. Listen to the people who are actually highly capable in this field, and by all means be skeptical of them to some constructive degree, but have the humility to accept that in all likelihood you don't have what it takes to legitimately challenge the heights that high-level professionals are operating on. I'm sure some amateurs exist who actually can come out of nowhere to succeed in such a task, but in all probability that's not you - as always: the more you learn about something, the more you appreciate just how out of your reach it actually probably is. It's obvious to me that neither of you have been through that and come out the other side - on this topic whole topic at least - but like I said, don't take my word for it. I might be a mathematician but I'm not a professional, so if you resent me for being straight with you, go straight to consulting the top instead of going through me.

[Magnus Anderson]

Yes it was equally unnecessary for you to admit you're not a mathematician, but it's relevant since we're talking about mathematics and it explains a lot.
I was being as unnecessary as you in letting you know that I am, and doing so was just as relevant for the same reason: that we're talking about mathematics.

Not really. There's a big difference between the two.

Just earlier you were saying this where you're literally questioning why Ecmandu was using the "standard algorithm" to add numbers together.

I merely asked him what's the purpose of talking about the standard algorithm of adding numbers together (which, by the way, is not the same as addition.)

You admit that addition can't add \(0.\dot0{1}\) to \(9.\dot9\)

First of all, I wasn't talking about addition in general (a distinct concept), I was talking about the standard algorithm of adding numbers together. This isn't the same as addition.

And I didn't admit that one cannot use standard algorithm of addition to find the sum of \(0.\dot0{1}\) to \(9.\dot9\). I merely asked a hypothetical question.

and conclude that this proves nothing about there being no difference between two representations of the same quantity \(10\), throwing addition itself into doubt rather than face the (already proven) possibility that you've been wrong this whole time. It's not too late to do the right thing, by the way.

There is a difference between the two numbers and there's a super simple proof for it. If it's the case that this proof is a real proof, which I firmly believe that it is, nothing else is relevant. Don't think you've presented a single argument against it.

I've said many times that every single thing you're saying is logically valid for finites only - but it's not sound. Infinites are not finites, they're the literal opposite - you've not once acknowledged this overwhelmingly simple but crucial and undeniable fact.

The word "finite" and the word "infinite" are opposites but the word "finite quantity" and the word "infinite quantity" have a lot in common given that both are quantities.

Yes, some of us are interested in what is logical rather than popular - including me and the professional mathematicians who agree with me.

Not quite. Not when it comes to this subject.

I have no doubt that you're interested in what's logical too, but that doesn't mean you're as capable at attaining and/or accepting it - at least in the field that you admit isn't your field. But over time even any such good intentions of yours have started to show signs of wavering since you're doubling down so hard that you don't seem to be able to even acknowledge the obvious truths that I keep correcting you with - lately you seem more concerned with pot-shots at me, fallacious argument and feeding your cognitive biases.

There are no obvious arguments.

Professionals and I only hold the popular view because it is logical.

Not quite.

Your fallacious argument here is to accuse me of "argumentum ad populam" and to apply the formal fallacy of "affirming the consequent" to it within the bounds of the informal fallacy of the "false dilemma", by treating "logic" and "popularity" as mutually exclusive, and concluding that since an argument isn't logical because it's popular, my argument being popular therefore entails that I'm not interested in the logical.

You have no clue what you're talking about, so maybe you should just stop it and instead stick to the topic, what do you think?

Instead of talking about who you are and how many people agree with you, how about you show us the flaws in our arguments? But without pretending that you did. Whatever you did wasn't enough. Accept it and either leave the thread or continue discussion.

Show people where they are wrong, don't tell them that they are wrong. And if you can't be bothered, because you estimated that it'd take too much effort, leave. Everything else is an expression of frustration with the fact that people aren't agreeing with you ("Oh why can't people accept what the Great Wise Man Silhouette has to say on the subject!?") As if people are supposed to behave according to the expectations of the Great Wise Silhouette (or anyone else.)

[Magnus Anderson]

*cough* I call bullshit!

Everything about math is what works in practice!

Edit: If they have different functions, they are not equal

Consider that, in practice, it might happen to be useful to be mistaken on this subject. I'm not saying that it is, but let's assume that it is. What would be the correct answer to the question posed in the OP? Would you say \(1 = 0.\dot9\) or would you say \(1 \neq 0.\dot9\)? For \(1 = 0.\dot9\) is more useful in practice whereas \(1 \neq 0.\dot9\) is more logical. If the subject of this thread is what is useful, the correct answer would have to be \(1 = 0.\dot9\). But I'm insisting that's not the subject of this thread. This thread is primarily about what is true.

[Silhouette]

First of all, I wasn't talking about addition in general (a distinct concept), I was talking about the standard algorithm of adding numbers together. This isn't the same as addition.

This is hilarious - you're consulting "Math for Kids" as the source to share your level of knowledge!

You're talking to someone who casually came up with their own original algorithm for addition (and other arithmetical operators) based on binary logical operators alone over 6 years ago.

The standard algorithm of adding numbers together is addition. I can't believe I'm having to repeatedly state a tautology for you. That is, it's a system of adding digits when numbers with more than 1 digit are involved. There's other algorithms that do the same thing, such as the one I invented myself - and guess what. They're all addition!

You keep insisting that my explanations are merely "telling you" rather than "showing you", and here you are simply making claims without explanation:

Not really. There's a big difference between the two.

I merely asked him what's the purpose of talking about the standard algorithm of adding numbers together (which, by the way, is not the same as addition.)"
There is a difference between the two numbers and there's a super simple proof for it. If it's the case that this proof is a real proof, which I firmly believe that it is, nothing else is relevant. Don't think you've presented a single argument against it.

Not quite. Not when it comes to this subject.

Not quite.

You have no clue what you're talking about, so maybe you should just stop it and instead stick to the topic, what do you think?

Instead of talking about who you are and how many people agree with you, how about you show us the flaws in our arguments? But without pretending that you did. Whatever you did wasn't enough. Accept it and either leave the thread or continue discussion.

Show people where they are wrong, don't tell them that they are wrong. And if you can't be bothered, because you estimated that it'd take too much effort, leave.

Mere claims. No explanation. Right there in front of us.

By contrast, you even quoted one of my explanations on the last page.

The word "finite" and the word "infinite" are opposites but the word "finite quantity" and the word "infinite quantity" have a lot in common given that both are quantities.

So opposites have a lot in common when they qualify the same concept.
"They have in common" the concept that they're qualifying only, opposites don't suddenly have a lot in common just because they're referring to the same thing.
"A and ¬A have a lot in common because ∃(A ⊂ B) ∧ ∃(¬A ⊂ B)"...
^this is an explanation of your logical error that you'll proceed to deny exists.

Your fallacious argument here is to accuse me of "argumentum ad populam" and to apply the formal fallacy of "affirming the consequent" to it within the bounds of the informal fallacy of the "false dilemma", by treating "logic" and "popularity" as mutually exclusive, and concluding that since an argument isn't logical because it's popular, my argument being popular therefore entails that I'm not interested in the logical.

You have no clue what you're talking about, so maybe you should just stop it and instead stick to the topic, what do you think?

1) Argumentum ad populam: a fallacious argument that concludes that a proposition must be true because many or most people believe it.
2) affirming the consequent: a formal fallacy of taking a true conditional statement invalidly inferring its converse even though the converse may not be true.
3) False Dilemma: a type of informal fallacy in which something is falsely claimed to be an "either/or" situation, when in fact there is at least one additional option.

You said: "Perhaps because some of us are interested in what is logical rather than what is popular?"
Let:
\(P\) denote "popular"
\(Q\) denote "logical"
In the above quote of yours you imply that I commit 1): \(P\rightarrow{Q}\)
"Logical rather than what is popular" takes the form of 3): \(P\lor{Q}, Q\vdash\lnot{P}\)
Combining 3) with a rejection of 1) takes the form of 2): \((\lnot(P\rightarrow{Q}),Q)\vdash\lnot{P}\)

Quite obviously I know what I'm talking about, so maybe you should just stop insisting I don't, what do you think?
I keep showing you the flaws in your arguments, but it's never enough, nor does it even count as showing apparently, just "telling". This fallacy that you're commiting is called Moving the goalposts such that no amount of explanation that I offer ever counts as explanation.
I expect people to respond to my clear logic, and of course I get frustrated when you keep just coming back with "nah, didn't happen, you just don't understand". I expect people to exhibit cognitive biases and to encounter the backfire effect, but what I'm encountering here just seems like complete and utter dumbness.

Stubborn as a mule.

[Magnus Anderson]

So you agree that the length of the lines is the same even though there are fewer 90-inch segments than 1-inch segments.

That's correct.

But that would imply that if you counted the segments in each, you would finish counting the 90-inch segments before the 1-inch segments (that's what "fewer" means).

When we say "The number of people on planet A is less than the number of people on planet B" it does not mean that we can "finish counting" the number of people on either planet.

I've demonstrated this before. If you have a planet populated by an infinite number of organism, and you remove one organism from it, you'd have fewer organisms, even though you can never "finish counting" the number of organism on that planet.

By definition, if you have a bunch of things in one place (regardless of their number) and you subtract one thing, you get a smaller number of things. To say otherwise is to say that you didn't really perform the operation of subtraction, which is a logical contradiction.

That in turn implies that at least the number of 90-inch segments is finite.

Not quite.

Yeah, 'cause heaven forbid we confuse conceptual matters with empirical ones. You'd get science!!!

You'd get confusion.

This thread is about concepts.

Really???

Yes, really.

Why are infinite quantities not quantities?

You're right that we know how much a centimeter is and how much a millimeter is, but if I tell you 1cm = n gwackometers, it suddenly becomes unknown (you don't know how much a gwackometer is).

Your \(n\) is a symbol representing an unknown. \(1cm\) is not representing an unknown. You're comparing a known value (\(1cm\)) to an unknown number (\(n\)) of gwackometers.

Not sure why any of this is relevant, to be honest.

The central insight in algebra is that it doesn't matter what the variables (or units) stand for--they might as well be unknowns--its the rules for manipulating them that matter. And that was my point about plugging \(\infty\) into algebraic equations--it doesn't matter what it stands for, it's just a symbol--manipulate to your hearts content. You don't end up prooving anything.

Of course I do.

Whatever it is you think this "similar" length is with respect to rays, you are once again on your own (and strictly speaking, we're talking about terms like "shorter" and "longer").

I am "on my own". That's a good argument.

"Infinite quantity" isn't really a quantity. That's just a phrase we use to talk about infinity.

Because Gib says so. Again, no arguments whatsoever. You are merely declaring that infinite quantity isn't really a quantity.

Does this prove anything about males? Do you think it proves males are quantities?

I am certainly not saying that the reason \(\infty\) represents quantity is because you can plug it into a mathematical equation.

That would be your misunderstanding.

Then you're saying \(\infty\) is both the unit and the quantity at the same time.

I am saying that every unit can be represented as a number of other, smaller, units. For example, \(m\) can be represented as \(100cm\).

Ok, Magnus, you're insisting that we treat \(\infty\) as an infinite quantity. Fine. I actually agree, \(\infty\) does mean "an infinite number of things". But then I deny your right to use it in math. You're trying to sneak it into math by, simultaneously, treat it as a unit--which, as I said, you can do but not while also treating it as a quantity.

I don't have to treat it like a unit and I don't. I'm simply saying that you can, if you want to.

What number does 3cm equal? That is, how do you get rid of the units so that you're left with only a number? 3cm = X. What is X? <-- That's what you're trying to do with \(\infty\). You're trying to say 3\(\infty\) = X, and since \(\infty\) is an infinitely large number, you can plug that number into \(\infty\), multiply it by 3, and solve for X.

By following a rule that says that all numbers used in an expression represent a quantity of the same unit. So instead of writing something like \(3cm + 10cm = 13cm\) you can simply write \(3 + 10 = 13\) because every number in the equation represents a number of units of the same kind (centimeters.)

What about the formula E = mc^2? We know what c is (speed of light), but what is m? It's a symbol for mass, but do you know what the mass is?

That's true. But how is that relevant?

[Magnus Anderson]

:lol: This is hilarious - you're consulting "Math for Kids" as the source to share your level of knowledge!

https://en.wikipedia.org/wiki/Carry_(arithmetic)

In elementary arithmetic, a carry is a digit that is transferred from one column of digits to another column of more significant digits. It is part of the standard algorithm to add numbers together by starting with the rightmost digits and working to the left. For example, when 6 and 7 are added to make 13, the "3" is written to the same column and the "1" is carried to the left. When used in subtraction the operation is called a borrow.

That's a Wikipedia article (not a "Math for Kids" website) describing a very specific way to find the sum of two quantities. It's definitely not the only way. You can add "6" and "7" together by placing six apples and seven oranges in front of yourself and then counting all of them. That's not the same as the above.

Addition is a much more general concept, and saying that I'm talking about addition in general is not the same as saying that I'm talking about some specific algorithm for adding numbers. I did not the do the former, I did the latter.

By contrast, you even quoted one of my explanations on the last page.

Yes, and it's wrong, and I explained why, even though it's unnecessary. (One doesn't have to prove more than it's necessary.)

So opposites have a lot in common when they qualify the same concept.

They don't and I never said that.

"They have in common" the concept that they're qualifying only, opposites don't suddenly have a lot in common just because they're referring to the same thing.

Correct. We're talking about infinite quantities, not the word "infinite" in general. The word "infinite" is not a quantity per se. Neither is "finite".

Quite obviously I know what I'm talking about, so maybe you should just stop insisting I don't, what do you think?

This is one of the most pointless things you can say in a forum discussion.

I keep showing you the flaws in your arguments, but it's never enough, nor does it even count as showing apparently, just "telling". This fallacy that you're commiting is called Moving the goalposts such that no amount of explanation that I offer ever counts as explanation.

You have to understand that this isn't the subject of this thread.

I expect people to respond to my clear logic, and of course I get frustrated when you keep just coming back with "nah, didn't happen, you just don't understand". I expect people to exhibit cognitive biases and to encounter the backfire effect, but what I'm encountering here just seems like complete and utter dumbness.

Frustration isn't something to be proud of.

[MagsJ]

You didn’t even get that I was defending Magnus the whole time either, neither did Magnus, and apparently neither did sil...

Apparently we had cross discussions occurring when I only thought it was one discussion, a triangulation of sorts:

Magnus : 0.9... /= 1, orders of infinity exist
Silhouette: 0.9... does equal 1, no orders of infinity exist

Ecmandu/me: 0.9.../=1, no orders of infinity exist

I really just wanted to come in and stomp all over the twee debate and polite conversation being had. There’s a report button at the top of this post, if you click on it someone will come to the aid of the twit.

[Ecmandu]

I’m going to get back to the .9.../1 part, I made a mental note in my head when I read the apple thing, that infinite apples would have infinite weight, this is really a physics argument. Each member being part of the whole set would have infinite weight as well, making it physically impossible to remove an apple in the first place.

It’s just a mental note, not completely thought through, but I thought you guys might find it interesting.

[Magnus Anderson]

infinite apples would have infinite weight, this is really a physics argument. Each member being part of the whole set would have infinite weight as well, making it physically impossible to remove an apple in the first place

It's not a physics argument. It does not matter what's physically possible or impossible. This discussion is entirely about concepts.

An infinite number of apples can have any weight whatsoever. If each one of the infinitely many apples weighed exactly one infinitesimal kilos, then the apples as a whole would weigh exactly 1 kilo.

[Ecmandu]

infinite apples would have infinite weight, this is really a physics argument. Each member being part of the whole set would have infinite weight as well, making it physically impossible to remove an apple in the first place

It's not a physics argument. It does not matter what's physically possible or impossible. This discussion is entirely about concepts.

An infinite number of apples can have any weight whatsoever. If each one of the infinitely many apples weighed exactly one infinitesimal kilos, then the apples as a whole would weigh exactly 1 kilo.

Not really. So there’s two philosophies to this as far as I can see.

Let’s say you count a trillion apples through, and you stop and count again. The first trillion apples would be outweighed by the rest of the set. Ones finite, ones infinite.

Now let’s think about this in terms of infinitely regressive sets - they’ll have the same weight.

[Ecmandu]

Like I said! These are just notes!

So, thinking about infinitely regressive sets further...

If you only count the first 9 in 9/10ths... the rest of the infinite sequence combined should also add up to 9/10ths.

So there’s two ways you can look at it 9/10ths equals 9/10ths, or they are different 9/10ths.

If they are different 9/10ths, then adding them gives you 18/10ths, which I’m sure you’re not trying to argue!

So... when I consider the variables here, I definitely side on them being equal but not separate as additive properties.

I hope you and sil can understand that!

[Ecmandu]

Yeah!

Oh, you guys are going to love this!

So, I was using my notes in the last two posts, and I realized a contradiction in convergence theory:

Get this!

So you have a number like 0.9... that converges to the number 1!

Here’s the deal. 0.09.... is converging to 1 as well.
Meaning: 0.199.... 0.009... is converging to 1 as well...

Meaning 0.01999...

What this means is that you end up with the “infinitesimal” before the 9’s necessarily have to start.

This means that as you count through the convergence, you either have infinitesimal 1 before the 9’s or you have the 9’s equaling 1.111...

[Ecmandu]

Yeah!

Oh, you guys are going to love this!

So, I was using my notes in the last two posts, and I realized a contradiction in convergence theory:

Get this!

So you have a number like 0.9... that converges to the number 1!

Here’s the deal. 0.09.... is converging to 1 as well.
Meaning: 0.199.... 0.009... is converging to 1 as well...

Meaning 0.01999...

What this means is that you end up with the “infinitesimal” before the 9’s necessarily have to start.

This means that as you count through the convergence, you either have infinitesimal 1 before the 9’s or you have the 9’s equaling 1.111...

Dammit! Ok, this is really important !!!

For 0.9... to equal 1, the solution is 3.1...!!!

Like I said before...

0.9... =1

That means that 0.01999... equals 0.2999...

0.2999... equals 3.1111...

[Ecmandu]

I’m getting there! Let me work through this process!

If 1=0.9...

Then 0.009... = 0.01

Which means that 0.9... equals 1.1...

When you add those up,

You get 2.000...

That’s it!

That’s the disproof!

[Ecmandu]

I’m getting there! Let me work through this process!

If 1=0.9...

Then 0.009... = 0.01

Which means that 0.9... equals 1.1...

When you add those up,

You get 2.000...

That’s it!

That’s the disproof!

So what I’m trying to explain here, is that in the proof that 0.999... equals 1; ALL nines repeating always equal 1, which is why I say that the solution to 0.999 actually equals, not just 1, but 1.111... (unless you have an infinitesimal which is impossible (the one never gets expressed)

As I stated earlier, when you carry from the right to left, that makes 0.999... equal 2, with no infinitesimal to flip the 9’s, you’re still left with 1.999... which equals 2.

[Ecmandu]

You want to get at the truth?

Work with me here.

I know I’m close.

Because an infinitesimal is impossible, you have to carry from left to right.

This causes all the 9’s to eventually be 1’s... which not equal to 1 but is equal to 1.1...

1.1... + .999... = 2

[Magnus Anderson]

Some clarifications:

Whatever it is you think this "similar" length is with respect to rays, you are once again on your own (and strictly speaking, we're talking about terms like "shorter" and "longer").

If you have an infinite line of people standing in front of you, and you remove the person standing at the end of the line (or at the beginning of the line, it's the same thing), you'd have a shorter line of people standing in front of you. So even though the queue is a ray, and not a line segment, you can still talk about its length.

"Infinite quantity" isn't really a quantity. That's just a phrase we use to talk about infinity. If you want to say you know what \(\infty\) stands for in the equation (namely infinity), then you're simply doing something you can't do. It doesn't stop it from being a symbol which you can plug into the equation and manipulate according to the rules of algebra. I can do the same with the symbol for male:

\(\frac{{6}\times{♂}}{3}\) = \({2}\times{♂}\)

Does this prove anything about males? Do you think it proves males are quantities?

It does not. That's because the symbol already has a meaning, one that has nothing to do with quantities. Even if, instead of using a known symbol, you used an unknown symbol to which you assigned no meaning, we'd still be talking about a symbol that does not represent a quantity.

The symbol \(\infty\) is by definition a quantity albeit not the kind of quantity we're used to. By definition, \(\infty\) is a number greater than every real number which is why we consider expressions such as \(\infty > 1\) as being true. You can't do something like that with your ♂ symbol. Indeed, the symbol has no defined relation to other numbers. Does \(♂ > 1\)? We can't know, the result is undefined.

The problem occurs with equations such as \(\infty + 1 = \infty\). True or false?

If what that equation means is "If you take some infinite quantity and add one to it, you'd get an infinite quantity as a result" then the equation is TRUE.

However, if what that equation means is "If you take some infinite quantity and add one to it, you'd get the same infinite quantity as a result", then the equation is FALSE.

In the first case, the symbol of infinity represents ANY INFINITE QUANTITY IN GENERAL. In the second case, the symbol of infinity represents THE SAME INFINITE QUANTITY WHEREVER IT OCCURS.

The meaning of the symbol of infinity is what has to be well-defined in order to avoid making pseudo-proofs such as this one:

$$
s = 9 + \underline{0.\dot9}\\
s \div 10 = \underline{0.\dot9}\\
s - s \div 10 = 9 + 0.\dot9 - 0.\dot9 = 9\\
s \times (1 - 1 \div 10) = 9\\
s \times 0.9 = 9\\
s = 9 \div 0.9\\
s = 10\\
$$

\((9 + \underline{0.\dot9}) \div 10 \neq \underline{0.\dot9}\)

The two underlined numbers aren't the same: the two infinite sums don't have the same infinite number of non-zero terms (or in plain terms, they don't have the same infinite number of 9's.)

That's why \(0.\dot9 - 0.\dot9\) in the third equation of the proof does not equal to \(0\).

The infinitesimal quantity has conveniently disappeared.

[gib]

When we say "The number of people on planet A is less than the number of people on planet B" it does not mean that we can "finish counting" the number of people on either planet.

Well, geez, man, not sure how else to put it. "Fewer" implies finitude. It implies an actual quantity relative to a larger quantity. I shouldn't need proof of this, that's just the common definition. If you need proof, what you're really asking is: why should I accept your definition when I have my customized one that fits with my theories on infinity. And to that I say: fine, we can go with either definition, but then you gotta help me understand your definition.

Let's try this: imagine the scenario I described earlier, the one with two infinite parallel lines. For all intents and purposes, the same length. Now remove every odd point from one of the lines. Then move all remaining point into the spots left behind by the points you removed. According to you, the line with the points removed is now "shorter". But since we moved all points into the spots left empty from the points we removed, the lines are perfectly identical. It's as if we didn't remove any points at all. We're back to the initial state of the scenario. So here's your chance to shine. Help me understand how the lines are different now. Help me understand what it means that the line we removed points from is now "shorter".

By definition, if you have a bunch of things in one place (regardless of their number) and you subtract one thing, you get a smaller number of things. To say otherwise is to say that you didn't really perform the operation of subtraction, which is a logical contradiction.

You are, once again, dismissing counter-examples. Nothing wrong with following a string of logic to arrive at certain conclusions, but when someone shows you how counter-examples can be drawn in special case (like when the number is infinite), you can do two things: 1) dismiss it and just reiterate your original logic, or 2) consider the counter-example and rethink your original logic. You can then show how your original logic still holds and give an explanation for why, or you can concede that your original logic doesn't actually hold in the special case of the counter-example. Seems to me like you're choosing 1).

Yeah, 'cause heaven forbid we confuse conceptual matters with empirical ones. You'd get science!!!

You'd get confusion.

This thread is about concepts.

Unless we're talking about Anderson's magical fantasy world of shorter and longer infinities, this thread is about reality. The only way confusion would arise is if we insist on having contradictions between the concepts and the empirical evidence while still maintaining the concepts represent reality. If I say the concept of a flat world implies that you'd fall off the edge of the world if you walked far enough, and empirical evidence shows that you don't, you'd only be confused if you insisted your flat-earth concept is still reality. The alternative is to accept that the concept is wrong. You can still say that the concept of a flat-earth implies that you'd fall off the edge if you walked far enough, but what good does holding onto that concept do if the empirical evidence shows that it's wrong?

Why are infinite quantities not quantities?

*Ugh* Ok, let's try this. Infinity means no end. That means surpassing all numbers. As soon as you reach a particular number, you know you're not at infinity because otherwise you could say you've reached the end of all numbers, and you know that just doesn't make sense (right?). An infinite quantity just means beyond quantity. <-- That's why it's not a quantity. Give me any quantity, any quantity at all, and I will show you it's not infinity.

Your n is a symbol representing an unknown. 1cm is not representing an unknown. You're comparing a known value (1cm) to an unknown number (n) of gwackometers.

So then tell me, how many gwackometers is 1cm?

The central insight in algebra is that it doesn't matter what the variables (or units) stand for--they might as well be unknowns--its the rules for manipulating them that matter. And that was my point about plugging ∞ into algebraic equations--it doesn't matter what it stands for, it's just a symbol--manipulate to your hearts content. You don't end up prooving anything.

Of course I do.

This should be good.

Because Gib says so. Again, no arguments whatsoever. You are merely declaring that infinite quantity isn't really a quantity.

I am often amazed at what people need proof of.

I am certainly not saying that the reason ∞ represents quantity is because you can plug it into a mathematical equation.

That would be your misunderstanding.

Glad we got that cleared up.

Then you're saying ∞ is both the unit and the quantity at the same time.

I am saying that every unit can be represented as a number of other, smaller, units. For example, m can be represented as 100cm.

So \(\infty\) stands for: an infinite number of Xs (organisms, train carts, points in line, whatever) where X is the unit, correct? Then I forbid you to use it in arithmetic. If \(\infty\) was the unit (as in 2\(\infty\) meaning 2 infinities), then I'd say run wild, have fun, but you're not saying that. You're saying you want to take the quantitative value of \(\infty\) and multiply it by 2, leaving whatever there is an infinity of as the unit, and to that I say: STOP IT! *slaps hand*

By following a rule that says that all numbers used in an expression represent a quantity of the same unit. So instead of writing something like 3cm + 10cm = 13cm you can simply write 3 + 10 = 13 because every number in the equation represents a number of units of the same kind (centimeters.)

No, no, no, you're just making the unit implicit. I mean, really get rid of it, as in you just have the number 13, not 13 centimeters, just 13. But this is a non-issue as I see above you don't mean to get rid of units entirely but simply use whatever object there is an infinity of as the unit. Again, I forbid you!!! *ominus voice echoing*

That's true. But how is that relevant?

You're insisting that because symbols serve to represent something, we know what those are. But I'm saying that the whole point of variables (which are symbols) in algebra is to have a way of doing arithmetic with unknowns. You can know them, but you don't have to. I was saying that if you want to use \(\infty\) as a variable, it has to stand for an unknown because the actual value of infinity is off limits.

[Ecmandu]

I know I sound like an incomrehensible crazy person...

I’m working again on the 0.999... /= 1 argument, and if as Magnus states, silhouette is the “ecmandu whisperer”...

It’s a lot to ask silhouette, but can you help me disprove your proof that 0.999... does equal 1??!!

My argument is this:

The number 1 never occurs with the infinitesimal argument ... thus, the 0.999... never falls like dominoes from right to left to create the whole number from the infinite sequence. Are we good so far??

So next, this means the only other option is to carry from left to right.

This means that you have to add 0.999... to 0.111...

The first time you do this you get 1.0999... and 1.0111...

The second time you do this, you get 2.10999...

See where I’m going with this sil???

Like I stated before, if you have no infinitesimal and the only other option is to carry from left to right, you have no equality for 1=0.999...

So, Silhouette! I’m asking you to “man up” here, and consider my argument is true.

[Magnus Anderson]

Let's try this: imagine the scenario I described earlier, the one with two infinite parallel lines. For all intents and purposes, the same length. Now remove every odd point from one of the lines. Then move all remaining point into the spots left behind by the points you removed. According to you, the line with the points removed is now "shorter". But since we moved all points into the spots left empty from the points we removed, the lines are perfectly identical. It's as if we didn't remove any points at all. We're back to the initial state of the scenario. So here's your chance to shine. Help me understand how the lines are different now. Help me understand what it means that the line we removed points from is now "shorter".

So we have two infinite parallel lines that are of the same length. We pick one of them and remove every odd inch from it. Then we fill in the gaps that we created using remaining inches. By doing this, the gaps disappear leaving the two lines looking perfectly identical.

The problem is that there isn't enough inches remaining to fill in the gaps without creating new gaps elsewhere. This illusion is created by moving the gaps out of our sight.

If you don't see it, it's not there.

And if you keep pushing things out of your sight, you can keep reassuring yourself they don't exist.
Especially if this process is an infinite one (:

Here's the line we started with:
\( \bullet \bullet \bullet \bullet \bullet \bullet \cdots \)

Now here's the line with odd inches removed from it:
\( \circ \bullet \circ \bullet \circ \bullet \cdots \)

There's an infinite number of inches out of our sight. We don't see them, we merely see the ellipsis "..." which tells us there is more to this line than what we see. What we want to do now is take three inches from the remaining inches that we don't see, so that we can fill in the gaps. We can do that, because there's still an infinite number of inches remaining, so we do that and we get:

\( \bullet \bullet \bullet \bullet \bullet \bullet \cdots \)

Voila! The line looks like the original one! They now appear to be identical! But what happened to those gaps? Where did they go? Well, they went out of our sight. They didn't magically vanish. We don't know exactly where they went, but they are somewhere out of our sight.

So the lines aren't really identical. They merely look like they are.

The gaps can't magically vanish. The only thing we can do is push them out of our sight forever thereby creating an illusion that the two lines are identical.

They are not.

This "paradox" is known as Hilbert's paradox of the Grand Hotel:
https://en.wikipedia.org/wiki/Hilbert%2 ... rand_Hotel

I think Carleas mentioned it somewhere at the beginning of this thread (40-50 pages ago . . .)