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

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

Yes, 1 = 0.999...
13
42%
No, 1 ≠ 0.999...
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Other
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### Re: Is 1 = 0.999... ? Really?

You know my pitch here. Whether in regard to living matter evolving into apples, the definitive understanding of all things mathematical or human minds capable of discussing either one, we are all embedded in that same profoundly mysterious and problematic gap between what we think we know about anything and all that there is to know about everything.

Besides, the discussion revolves not around whether 6 X 7 = 43 -- who here would get into a fierce debate about that -- but 1 either equaling or not equalling 0.999....
The point is that you don't need to know everything in order to come to some reasonable and true conclusions.
Okay, but as long as the exchanges revolve around words talking about numbers pertaining only to more words still, it's hard for folks like me to grasp the relevance of the debate as it might be applicable to, say, technology and engineering. Again, it's actual use value and exchange value in human relationships.
If you want to engineer something, then you can't be flip flopping around.

Take a stand and see where it goes. That's the lesson to be taken away from this argument.
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### Re: Is 1 = 0.999... ? Really?

Phyllo,

Engineers only use pi to 6 decimal places.

That’s not what we’re talking about here.
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### Re: Is 1 = 0.999... ? Really?

I'm talking about how to solve problems without getting all fucked up.

Something like 1=0.999... is just one example.
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:The short-cut isn't necessarily wrong, multiplication by 10 in base 10 shifts the decimal point to the right. But $$123 = 123.0$$, so we aren't adding a zero, the zero was already there. That's not the case for $$0.\dot9$$ (if there are $$L$$ decimal places, there can't be any number in the $$L+1$$th place, because $$L+1$$ is undefined).

You're preeching to the choire.

Carleas wrote:I also suspect that we can prove the short-cut as a general theorem, since it's the case that for any number in base x, multiplying by x shifts the decimal point one place to the right. (Tangent: is there a base-agnostic word for the 'decimal' point?)

How 'bout the "base"?

Carleas wrote:I agree this isn't the construction we need to convince skeptics, but I'm not sure what construction the skeptics are using to define $$0.\dot9$$. Again, without that construction, I'm not sure that we all mean the same thing when we say $$0.\dot9$$.

Magnus likes to use $$\sum_{i=1}^{\infty}\frac{9}{10^i}$$.

Magnus Anderson wrote:Absolutely. Numbers aren't closed on addition? A number where addition is defined for only half the number line (all negative numbers, but no positive numbers)? An end to all decimal expansions, such that there is uncertainty about what happens to any infinite expansion when multiplied by 10? Does that make a whole class of rational numbers that aren't closed on multiplication? Is multiplication by $$L$$ defined?

My impression is that the existence of $$L$$ is ad hoc, unnecessary, and has a lot of unintended consequences. I don't believe it can be proven from other standard axioms, so we'd be adding it as an additional axiom, and it isn't clear why.

Yeah, I'm not sure why Magnus brought it up, nor did he care to explain what he meant. He did say it was a "special" kind of number. Not a natural, not an integer, not a rational, and not a standard real. He dropped the term "non-standard real" at one point, but he has yet to confirm this is what he meant. We've also been discussing hyperreals (infinitely large and infinitely small) and he might have this in mind, but if so, I'll dispute it as I see no reason to assume there is a largest hyperreal number (hell, they're already larger than infinity, what would stop them beyond that?).

iambiguous wrote:In the interim, mere mortals such as ourselves carry on as best we can. I'm just curious as to why those here who obviously do have an enormous amount of understanding with respect to math still can't pin this down conclusively.

I don't think it's a matter of being able to pin it down, I think it's a matter of unanimity. As far as I'm concerned, the matter was pinned down the minute someone posted this proof:

X = $$0.\dot9$$
10X = $$9.\dot9$$
10X = 9 + $$0.\dot9$$
10X = 9 + X
9X = 9
X = 1
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### Re: Is 1 = 0.999... ? Really?

gib wrote:I'm not sure what Magnus's objection to this step was, but I seem to recall something about a 0 appearing at the end of the string of 9s.

Carleas wrote:I don't know that Magnus has taken that position

I didn't.

Carleas wrote:There is precedent for this, in that division works for all numbers except $$0$$. We could similarly say that addition is undefined with respect to $$L$$. I suspect that changing the definition of addition in that way would have other repercussions, but I'm not sure. I think all operations would have to be undefined with respect to $$L$$.

$$L + 1$$ refers to a number larger than the largest number which is a contradiction in terms. On the other hand, $$L - 1$$ is perfectly fine (i.e. no contradiction.)

And there's no need to change the definition of addition.

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

"If two calculations both result in ∞, that doesn't mean that these outcomes are equal." ("Als uit twee berekeningen allebei oneindig komt, dat betekent niet dat die uitkomsten gelijk zijn.")
- The Elder Milikowski, VLIW architect and physicist

"Thus ∞ is not a number."
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### Re: Is 1 = 0.999... ? Really?

gib wrote:How 'bout the "base"?

"Base point"? I don't see that used anywhere. A bit of wikiing brought me to "radix point", which I've never heard before but seems to be what I'm looking for.

gib wrote:Magnus likes to use $$\sum_{i=1}^{\infty}\frac{9}{10^i}$$.

I don't think there's a non-question-begging way to proceed from there.

$$10 * \sum_{i=1}^{\infty}\frac{9}{10^i} = \sum_{i=1}^{\infty}\frac{90}{10^i} = \sum_{i=1}^{\infty}\frac{9}{10^{i-1}}$$, but I don't see that leading to anything that's not question-begging: if $$\infty$$ is finite (what?), then the latter two sums 'end' before the sum on the left side.

But what's the difference between $$\infty$$ and $$L$$?
$$\sum_{i=1}^{\infty}\frac{9}{10^i} \stackrel{?}{=} \sum_{i=1}^L\frac{9}{10^i}$$

Magnus Anderson wrote:$$L + 1$$ refers to a number larger than the largest number which is a contradiction in terms. On the other hand, $$L - 1$$ is perfectly fine (i.e. no contradiction.)

And there's no need to change the definition of addition.

What kind of number is $$L$$? It can't be a natural number, an integer, a rational number, or a real number without redefining every basic operation.

And everything just seems to behave strangely around $$L$$:
$$\frac{L}{2} + \frac{L}{2} = L$$
But multiplying both sides by 2, as we can usually do with an equation, yields
$$L + L = 2L$$
A contradiction by definition. But this is hard to escape: if we can put $$L$$ into any equation, and if we can use any standard operator on it, ever, you get weird things. Like
$$L - 1 + 1 = L + (-1) + 1 = L + 1 - 1$$
That should be permitted, and if it's not then addition involving $$L$$ doesn't mean what addition usually means.
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### Re: Is 1 = 0.999... ? Really?

gib wrote:I don't think it's a matter of being able to pin it down, I think it's a matter of unanimity. As far as I'm concerned, the matter was pinned down the minute someone posted this proof:

X = $$0.\dot9$$
10X = $$9.\dot9$$
10X = 9 + $$0.\dot9$$
10X = 9 + X
9X = 9
X = 1

phyllo wrote:If you want to engineer something, then you can't be flip flopping around.

Take a stand and see where it goes. That's the lesson to be taken away from this argument.

But, as someone not able to follow the mathemtics here with any real sophistication, how important, in regard to creating technology and accomplishing engineering feats, is it to get this right?

That's what I can't wrap my head around. Is the above in some respect the equivalent of this: "You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on."

Intellectually, that seems to make sense. But if point A is my front door and point B is the mail box, I make it every time. And then back again.

So I'm wondering if, "for all practical purposes", 1 = .999...has any real meaning in our lives. Such that, for example, if someone assumed they were not equal, and this turned out to be true, it would actually make their life different in some way.

But, here again, I am more than willing to concede this revolves entirely around my ignorance of the math. Analogous, perhaps, in a larger sense, to someone who does understand Einstein's space-time continuum wondering how his or her life might be different if Einstein turned out to be wrong.
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

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

Carleas wrote:What kind of number is $$L$$?

What kind of question is that? What exactly are you asking?

It can't be a natural number, an integer, a rational number, or a real number without redefining every basic operation.

It's neither of those.

And everything just seems to behave strangely around $$L$$:
$$\frac{L}{2} + \frac{L}{2} = L$$
But multiplying both sides by 2, as we can usually do with an equation, yields
$$L + L = 2L$$

Well, you multiplied $$L$$ by $$2$$. Not sure what you expected.

But this is hard to escape: if we can put $$L$$ into any equation, and if we can use any standard operator on it, ever, you get weird things. Like
$$L - 1 + 1 = L + (-1) + 1 = L + 1 - 1$$
That should be permitted, and if it's not then addition involving $$L$$ doesn't mean what addition usually means.

$$L + 1 - 1$$ is fine.

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

But, as someone not able to follow the mathemtics here with any real sophistication, how important, in regard to creating technology and accomplishing engineering feats, is it to get this right?
The "result" isn't useful in itself. It's a curiosity and an exercise in problem solving and reasoning.
That's what I can't wrap my head around. Is the above in some respect the equivalent of this: "You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on."

Intellectually, that seems to make sense. But if point A is my front door and point B is the mail box, I make it every time. And then back again.
That particular problem produces a couple of interesting conclusions about mathematics and the universe.

Either infinite series converge - which is useful to know for solving other problems (and it would lay to rest some of the arguments in this thread). Or the universe is not infinitely divisible - at some point trying to go "half the distance" produces a quantum jump to the end point.
So I'm wondering if, "for all practical purposes", 1 = .999...has any real meaning in our lives. Such that, for example, if someone assumed they were not equal, and this turned out to be true, it would actually make their life different in some way.
In itself, it means nothing in our lives.
But, here again, I am more than willing to concede this revolves entirely around my ignorance of the math. Analogous, perhaps, in a larger sense, to someone who does understand Einstein's space-time continuum wondering how his or her life might be different if Einstein turned out to be wrong.
Science turns out to be wrong all the time.

Life goes on in spite of that. But there are repercussions in wasted effort, money and lives.

For example, useless medical procedures and treatments that either do nothing or cause harm. Money wasted on medical research that is going in the wrong direction.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Carleas wrote:What kind of number is $$L$$?

What kind of question is that? What exactly are you asking?

He's asking exactly the same question I was asking, which you refuse to answer. You yourself tell us: L is not a nature, it's not an integer, it's not a rational, and it's not a standard real. Yet you seem to be obstinately illusive when it comes to explanation what kind of number L is.

Magnus Anderson wrote:
And everything just seems to behave strangely around $$L$$:
$$\frac{L}{2} + \frac{L}{2} = L$$
But multiplying both sides by 2, as we can usually do with an equation, yields
$$L + L = 2L$$

Well, you multiplied $$L$$ by $$2$$. Not sure what you expected.

Can't speak for Carleas, but I would expect that multiplying L by 2 is impossible. If you can't even add 1 to it, how can you multiply it by 2. Or is multiplying it by 2 possible despite not being able to add 1. <-- Now that would be bizarre indeed!

I think Carleas's point is that there should be nothing wrong with dividing L by 2. And there should be nothing wrong with adding half L to half L. Combining those allows you to do: $$\frac{L}{2} + \frac{L}{2} = L$$.

Then there should be nothing wrong with multiplying half L by 2. Thus you should be able to do it to both fractions in the addition, which gives you: L + L.

But then you get a result which, by definition, you shouldn't get: L + L = 2L.

Think of it this way: I have L cats and L dogs--that's possible, right?--but then I have 2L pets.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:What kind of question is that? What exactly are you asking?

There are numbers that function something like the way you're describing $$L$$, but you're avoiding identifying $$L$$ as any of those. It clearly doesn't function like other numbers (as you've acknowledged), but you're really only providing an accounting of how it does function when it's function allegedly prevents what we're trying to do.

You're claiming that a thing exists which makes an equality false, and there are two responses to that: 1) the thing doesn't exist, or 2) the thing doesn't make the equality false. To explore either, we need a rigorous definition of the thing you're positing.

Magnus Anderson wrote:$$L + 1 - 1$$ is fine.

How can that be? L+1 is undefined! You can't subtract 1 from an undefined quantity and get a defined quantity.
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
But, as someone not able to follow the mathemtics here with any real sophistication, how important, in regard to creating technology and accomplishing engineering feats, is it to get this right?
The "result" isn't useful in itself. It's a curiosity and an exercise in problem solving and reasoning.

Okay, if everyone sophisticated in math here can agree, it's good to know.

That's what I can't wrap my head around. Is the above in some respect the equivalent of this: "You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on."

Intellectually, that seems to make sense. But if point A is my front door and point B is the mail box, I make it every time. And then back again.

phyllo wrote: That particular problem produces a couple of interesting conclusions about mathematics and the universe.

And the part "the human condition" plays in that universe?

phyllo wrote: Either infinite series converge - which is useful to know for solving other problems (and it would lay to rest some of the arguments in this thread). Or the universe is not infinitely divisible - at some point trying to go "half the distance" produces a quantum jump to the end point.

It's the extent to which these "solved problems" ramify on human interactions at the juncture of science, philosophy and theology, that most interest me. Given the distinctions that I make between the either/or and the is/ought worlds.

So I'm wondering if, "for all practical purposes", 1 = .999...has any real meaning in our lives. Such that, for example, if someone assumed they were not equal, and this turned out to be true, it would actually make their life different in some way.

phyllo wrote:In itself, it means nothing in our lives.

Given the extent that you or any one of us here can possibly know this.

But, here again, I am more than willing to concede this revolves entirely around my ignorance of the math. Analogous, perhaps, in a larger sense, to someone who does understand Einstein's space-time continuum wondering how his or her life might be different if Einstein turned out to be wrong.

phyllo wrote:Science turns out to be wrong all the time.

Meaning, of course, there is a path that one can take to be right. And, in regard to the either/or world, it always intrigues me when confronted with seeming antinomies...from the existence of something instead of nothing, to quandaries embedded in the determinism/free will debate, to the evolution of lifeless matter into living matter into self-conscious matter, to all of the fascinating speculations about sim worlds and dream worlds and solipsism and a matrix reality.

And, in a way that both intrigues and baffles me, this.
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

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

Thinking more about $$L$$, I have some additional objections.

The proof that there are infinite primes goes like this: let's ask if $$p_n$$, the $$n$$th prime, is the largest prime. In commonly accepted math, we can see that it can't be the largest because $$(p_1 * p_2 * p_3 * ... * p_n)+1$$ must be divisible by a prime number not included in the first $$n$$ prime numbers, and thus greater than $$p_n$$.

What's problematic about this in your system is that the response to this proof is that at some point, $$\prod_{i=1}^{n}p_n > L$$, i.e. the product of the first $$n$$ primes is greater than the greatest number. What's strange about that is that this is a normal multiplication, it's not adding something to $$L$$ in a way that's clearly undefined, and yet we're saying it's running into the ceiling. But if $$L$$ isn't specified, how can we know if some product is too big?

It seems that $$L$$ must be unreachable by operations on the real numbers, since any operation multiplication or addition on the real numbers produces a real number, and L is greater than all real numbers. And consequently, no operation on $$L$$ can produce a real number, because then reversing the operation would produce $$L$$ (so if $$L - 1 = r, r\in\mathbb{R}$$ then $$r+1 = L$$.

I'd say that, in that case, both $$0.\dot9$$ and $$9.\dot9$$ have the same number of decimal places, even if $$L$$ exists: they both have $$L$$ nines.
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### Re: Is 1 = 0.999... ? Really?

Consider: at the point of lack of differentiabiliry of the hypothetical -mathematical absolute -proximal , the functional difference is no longer either .9999999999999-99999=1 OR .9999999999999-9999 not = 1, since it is a non function.
There principles of mathematics demand that both =1 and (-=1) to be functional equivalents.

That is the whole point of Leibnitz' supposition.
If we further enquire by the validity of the principle, (-&+), we fall into the same mathematical trap.
So I would vote not either yes, or no on this .
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### Re: Is 1 = 0.999... ? Really?

Note: I think voting has been timely expired. This type of census should temporally be left open.
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### Re: Is 1 = 0.999... ? Really?

How can that be? L+1 is undefined! You can't subtract 1 from an undefined quantity and get a defined quantity.

I suppose it has something to do with the fact that you can subtract $$1$$ from $$1$$ to get $$0$$?

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

gib wrote:Can't speak for Carleas, but I would expect that multiplying L by 2 is impossible. If you can't even add 1 to it, how can you multiply it by 2. Or is multiplying it by 2 possible despite not being able to add 1. <-- Now that would be bizarre indeed!

Not sure what you mean when you say that you expect that "Multiplying $$L$$ by $$2$$ is impossible".

The point is that it's a logical contradiction to speak of a number that is twice the size the largest number. It's not impossible to speak of such a number (indeed, it's not impossible to contradict yourself) but it's a logical contradiction to do so.

I can say that Socrates is a man and then later on say that he's a woman. Nothing impossible about that. But it's a logical contradiction to do so. Either Socrates is a man or he's a woman. You can't have it both ways. Either there is no number larger than $$L$$ or there is a number larger than $$L$$. You can't have it both ways.

$$2L$$ is a reference to a number larger than the largest number. Nothing impossible about saying such a thing. But it's a logical contradiction to do so.

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

Carleas wrote:I'd say that, in that case, both $$0.\dot9$$ and $$9.\dot9$$ have the same number of decimal places, even if $$L$$ exists: they both have $$L$$ nines.

It is not my argument that $$9 + 0.\dot9 \neq 9.\dot9$$. Rather, my argument is that $$10 \times 0.\dot9 \neq 9.\dot9$$.

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

Magnus Anderson wrote:I suppose it has something to do with the fact that you can subtract $$1$$ from $$1$$ to get $$0$$?

Right, so $$L + (1-1) \neq (L+1)-1$$. That means $$L$$ isn't a real number, because addition and multiplication aren't associative on it.

So, $$L$$ is some special type of number, it doesn't follow normal rules of arithmetic. So it's possible that $$L-1 = L$$, we can't really say because we don't really have a definition of what $$L$$ is.

Magnus Anderson wrote:The point is that it's a logical contradiction to speak of a number that is twice the size the largest number. It's not impossible to speak of such a number (indeed, it's not impossible to contradict yourself) but it's a logical contradiction to do so.

I think we could say the same thing about the idea of the largest number itself: it's not impossible to speak of such a number, but it contradicts a significant part of standard math.

Magnus Anderson wrote:It is not my argument that $$9 + 0.\dot9 \neq 9.\dot9$$. Rather, my argument is that $$10 \times 0.\dot9 \neq 9.\dot9$$.

Let's say that there are $$L$$ $$9$$s following the decimal point in $$0.\dot9$$. How many digits are there in $$10 \times 0.\dot9$$ ? $$L+1$$? If there are $$L$$, are there only $$L-1$$ decimal places? What's in the $$L$$th decimal place?

What if we go the other way:
$$\frac{9 + 0.\dot9}{10} \stackrel{?}{=} .\dot9$$
$$\frac{9.\dot9}{10} \stackrel{?}{=} 9.\dot9 - 9$$
If so, it seems we can proceed and conclude $$0.\dot9 = 1$$. If not, why not? One of the $$9$$s gets pushed of the end, but we don't have the $$L+1$$ problem.
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Carleas
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:
Magnus Anderson wrote:It is not my argument that $$9 + 0.\dot9 \neq 9.\dot9$$. Rather, my argument is that $$10 \times 0.\dot9 \neq 9.\dot9$$.

A further thought on this: I think this breaks the associative property of multiplication for all numbers, not just $$L$$. Because
$$(0.\dot9\ * 10) * \frac{1}{10} \neq 0.\dot9\ * (10 * \frac{1}{10})$$
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### Re: Is 1 = 0.999... ? Really?

Magnus, Carleas...

I’m glad Carleas can speak in your latex language, because you’ll be more likely to understand it.

I already gave you my disproof Magnus, for “completed infinities” (I am the originator of this disproof and you ignored it)

If you take any real number... (I’ll use the number 1)

If you take the number 1 and divide it by 1/2, it equals 1/2+1/2!

If you continue the sequence ... 1/2+1/2=1

If you continue the sequence 1/2+1/2=1/4+1/4+1/4+1/4

If you continue this sequence to convergence!!!

1=0+0+0....

1=0

You can do this for ANY real number if completed infinities exist.

Completed infinities don’t exist.
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:So it's possible that $$L-1 = L$$, we can't really say because we don't really have a definition of what $$L$$ is.

Actually, $$L - 1 < L$$. And we do have a definition of $$L$$. It's a number larger than every other number.

I think we could say the same thing about the idea of the largest number itself: it's not impossible to speak of such a number, but it contradicts a significant part of standard math.

Except that "A number larger than every other number" is not a contradiction in terms.

Let's say that there are $$L$$ $$9$$s following the decimal point in $$0.\dot9$$. How many digits are there in $$10 \times 0.\dot9$$ ? $$L+1$$? If there are $$L$$, are there only $$L-1$$ decimal places? What's in the $$L$$th decimal place?

Let's say that the number of $$9$$s in $$0.\dot9$$ is some infinite number $$a$$. The number of $$9$$s in $$10 \times 0.\dot9$$ would be $$a - 1$$.

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

Magnus Anderson wrote:
Carleas wrote:So it's possible that $$L-1 = L$$, we can't really say because we don't really have a definition of what $$L$$ is.

Actually, $$L - 1 < L$$. And we do have a definition of $$L$$. It's a number larger than every other number.

I think we could say the same thing about the idea of the largest number itself: it's not impossible to speak of such a number, but it contradicts a significant part of standard math.

Except that "A number larger than every other number" is not a contradiction in terms.

Let's say that there are $$L$$ $$9$$s following the decimal point in $$0.\dot9$$. How many digits are there in $$10 \times 0.\dot9$$ ? $$L+1$$? If there are $$L$$, are there only $$L-1$$ decimal places? What's in the $$L$$th decimal place?

Let's say that the number of $$9$$s in $$0.\dot9$$ is some infinite number $$a$$. The number of $$9$$s in $$10 \times 0.\dot9$$ would be $$a - 1$$.

Carleas was trying to show you through proof contradiction that infinity cannot act upon real numbers. He was not saying that L-1 was less that L.

He was showing you the absurdity of adding operators to infinity.

Anyways.... answer my last post. You believe orders of infinity exist because of convergence (completed infinities) take down that argument !!

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

Magnus Anderson wrote:"A number larger than every other number" is not a contradiction in terms.

That phrase is pretty vague, but the way you are using it, it is a contradiction of much of standard math, e.g. the associative property of multiplication on the real numbers, the property that the set of real numbers is closed under addition and multiplication, etc.

I've given you a lot of examples of how it screws with standard math. You haven't provided any response to those. Can you?

Magnus Anderson wrote:some infinite number $$a$$

What operations work on the "infinite number[s]"? What are the properties of those operations? Can you add real and infinite numbers? Is the sum or difference always an infinite number? Can you add real numbers together and eventually get an infinite number? What does it mean for an infinite number to be larger than another infinite number? Is L larger than all infinite numbers?
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Carleas
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