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

So for what it’s worth, there’s some Freudian/Jungian psychology for the day.

Thank you for that edifying discourse, E. I was intrigued, and feel I should now risk the following hypothesis; all the great mathematicians were great because they were gay.

You heard em, Andy. Either make some significant changes to your sexual orientations, or continue to flunk this thread. Makes no difference to me.
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Harris vs. Peterson; a wittgensteinian exercise in philosophical comedy
promethean75
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:Tell me, why is it that two infinite quantities cannot be added together to get a bigger infinite quantity?

I can answer that one for James. He clearly stated that in order to use maths with infinities, you first have to define a standard infinity. That's where his infA originated (as opposed to infB). And his reasoning was simply that trying to say "2 * infinity" is like saying "2 * length". For it to make any sense, you have to be more specific.
You have been observed.
obsrvr524

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

obsrvr524 wrote:
Magnus Anderson wrote:Tell me, why is it that two infinite quantities cannot be added together to get a bigger infinite quantity?

I can answer that one for James. He clearly stated that in order to use maths with infinities, you first have to define a standard infinity. That's where his infA originated (as opposed to infB). And his reasoning was simply that trying to say "2 * infinity" is like saying "2 * length". For it to make any sense, you have to be more specific.

Ecmandu and SIlhouette are telling us that even if you define a standard infinity, you cannot add two infinities together to get a bigger infinity.

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

Magnus Anderson wrote:
obsrvr524 wrote:
Magnus Anderson wrote:Tell me, why is it that two infinite quantities cannot be added together to get a bigger infinite quantity?

I can answer that one for James. He clearly stated that in order to use maths with infinities, you first have to define a standard infinity. That's where his infA originated (as opposed to infB). And his reasoning was simply that trying to say "2 * infinity" is like saying "2 * length". For it to make any sense, you have to be more specific.

Ecmandu and SIlhouette are telling us that even if you define a standard infinity, you cannot add two infinities together to get a bigger infinity.

Magnus. Good. We’re still on the same page.

I thought it was an interesting argument that there are more numbers than decimals; however, anytime you use a ratio, decimals are implied. Can you contradict this?

Silhouette and I have provided a few compelling arguments already that you haven’t addressed as to WHY 2*infinity is a contradiction.

You keep stating it, but you avoided our proofs through contradiction.

Maybe you think there’s a backdoor that circumvents our “proofs”, that’s fine, but I haven’t seen it yet.

I personally used the “cheat” argument and the “2 gods” argument and the “quantum flux” argument.

I can let silhouette speak for his own arguments, the “straw/box” argument of his is really good.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:I thought it was an interesting argument that there are more numbers than decimals; however, anytime you use a ratio, decimals are implied. Can you contradict this?

There is no decimal representation of $$\frac{1}{3}$$, for example. $$0.333\dots$$ isn't it. That's a different number.

If you're asking for a rigorous proof, I don't have one. And I don't think it's necessary. You can't demand of me to prove more than it's necessary to prove in order to show that $$0.999\dotso$$ does not equal $$1$$. I sometimes do so, but it's only because it's fun, not because it's necessary.

On the other hand, since you are the one claiming that the set of decimal numbers is the set of all numbers, it's up to you to prove it.

Silhouette and I have provided a few compelling arguments already that you haven’t addressed as to WHY 2*infinity is a contradiction.

You keep stating it, but you avoided our proofs through contradiction.

You provided no compelling arguments, no proofs. You merely defined the word "infinity" in your own way and used the logical implications of such a concept to "prove" that I am wrong even though that's not the way I define the word "infinity". (In fact, that's not the standard definition of the word.)

the “straw/box” argument of his is really good

The "straw/box" argument merely shows that he does not understand the simple fact that there are no inherently correct definitions of words.

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

So here’s the deal Magnus !

You stated outright that you don’t believe that 1=0.999...

So what are we arguing about?

You don’t believe in convergence theory and convergence theory is necessary to quantify infinity.

You know? There’s a big difference between infinity and infinite sequences.

1/3 * 3 equals nine in base 10; 10/3 equals 3*3 which equals 9 to get back to ten.

So.. really.. we have an operator trick.

Now, if as you state, infinity doesn’t converge, then there are no orders of infinity that can be used, because as non convergence infinity ceases to be applied as a unit of measurement which is subject to operators such as additions and powers.
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:Isn't it by definition that there is always a number smaller than 1-0.999...? Infinity isn't a destination. Infinite means there is always more. That "always more" is the exact difference between 1 and 0.999....

It is part of the very definition of 0.999... that it is not 1.0.

It's by definition that there's always another $$9$$ in $$0.\dot9$$ and therefore never ever any room for anything at all between $$1$$ and $$0.\dot9$$ no matter how hard you go.
"Infinite means there is always more" - sure: just more 9s in this case though. The recurring "dot" is applying to the 9 it's on top of, not anything else.
The result of doing this is that there's "infinitely no space left": an infinite lack of space for even one finite infinitesimal - never mind an infinite number of them!

The idea that dividing one by 3 and then multiplying the result by 3 leaves an infinite gap between what you started with and finished with... - this should be a pretty good clue that you're inverting the sense of what infinity is if you think that. There's no infinite gap when dividing 3 by 3 and then multiplying the result by 3 - be consistent.
The idea that the closest possible decimal quantity to $$\frac{1}3$$ is $$0.\dot3$$, therefore there's an infinite gap between them should sound some alarm bells too.

obsrvr524 wrote:I can answer that one for James. He clearly stated that in order to use maths with infinities, you first have to define a standard infinity. That's where his infA originated (as opposed to infB).

This is perfectly circular by James here.
To have the conditions to do something (use maths with infinities) you need to have those conditions (a defined infinity).
I can clearly state that in order for black to be white, you first have to define a black white.
The conclusion that the in-finite can be de-fined mathematically is assumed in the premise that a standard in-finite can be de-fined.

obsrvr524 wrote:And his reasoning was simply that trying to say "2 * infinity" is like saying "2 * length". For it to make any sense, you have to be more specific.

Again with the treatment of the quality of having no quantity as quantity.
"For non-specified ending to make any sense, you just have to be specific with the non-specified ending"...

Magnus Anderson wrote:Ecmandu and SIlhouette are telling us that even if you define a standard infinity,

I am saying that defining (giving ends) to infinity (endlessness) is a contradiction in terms.

Magnus Anderson wrote:you cannot add two infinities together to get a bigger infinity.

This is correct though.

Magnus Anderson wrote:The difference between $$1 - 0.\dot9$$ can be represented using a number. James's $$0.000...1$$ is a number. Indeed, the very expression $$1 - 0.\dot9$$ can be considered a number in the same way that a fraction of two integers is considered a number (a rational number.)

I forgot to mention that numbers have to avoid contradictions in terms, how silly of me.
"0.000" defines the start and "1" defines the end. With all ends accounted for, the "..." attempting to imply infinity is a contradiction in terms.
This is what happens when you try to define infinity by giving it ends: you make it finite (the opposite of what it is).
If there could be an infinity of something, e.g. 0 digits entirely enclosed by finite bounds, then the "finite" bounds would be indefinitely separated such that they'd never come to exist - just like the end of infinity never comes to exist by both definition and derivation.

$$0.000...1$$ is a numerical representation of an invalid number. Just goes to show that just because you can represent something numerically, doesn't mean it's a number. Fortunately my logic went in the other direction, that to be a number, you have to be able to represent it numerically. Let's avoid "affirming the consequent" here, eh?

Magnus Anderson wrote:What you cannot do is you cannot represent the difference using a decimal number. And that's exactly what your so-called argument against the existence of numbers smaller than $$0.000...1$$ boils down to. You're basically saying that if a quantity cannot be represented as a decimal number, it does not exist. But that's simply not true. "Decimal number" is not synonymous with "number".

So as I was saying, numbers are necessary in any valid format. The invalid $$0.000...1$$ therefore doesn't count.
I'd accept any valid numerical representation, decimal or otherwise.
$$\sum_{n=1}^\infty\frac9{10^n}$$ or $$0.\dot9$$ are fine because here the infinity isn't finite nor the finite infinity. The finites and the infinites are separate: the finites are defined in a very specific way such that the infinity can then say "now go do that defined thing an infinite number of times in the one way that infinity can be infinite".
$$\frac1\infty$$ is not fine because the infinity is standing in for a finite, or it is represented as a finite - they are not separated. Infinity alongside any numerical format of finites is like an operation, as Ecmandu said. It's an instruction to operate on defined finites in an ongoing way - it's never "the end" of having been operated on - because obviously "the end" i.e. finitude is the opposite of endlessness. $$2\times\infty$$ is invalid for the same reason.

Just be consistent! That's all you need. Math wouldn't be math if it wasn't consistent.

Magnus Anderson wrote:Consider that there is no decimal representation of $$\frac{1}{3}$$. Yet, $$\frac{1}{3}$$ represents a real quantity. It's a quantity that can be represented as a rational number ($$\frac{1}{3}$$) as well as a base-3 number ($$0.1$$). The fact that there is no decimal representation of $$\frac{1}{3}$$ does not mean it's not a representation of a quantity.

Again, this violates consistency.
Why is it okay to divide 9 by 3 to get 3 and then multiply that by 3 to get back to 9, but not ok to divide 10 by 3 and/or then multiply the result back by 2 to get $$10$$ or $$9.\dot9$$ by using the exact same method?

You'll no doubt accept that 10/4 is 0.25 but somehow as soon as the remainders don't resolve to 0, it's not valid?
You probably didn't even stop to think that your logic here would invalidate the irrational and transcendental numbers, right? I bet you think they're okay, but the exact same thing isn't okay when it goes against your point.

Magnus Anderson wrote:What makes you think that hyperreal numbers aren't numbers?

You can call black "white" and it would still be what it is and and not its opposite.
I keep saying it can be interesting to see what happens if you treat infinites as finites, but that doesn't make it true - only useful.
Only count numbers as numbers and you'll be up to speed.

Magnus Anderson wrote:If you're asking for a rigorous proof, I don't have one. And I don't think it's necessary.

But then this says it all really.

I'm trying to get you to sufficiently delve into the details but if rigor isn't necessary to you, there's not been any point in me trying.

Magnus Anderson wrote:You provided no compelling arguments, no proofs. You merely defined the word "infinity" in your own way and used the logical implications of such a concept to "prove" that I am wrong even though that's not the way I define the word "infinity". (In fact, that's not the standard definition of the word.)

Infinity is the quality of having no quantity is something I've said over and over, which is exactly how it's both defined and derived. This isn't "my own way", it is "the" way.
These kinds of resorts of yours are pretty low - to just deny that all this work people have gone to for your sake was never even done. It's at the very least rude, if not outright ignorant.

Magnus Anderson wrote:The "straw/box" argument merely shows that he does not understand the simple fact that there are no inherently correct definitions of words.

I'm sorry, I don't accept the definitions of the words you've used to say there are no inherently correct definitions of words.

You see how self-annihilating this kind of argument of yours can get?
If you can't legitimately define the terms for your argument to be valid, then throw out words altogether, right?

Try imagining the possibility that you're wrong about something and think about how you'd react and resolve it. If the answer is to double down indefinitely, then if neither of us ties up this resolved issue, this debate will go on endlessly. Defining a finite resolution to such an infinite scenario would be invalid - is invalidity what you're aiming for?

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

Suppose that there's a half an apple in front of you.

Obligatory picture (can't embed, too big):
https://riversedgecurriculum.files.word ... 215396.jpg

Now, suppose that you want to quantify (i.e. represent using some sort of symbol) the number of whole apples that is in front of you.

And suppose that you're only familiar with the set of natural numbers.

Tell me, how would you represent what you see in front of yourself?

Note that we're not interested in approximate values, only in exact values. How would you answer the question "What's the exact number of whole apples in front of you?"

No natural number can describe the quantity of whole apples that is in front of you. The set of natural numbers allows you to represent one apple, two apples, three apples and so on, but what's in front of you doesn't quite fit any such description.

What does this mean? Does this mean that the number of whole apples in front of you is equal to one (unfortunately, no zero among natural numbers) or does it mean that your vocabulary is limited?

It would be really strange to say that the number of whole apples in front of you is one for you would be denying your senses (or at the very least, you would be contradicting yourself.)

That would be a very backward kind of reasoning, one that I've found to be frequently used in sophistical arguments.

What the above situation indicates is not that your senses are deceiving you, nor that your prior assumptions are incorrect, but quite simply that your vocabulary is limited i.e. you have no symbols with which you can represent the quantity that is in front of you.

That's why you have to extend your vocubalury with new kinds of numbers, such as rational numbers.

With rational numbers you can say there is exactly $$\frac{1}{2}$$ whole apples in front of you.

The set of natural numbers contains no numbers smaller than $$1$$. That does not mean that numbers smaller than $$1$$ do not exist, right? It simply means that the set of natural numbers is a limited vocabulary, and hence, not the set of all numbers that can be imagined.

Suppose now that there is an infinite number of apples in front of you and that you're only familiar with real numbers. You'd be facing a similar situation. The set of real numbers contains no such number, and yet, the quantity that is in front of you requires it. What would you do? You'd do the same thing: you'd extend your vocabulary.

And what if you had this same infinity of apples in two different places? How would you represent that? "Infinity" wouldn't be enough because you'd want to represent the obvious difference between the two endless quantities. For this isn't the same infinite quantity you saw before. It's clearly twice the size. You can say the quantity is infinite but only in the general sense of the word. But if you wanted to be specific, you'd have to say something like $$2\times\infty$$ or $$2\times\omega$$ or $$2\times \text{infA}$$. You'd have to pick a standard against which infinite quantities would be measured.

The same applies to the difference between $$0.\dot9$$ and $$1$$. You'd have to extend your existing vocabulary by introducing infinitesimals.

The lesson to be taken is that JUST BECAUSE YOU HAVE NO SYMBOLS WITH WHICH YOU CAN REPRESENT SOME KIND OF REALITY, IT DOES NOT MEAN THAT THAT KIND OF REALITY DOES NOT EXIST.

Ecmandu and Silhouette argue that it's impossible to add two infinite quantities together and get a bigger infinity quantity because they work with their own concept of infinity according which the word "infinite" means "the property of having no ends of any kind" where "end of any kind" has a very, very, very, very broad meaning. Ultimately, the only thing they are showing is that THEIR concept of infinity cannot be added to, subtracted from, multiplied, divided, etc. None of that has any relevance to the subject at hand.

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

So, Magnus , you’re saying that all infinities can be bound, you’re saying that infinity can be bound in general? You’re saying that the endless ends every time? (Contradiction)?
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:It's by definition that there's always another $$9$$ in $$0.\dot9$$ and therefore never ever any room for anything at all between $$1$$ and $$0.\dot9$$ no matter how hard you go.

It's by definition that there's always another $$9$$ in $$0.\dot9$$ and that means that there's always a gap between $$1$$ and $$0.\dot9$$. This is obvious as hell, the fact that you have the guts to deny it is worrisome.

1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 = 0.001
etc.

There's $$1$$ at the end of EVERY partial sum.

1 - 1.0 = 0
1 - 1.00 = 0
1 - 1.000 = 0
etc.

There's $$0$$ at the end of EVERY partial sum.

The result of doing this is that there's "infinitely no space left": an infinite lack of space for even one finite infinitesimal - never mind an infinite number of them!

Not really.

The idea that dividing one by 3 and then multiplying the result by 3 leaves an infinite gap between what you started with and finished with... - this should be a pretty good clue that you're inverting the sense of what infinity is if you think that. There's no infinite gap when dividing 3 by 3 and then multiplying the result by 3 - be consistent.

That's correct but it's a strawman (a product of your imagination.) Noone said, and noone's logic implied, that dividing one by 3 and then multiplying the result by 3 leaves a gap between what we started with and finished with.

The idea that the closest possible decimal quantity to $$\frac{1}3$$ is $$0.\dot3$$, therefore there's an infinite gap between them should sound some alarm bells too.

I assume what you mean by "infinite gap" is compatible with "infinitely small gap". If so, that's true: there is an infinitely small gap between the two values. If you think that this should sound some alarm bells, you should explain why.

This is perfectly circular by James here.
To have the conditions to do something (use maths with infinities) you need to have those conditions (a defined infinity).
I can clearly state that in order for black to be white, you first have to define a black white.
The conclusion that the in-finite can be de-fined mathematically is assumed in the premise that a standard in-finite can be de-fined.

The point is that if you want to determine whether $$2\times\infty = \infty$$ is necessarily true or false, you have to put the two infinite quantities in relation to each other. Otherwise, the answer is indeterminate: it might be true but it also might be false. The same applies to expressions such as $$2\times\text{finite number} = \text{finite number}$$. If you want to know whether such an equation is necessarily true or necessarily false, you have to define the relation between the two finite numbers e.g. if you say that the second finite number is twice the size the first number, then the equation is necessarily true. Otherwise, the answer is indeterminate.

Your argument is that no relation can be defined between two infinite quantities. Which is a ridiculous claim.

I am saying that defining (giving ends) to infinity (endlessness) is a contradiction in terms.

And that means exactly what? How do you give "ends" to an infinite set? Don't talk about sequences, that's irrelevant. Take the set of natural numbers. The set is infinite. How do you give "ends" to it? You can give it an "end" by making it finite, but by doing so, it'd no longer be infinite. (And noone is doing this, anyways.)

Your problem is with statements such as "There are two planets populated by an infinite number of organisms existing at two different points in space." You have this weird idea that such statements are contradictory because they speak of quantities of things that are spatially bounded i.e. things that do not exist everywhere but only within a limited portion of space. You have a problem with the idea that there can be an infinite number of things within a finite portion of the universe. You think this is a contradiction because it does not abide by your concept of "true" infinity, the concept of infinity as "truly endless". The fact is that noone cares about your concept of infinity. It's not the one that most people work with. And it's certainly not the one I am using. You can't say I am contradicting myself if I am not saying what you think I'm saying (and I'm not saying what you think I'm saying because I'm not working with your concept of infinity.)

I forgot to mention that numbers have to avoid contradictions in terms, how silly of me.
"0.000" defines the start and "1" defines the end. With all ends accounted for, the "..." attempting to imply infinity is a contradiction in terms.

That would be you channeling the spirit of Ecmandu and also the spirit of Carleas and many other people who make the same exact silly claim because they did not bother to understand what the symbol represents. Basically, you're too confident about how well you understand other people's positions.

There's no $$1$$ at the end of the infinite sequence of $$0$$s. How many times must it be said?

$$0.000...1$$ is a numerical representation of an invalid number.

Sure, it's an "invalid" number, whatever that means. And if you're saying it's not a representation of a quantity that exists, why is it so? Because you say so?

Just goes to show that just because you can represent something numerically, doesn't mean it's a number.

It's a number, it has been shown to be a number, it's just that you don't want to accept it.

So as I was saying, numbers are necessary in any valid format. The invalid $$0.000...1$$ therefore doesn't count.

Why is that an invalid format? Because you do not understand it?

I'd accept any valid numerical representation, decimal or otherwise.

You'd accept anything familiar to you and reject anything that is not. Sort of like rejecting rational numbers because only natural numbers are familiar to you.

$$\sum_{n=1}^\infty\frac9{10^n}$$ or $$0.\dot9$$ are fine because here the infinity isn't finite nor the finite infinity.

As if $$0.000\dotso1$$ is a finite infinity i.e. a contradiction in terms.

$$\frac1\infty$$ is not fine because the infinity is standing in for a finite, or it is represented as a finite - they are not separated.

Basically, you think it's a contradiction in terms. That's all you said. Of course, without any argument whatsoever.

Infinity alongside any numerical format of finites is like an operation, as Ecmandu said. It's an instruction to operate on defined finites in an ongoing way - it's never "the end" of having been operated on - because obviously "the end" i.e. finitude is the opposite of endlessness.

No, infinity is not an operation. That's the spirit of Ecmandu taking you over. Don't let it take you over.

$$2\times\infty$$ is invalid for the same reason.

Yes, it's invalid because you think it's a contradiction in terms because you don't understand what it stands for.

Just be consistent! That's all you need. Math wouldn't be math if it wasn't consistent.

That's exactly thet point of this thread. $$0.\dot9 = 1$$ betrays consistency.

Again, this violates consistency.
Why is it okay to divide 9 by 3 to get 3 and then multiply that by 3 to get back to 9, but not ok to divide 10 by 3 and/or then multiply the result back by 2 to get $$10$$ or $$9.\dot9$$ by using the exact same method?

$$9\div3\times3 = 9$$ and $$10\div3\times3 = 10$$ are true. What's not true is that $$10 = 9.\dot9$$. Again, a strawman.

You'll no doubt accept that 10/4 is 0.25 but somehow as soon as the remainders don't resolve to 0, it's not valid?

Because it's obvious that $$10\div3 \neq 3.\dot3$$. That's how you can know that the division algorithm that we use does not give us correct results when dividing certain numbers.

Infinity is the quality of having no quantity is something I've said over and over, which is exactly how it's both defined and derived. This isn't "my own way", it is "the" way.

The word "infinite" simply means "endless". I have no idea what "the quality of having no quantity" means. "No quantity" sounds more like "zero" than "infinity". But "zero" is also considered a quantity. I am not really sure there's anyone beside you (other than Ecmandu) who defines infinity that way.
Last edited by Magnus Anderson on Tue Jan 07, 2020 1:32 am, edited 1 time in total.

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

Ecmandu wrote:So, Magnus , you’re saying that all infinities can be bound, you’re saying that infinity can be bound in general? You’re saying that the endless ends every time? (Contradiction)?

Is it a contradiction in terms to say that an infinite number of things exist within a limited portion of the universe? Depends on what you mean by "infinite". In general, when we say "an infinite number of things" we mean endless in terms of quantity i.e. we want to say that the number of things is endless. The fact that they are located within a limited portion of the universe (which means they have spatial ends) does not change the fact that their number is infinite (that they have no end when it comes to their quantity.) So there is no contradiction whatsoever. Similarly, saying that there is an infinite quantity of things in one place and an infinity quantity of things in another place is not contradictory. It's only contradictory if you're working with a different notion of the infinite. But such a notion is irrelevant.

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

If my calculations are right . . .

$$0.000\dotso1 = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \dots = (\frac{1}{10})^\infty = \frac{1}{10^\infty} = 10^{-\infty}$$

$$0.\dot9 + 10^{-\infty} = 1$$

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

Magnus Anderson wrote:If my calculations are right . . .

$$0.000\dotso1 = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \dots = (\frac{1}{10})^\infty = \frac{1}{10^\infty} = 10^{-\infty}$$

$$0.\dot9 + 10^{-\infty} = 1$$

How might this be applicable to, say, an apple?
Objectivists: Like shooting fish in a barrel!

He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles

Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382

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

It might not be.

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

Magnus Anderson wrote:It might not be.

Let alone to one of us?

Almost forgot:
Objectivists: Like shooting fish in a barrel!

He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles

Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382

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

Magnus Anderson wrote:
gib wrote:There would be an infinite number of 60-inch-segments, so $$\frac{\infty}{50}$$ 60-inch-segments would do, as would $$\frac{\infty}{70}$$ 60-inch-segments, or any other number in the denominator. Likewise, there would be $$\frac{\infty}{60}$$ 50-inch-segments, $$\frac{\infty}{60}$$ 70-inch-segments, and $$\frac{\infty}{60}$$ whatever-inch-segments.

Your argument amounts to "Because $$\frac{\infty}{50}$$, $$\frac{\infty}{60}$$ and $$\frac{\infty}{70}$$ are non-finite numbers, it follows they are necessarily equal". That's exactly the same as saying "Because $$2$$, $$4$$ and $$6$$ are finite numbers, it follows they are necessarily equal". It's the same exact logic. Beg the question much? You will, of course, deny it under the banner that what applies to infinite numbers does not apply to finite numbers. Damn straight. But what will you do to prove this claim? What have you done so far? As far as I can tell, nothing. It's a mere assertion.

...

Again, what you're saying is that the logic of finite sets does not apply to infinite sets, specifically, that subtraction operates on infinite quantities in a different way than it does so on finite sets. You didn't prove this, you didn't explain why, you merely asserted it. You want me to question my beliefs without providing any arguments against them.

Well, I've been doing a bit more than just asserting. I gave some counter-examples as a means of proof by contradiction:

gib wrote: I certainly would not have accepted that train B is longer because its carts are longer. Imagine two infinitely long sticks. They're both infinitely long. For all intents and purposes, equal. Now imagine chopping up one stick into 90 inch long segments and the other into 93.75 inch long segments (what an odd number to choose). Is the one with 93.75 inch long segments suddenly longer?

gib wrote:To say each resultant queue is half the size of the original queue is to say it's only half as long, or "shorter". I just don't know how to make sense out of that. For something to be "shorter" in length is to imply it has a beginning and an end. You'd have to imagine putting it next to something else and observing that it ends before the other thing ends. But we both agreed that the two resultant queues are still infinite. To me, that means I can't imagine putting those queues next to the original one and seeing that they're shorter. They would still seem to be the same length.

gib wrote:To say you're two steps close to something that's an infinite distance away is to imply the distance between you and that something has gotten shorter. But that implies there is an end to the distance, a point you are coming close to...

Not exactly proof that $$\frac{\infty}{50}$$ = $$\frac{\infty}{60}$$ = $$\frac{\infty}{70}$$, but definitely proof that $$\frac{\infty}{50}$$ > $$\frac{\infty}{60}$$ > $$\frac{\infty}{70}$$ can't be true.

I also tried to explain why you have no right to switch between contexts when talking about infinity:

gib wrote:I don't deny that there can be different orders of infinity (which is what I think James is getting at), but you don't get there just by adding 1 to infinity. It's more complicated than that. Take the example of the two parallel lines Ecmandu brought up. He says that since there is an infinite number of points in the first line, adding the second line, which also has an infinite number of points, doubles the number of points. Whether the arithmetic works like that or not (I don't think it does), that's not an example of a higher order of infinity. A higher order of infinity is more like a plane compared to a line--something you arrive at by compounding an infinity of infinities. The lines are only infinities of points, not infinities of infinities. But a plane is an infinity of infinities because it is an infinity of lines which in turn is an infinity of points.

The idea is like this: infinity, if you want to imagine it as something that you can somehow reach, is a "transcendent" object. To get there, you have to transcend all finite things (in the case of numbers, all numbers). It's impossible, just like transcending space and time is impossible for physical beings like us, trapped within space and time. No matter how high you count, you're no closer to infinity, just as no matter how far through space you travel, you're no closer to being outside space. But if you want to suppose you somehow could reach infinity, you can imagine skipping the journey of counting (or traveling through space) and magically arriving there. In that case, you must objectify infinity--meaning you must now think of it as an object--i.e. a finite thing--this is your new unit, your new building block, your new fundamental particle in a higher universe--it is your new point. It's like when you transcend all points in the line, you get the line itself. You can then treat the line as the new unit and start over adding lines together. Now counting consists of counting these lines, these infinities, and the new infinity to strive towards is the plane, the new transcendent object in this higher universe.

Going back to counting points, as in the case of counting up the points in the second line, is to go back below the first order infinity. You may think of it as going back to a different universe (i.e. a different line), but this is not the same as a different order of infinity. Adding the first infinity to the first point in the second infinity is not valid. It does not equal
∞ + 1. It's adding apples and oranges. The infinity and the point are not only completely incommensurate objects, but they, in a sense, don't even exist relative to each other (that's why infinity is "transcendent"--it is "beyond" the universe of points--the point relative to the infinity could be thought of as the infinitesimal). You can add up objects in a box, but you cannot add the box to the objects (for example, 2 apples + 3 apples = 5 apples; but what about 2 apples + the box the apples came in? What does that equal?). But you can add several boxes together. The arithmetic works only in the same universe, not across universes.

^ That, and I did argue earlier in this thread that infinity is not a number, and therefore arithmetic doesn't necessarily apply to it. I know you can take the symbol $$\infty$$ and plug into mathematical equations and do algebra with it, which is probably what you're getting at when you say you can do arithmetic with it, but $$\infty$$ in that case plays the role of a variable, an unknown, not an actual infinite quantity; you can put whatever you want in it's place and the same rules would apply: $$\alpha$$/60, $$\phi$$/60, /60. $$\infty$$ in this case doesn't mean "the highest number possible", it means "some undetermined quantity"... but it has to be a quantity, otherwise it doesn't apply. The only sense I agreed with you that you can do arithmetic on $$\infty$$ is if you treat $$\infty$$ as a unit (where the quantity would be 1... like 1cm <-- cm is the unit, 1 is the quantity). This is why I said you shouldn't switched contexts willy-nilly--you're switching from saying "1 of something symbolised by $$\infty$$" to "and infinity of something else (like train carts)". When you do that, the math suddenly doesn't work. $$\infty$$ in this case is not simply defined as "an infinite number of things", it's defined as "some undetermined thing" (like a variable). But if you want to say you can do arithmetic on infinity (as an infinite number of things), I deny your right to do so.

Gib wrote:And you're also claiming that the word "shorter" implies an end. Again, without anything to back it up.

That's pretty damn basic. I'd even say definitional... as in, I never thought you would have asked for proof given that that's essentially how we define "shorter". If Max is shorter than Harry, that means the tip of Max's head is below that of Harry's. If neither has a tip at the top of their head (because they're infinitely tall or whatever), what does it mean to say one shorter than the other?
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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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gib
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:Suppose that there's a half an apple in front of you.

Obligatory picture (can't embed, too big):
https://riversedgecurriculum.files.word ... 215396.jpg

Cool story, bro.

Look, this dumbing-down approach of "you're just denying yourself/lacking the language for it" is sadly wasted on me - I know all too well how it is to deal with people whose vocabulary is all too conservative and restricted.
In fact, I love thinking outside the box - unreasonably so. I love the opportunity to do so.
I also love the rigorous rationalisation of exactly how you get outside the box, and the appreciation of every tiny step with all the exact minutae involved to get there, including the illegitimate ones.

I know what you think you're explaining and exactly how you're getting there - but with the apparent lack of telepathy at my disposal, I don't seem to be able to get you to understand that I understand every single thing you're doing and saying, whilst also rejecting it.

You tell me:
how do I communicate the fact that I don't need apple analogies to understand what you're saying, whilst still rejecting it?
Tell me how, because you're just wasting time, keystrokes and computer bits for both of us.

By the way, if you had "infinite apples in front of you", you'd be an apple and there would be nothing but apple everywhere.
You can't do that "twice". Everywhere is already taken.
If you had an infinite line of apples extending out "in front of you", you'd not be able to see beyond a small number of them, you'd never be able to get to $$\frac{1}2$$ way along to even divide it into 2, nor would you ever be able to divide all them into any finite number of apples. There would be a finite limit on apples going upwards, downwards, left, right, even backwards in this situation - just not forwards: the same as the natural numbers. Thus the line of apples is distinguished from an entire universe of apples by virtue of its finite constraints up, down, left, right, back, and in all other ways other than forwards. So any "size" of such a line of apples would be dictated by the size of the finite constraints, not the infinity that stretched before you. The infinity has only 1 type (quality), and no tokens (quantities): I am being very very very specific about the kind of ends that are going on here - don't give me your shit about how I'm being broad.
You can imagine "apples in front of you" in any way you like, and the same principle applies - you cannot get to the "end" of that line and add any more.
But pay attention to the following:
You could even set up a line of apples directly underneath the first, and there would still be infinite apples. The fact that you started the line with 2 apples on top of each other doesn't give the infinite line "more" or "less" apples. You changed a finite constraint. The infinity is just as unforgivingly endless in its monolithic single extension either way. There was no end before, and there's still no end - quantity is just as meaningless either way.
You regard the change in the finite number of apples that start the infinity as a change in the infinity.
This is just wrong, I'm sorry.
It's the finitude that governs the size. Any infinity has only one way of being infinite, and it defies all possible quantification, or operation using quantities.

You can "imagine" what it would be like "hey, let's say it was the infinity was affected by changes to finite constraints" and go on from there, and you have your arguments.
Fine.
Do that.
It's not legitimate, but let's see what happens.

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

Magnus Anderson wrote:If my calculations are right . . .

$$0.000\dotso1 = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \dots = (\frac{1}{10})^\infty = \frac{1}{10^\infty} = 10^{-\infty}$$

$$0.\dot9 + 10^{-\infty} = 1$$

Hey, there you go. That works. Good job!
You have been observed.
obsrvr524

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

Silhouette wrote:By the way, if you had "infinite apples in front of you", you'd be an apple and there would be nothing but apple everywhere.

That's not true. You have to understand that I am not using the word "infinite" the way that you do. In fact, apart from a handful of people, noone is.

You can't do that "twice". Everywhere is already taken.

That would be true if what I meant by "infinite number of apples" is "there is nothing in space other than apples". But that's not what I meant. The word "infinite" simply means "endless". It does not mean "endless in every single way one can think of". You can have an endless number of things within a portion of the universe that is not endless in certain regards. Of course, in order to accomodate all of the infinitely many things, the portion itself must have enough room and that means it must have infinite room, so the portion must be infinite in at least one direction but there is no requirement to be infinite in all directions.

Again, the word "infinite" does not mean "the property of being the only type of thing that exists".

If you had an infinite line of apples extending out "in front of you", you'd not be able to see beyond a small number of them

That's irrelevant.

I'm afraid you're confusing conceptual matters with empirical ones. Here in this thread, we're discussing concepts. We're not talking about whether it's possible, nor how is it possible, to determine whether things that can be represented by our concepts exist. So if you're claiming that there's no infinite quantities in the universe, or that it's impossible to establish whether infinite quantities exist, you're wrong, to be sure, but most importantly, what you're saying is irrelevant.

you'd never be able to get to $$\frac{1}2$$ way along to even divide it into 2

You can divide the infinite line of apples into two smaller equally-sized infinite lines of apples by removing every second apple from it and placing it elsewhere.

nor would you ever be able to divide all them into any finite number of apples

Of course you can.

There would be a finite limit on apples going upwards, downwards, left, right, even backwards in this situation - just not forwards: the same as the natural numbers.

The set of natural numbers does not have such limits. That's because it's a set.

The infinite line of apples does because it's a physical object, not merely a set, and the fact that it's infinite in only one direction (and not all) is completely irrelevant to the subject at hand. This is because we're talking about THE QUANTITY OF APPLES, not about THE PHYSICAL LINE OF APPLES. The quantity of apples can only be infinite in ONE direction, and if it is infinite in this one direction, then it IS infinite.

Thus the line of apples is distinguished from an entire universe of apples by virtue of its finite constraints up, down, left, right, back, and in all other ways other than forwards. So any "size" of such a line of apples would be dictated by the size of the finite constraints, not the infinity that stretched before you.

If a hungry man walked past the infinite line of apples, grabbed one and ate it, that act alone would tell us that the infinite line of apples has one apple less than before.

You can imagine "apples in front of you" in any way you like, and the same principle applies - you cannot get to the "end" of that line and add any more.

You don't have to go the "end" of the line in order to add more apples. You can add them elsewhere (just like with finite sets.)

You're trying to apply to infinite quantities what applies only to finite quantities (which makes you guilty of your own accusations.)

This is just wrong, I'm sorry.

I feel sorry for you too.

It's the finitude that governs the size. Any infinity has only one way of being infinite, and it defies all possible quantification, or operation using quantities.

Yes, because you say so.

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

gib wrote:I gave some counter-examples as a means of proof by contradiction.

Imagine two infinitely long sticks. They're both infinitely long. For all intents and purposes, equal. Now imagine chopping up one stick into 90 inch long segments and the other into 93.75 inch long segments (what an odd number to choose). Is the one with 93.75 inch long segments suddenly longer?

An infinite number of inches is equal to a smaller infinite number of 90 inch long segments and an even smaller infinite number of 93.75 inch long segments. There is no contradiction. You're just doing to the math the wrong way.

To say each resultant queue is half the size of the original queue is to say it's only half as long, or "shorter". I just don't know how to make sense out of that. For something to be "shorter" in length is to imply it has a beginning and an end. You'd have to imagine putting it next to something else and observing that it ends before the other thing ends. But we both agreed that the two resultant queues are still infinite. To me, that means I can't imagine putting those queues next to the original one and seeing that they're shorter. They would still seem to be the same length.

The only reason you think that length implies a beginning and an end is because you're used to working with line segments.

The other problem is that you're confusing conceptual matters with empirical ones. The same mistake that Silhouette is making.

To say you're two steps close to something that's an infinite distance away is to imply the distance between you and that something has gotten shorter. But that implies there is an end to the distance, a point you are coming close to...

It does not.

Very convenient.

I did argue earlier in this thread that infinity is not a number, and therefore arithmetic doesn't necessarily apply to it.

Why is infinity not a number?

I know you can take the symbol $$\infty$$ and plug into mathematical equations and do algebra with it, which is probably what you're getting at when you say you can do arithmetic with it, but $$\infty$$ in that case plays the role of a variable, an unknown, not an actual infinite quantity; you can put whatever you want in it's place and the same rules would apply: $$\alpha$$/60, $$\phi$$/60, /60. $$\infty$$ in this case doesn't mean "the highest number possible", it means "some undetermined quantity"... but it has to be a quantity, otherwise it doesn't apply.

It's not an unknown. And the word "infinity" does not mean "the highest number possible".

The only sense I agreed with you that you can do arithmetic on $$\infty$$ is if you treat $$\infty$$ as a unit (where the quantity would be 1... like 1cm <-- cm is the unit, 1 is the quantity). This is why I said you shouldn't switched contexts willy-nilly--you're switching from saying "1 of something symbolised by $$\infty$$" to "and infinity of something else (like train carts)". When you do that, the math suddenly doesn't work. $$\infty$$ in this case is not simply defined as "an infinite number of things", it's defined as "some undetermined thing" (like a variable). But if you want to say you can do arithmetic on infinity (as an infinite number of things), I deny your right to do so.

Meters, centimeters, inches, feet, etc are not undetermined things. $$1m = 100cm$$ is not the same as $$1 \times x = 100 \times y$$ where $$x$$ and $$y$$ are unknowns. Units $$\neq$$ unknowns.

That's pretty damn basic. I'd even say definitional... as in, I never thought you would have asked for proof given that that's essentially how we define "shorter". If Max is shorter than Harry, that means the tip of Max's head is below that of Harry's. If neither has a tip at the top of their head (because they're infinitely tall or whatever), what does it mean to say one shorter than the other?

That's what it means with respect to line segments. And if we assume that the concept of length applies only to line segments, it does not follow that rays don't have something similar. Either way, you've got nothing.

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

Expanding upon:

gib wrote:I know you can take the symbol $$\infty$$ and plug into mathematical equations and do algebra with it, which is probably what you're getting at when you say you can do arithmetic with it, but $$\infty$$ in that case plays the role of a variable, an unknown, not an actual infinite quantity; you can put whatever you want in it's place and the same rules would apply: $$\alpha$$/60, $$\phi$$/60, /60. $$\infty$$ in this case doesn't mean "the highest number possible", it means "some undetermined quantity"... but it has to be a quantity, otherwise it doesn't apply.

The symbol $$\infty$$ represents an infinite quantity; believe it or not, even when I'm doing arithmetic with it. It certainly does not play the role of an unknown because the quantity is known -- it's not unknown. And yes, you can use any symbol you want (e.g. you can use $$\omega$$ if you're into hyperreal numbers or $$\text{infA}$$ if you prefer James's notation) but that does not mean we've somehow changed what we're representing. We're representing the same thing: infinite quantity.

The only sense I agreed with you that you can do arithmetic on $$\infty$$ is if you treat $$\infty$$ as a unit (where the quantity would be 1... like 1cm <-- cm is the unit, 1 is the quantity). This is why I said you shouldn't switched contexts willy-nilly--you're switching from saying "1 of something symbolised by $$\infty$$" to "and infinity of something else (like train carts)". When you do that, the math suddenly doesn't work.

Saying "1 of something symbolized by $$\infty$$" is exactly the same as saying "an infinite number" (such as an infinite number of train carts) because $$\infty$$ means "infinite number".

And the math doesn't stop working. Quite the contrary.

$$\infty$$ in this case is not simply defined as "an infinite number of things", it's defined as "some undetermined thing" (like a variable).

That's exactly what $$\infty$$ means. It means "an infinite number of things". The fact that I'm treating it like a unit DOES NOT change that fact.

When I use the symbol $$m$$ to represent one hundred centimeters, does it suddently stop representing one hundred centimeters? Of course not. THAT'S EXACTLY THE PURPOSE OF SYMBOLS.

$$m$$ is not an unknown and neither is $$\infty$$.

But I can see the spirit of Ecmandu has taken over . . .

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

obsrvr524 wrote:
Magnus Anderson wrote:If my calculations are right . . .

$$0.000\dotso1 = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \dots = (\frac{1}{10})^\infty = \frac{1}{10^\infty} = 10^{-\infty}$$

$$0.\dot9 + 10^{-\infty} = 1$$

Hey, there you go. That works. Good job!

Nah, unfortunately, that doesn’t work.

10-infinity is not a quantity that’s defined.

It’s undefined in this equation.

Think of it this way... walk up to anyone, even mathematicians and say, “dude! 10-infinity” or “dude! 10 minus the power of infinity!”

They’ll be like, “dude! what the fuck are you talking about?”

You’ll have to explain this to everyone because it makes no sense.
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:I forgot to mention that numbers have to avoid contradictions in terms, how silly of me.
"0.000" defines the start and "1" defines the end. With all ends accounted for, the "..." attempting to imply infinity is a contradiction in terms.
This is what happens when you try to define infinity by giving it ends: you make it finite (the opposite of what it is).
If there could be an infinity of something, e.g. 0 digits entirely enclosed by finite bounds, then the "finite" bounds would be indefinitely separated such that they'd never come to exist - just like the end of infinity never comes to exist by both definition and derivation.

$$0.000...1$$ is a numerical representation of an invalid number. Just goes to show that just because you can represent something numerically, doesn't mean it's a number. Fortunately my logic went in the other direction, that to be a number, you have to be able to represent it numerically. Let's avoid "affirming the consequent" here, eh?

So as I was saying, numbers are necessary in any valid format. The invalid $$0.000...1$$ therefore doesn't count.
I'd accept any valid numerical representation, decimal or otherwise.
$$\sum_{n=1}^\infty\frac9{10^n}$$ or $$0.\dot9$$ are fine because here the infinity isn't finite nor the finite infinity. The finites and the infinites are separate: the finites are defined in a very specific way such that the infinity can then say "now go do that defined thing an infinite number of times in the one way that infinity can be infinite".
$$\frac1\infty$$ is not fine because the infinity is standing in for a finite, or it is represented as a finite - they are not separated. Infinity alongside any numerical format of finites is like an operation, as Ecmandu said. It's an instruction to operate on defined finites in an ongoing way - it's never "the end" of having been operated on - because obviously "the end" i.e. finitude is the opposite of endlessness. $$2\times\infty$$ is invalid for the same reason.

Just be consistent! That's all you need. Math wouldn't be math if it wasn't consistent.

$$0.000\dotso1$$ is equivalent to $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$. Represent it using Pi notation and I believe you'd be fine with it.

Similarly, $$\infty$$ is equivalent to $$1 + 1 + 1 + \cdots$$. Represent it using Sigma notation and I believe you'd be fine with it.

So, both $$\infty$$ (or $$\text{infA}$$ or $$\omega$$) and $$0.000\dotso1$$ are "valid" numerical representations, in the sense that, Silhouette can now understand their meaning.

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

Ecmandu wrote:Nah, unfortunately, that doesn’t work.

10-infinity is not a quantity that’s defined.

It’s undefined in this equation.

Think of it this way... walk up to anyone, even mathematicians and say, “dude! 10-infinity” or “dude! 10 minus the power of infinity!”

They’ll be like, “dude! what the fuck are you talking about?”

You’ll have to explain this to everyone because it makes no sense.

Doesn't work because you don't understand it?

You can replace $$10^{-\infty}$$ with $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$. It's the same quantity expressed differently. And it should have been obvious. $$10^{-\infty}$$ is a compact representation and it's akin to a hyperreal number $$10^{-\omega}$$.

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

Magnus Anderson wrote:
Silhouette wrote:I forgot to mention that numbers have to avoid contradictions in terms, how silly of me.
"0.000" defines the start and "1" defines the end. With all ends accounted for, the "..." attempting to imply infinity is a contradiction in terms.
This is what happens when you try to define infinity by giving it ends: you make it finite (the opposite of what it is).
If there could be an infinity of something, e.g. 0 digits entirely enclosed by finite bounds, then the "finite" bounds would be indefinitely separated such that they'd never come to exist - just like the end of infinity never comes to exist by both definition and derivation.

$$0.000...1$$ is a numerical representation of an invalid number. Just goes to show that just because you can represent something numerically, doesn't mean it's a number. Fortunately my logic went in the other direction, that to be a number, you have to be able to represent it numerically. Let's avoid "affirming the consequent" here, eh?

So as I was saying, numbers are necessary in any valid format. The invalid $$0.000...1$$ therefore doesn't count.
I'd accept any valid numerical representation, decimal or otherwise.
$$\sum_{n=1}^\infty\frac9{10^n}$$ or $$0.\dot9$$ are fine because here the infinity isn't finite nor the finite infinity. The finites and the infinites are separate: the finites are defined in a very specific way such that the infinity can then say "now go do that defined thing an infinite number of times in the one way that infinity can be infinite".
$$\frac1\infty$$ is not fine because the infinity is standing in for a finite, or it is represented as a finite - they are not separated. Infinity alongside any numerical format of finites is like an operation, as Ecmandu said. It's an instruction to operate on defined finites in an ongoing way - it's never "the end" of having been operated on - because obviously "the end" i.e. finitude is the opposite of endlessness. $$2\times\infty$$ is invalid for the same reason.

Just be consistent! That's all you need. Math wouldn't be math if it wasn't consistent.

$$0.000\dotso1$$ is equivalent to $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$. Represent it using Pi notation and I believe you'd be fine with it.

Similarly, $$\infty$$ is equivalent to $$1 + 1 + 1 + \cdots$$. Represent it using Sigma notation and I believe you'd be fine with it.

So, both $$\infty$$ (or $$\text{infA}$$ or $$\omega$$) and $$0.000\dotso1$$ are "valid" numerical representations, in the sense that, Silhouette can now understand their meaning.

That’s why I gave you the quantum argument, infinity is actually x (x operator), x (x operator) etc...

It’s undefined.

1+1+1... is not undefined.

Now! Infinite sequences are different than infinity.

But when you substitute those “x’s”. It’s quantum.

To say that infinity is 1+1+1... is absurd.
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