## 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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33%
No, 1 ≠ 0.999...
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53%
Other
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13%

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

obsrvr524 wrote:Do they have proof that 0 * ∞ = 0?

I'm not so sure that it does. How could that be proven?

And if it doesn't, they can only say that a/0 is undefined, not undefinable.

Magnus Anderson wrote:Infinity + 1 either equals infinity or it does not.

"Infinity" doesn't exist. An infinite thing can exist (and does I'm sure) but infinity is defined as the point where parallel lines meet. Since parallel lines do not meet, infinity cannot exist.

James' infA exists as a specific infinite series. That is not the same as infinity.

"They" have a proof that if $$0\times{n}=0$$, "n" can be any number you want since all numbers multiplied by zero equal zero. Therefore dividing both sides by 0 to get $$\frac0{0}$$ it's not $$\infty$$ that this equals, but "undefined".
Except "they" is you too. Maths is objective so it's your proof just as much as it is anyone else's.

And as I just said to Magnus, this doesn't mean temporarily undefined. It means it'll always be undefinable.

Indeed infinity does not exist. InfA doesn't either, because it contains within it the "..." that implies "infinitely more times", which as we've just learned is an undefined number of times, which could only lead to an undefined result if it could ever reach any result at all. It has a divergent limit that cannot be definitely specified one way or another - and calling it "infA" doesn't solve this.

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

Silhouette wrote:
Magnus Anderson wrote:I make no such mistake. In fact, it is people who claim that $$0.\dot9 = 1$$ that make that mistake over and over again.

Just repeat your debunked claim. Provide no explanation.
I've been saying all along that infinity having no destination is exactly why $$0.\dot9 = 1$$

I think that Magnus is exactly right about that and that your claim is exactly wrong.
You have been observed.
obsrvr524

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

obsrvr524 wrote:
Silhouette wrote:
Magnus Anderson wrote:I make no such mistake. In fact, it is people who claim that $$0.\dot9 = 1$$ that make that mistake over and over again.

Just repeat your debunked claim. Provide no explanation.
I've been saying all along that infinity having no destination is exactly why $$0.\dot9 = 1$$

I think that Magnus is exactly right about that and that your claim is exactly wrong.

After we debunk orders of infinity, I’m coming back after silhouette.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:
Silhouette wrote:
Magnus Anderson wrote:I make no such mistake. In fact, it is people who claim that $$0.\dot9 = 1$$ that make that mistake over and over again.

Just repeat your debunked claim. Provide no explanation.
I've been saying all along that infinity having no destination is exactly why $$0.\dot9 = 1$$

I think that Magnus is exactly right about that and that your claim is exactly wrong.

That's nice.

I mean, I just disproved it (again) in the post right at the end of the previous page:

Silhouette wrote:$$0.\dot0{1}$$ never gets to $$0$$ also means no gap can ever come into existence between $$0.\dot0{1}$$ and $$0$$.

The undefinability of any "gap" means there's no grounds to distinguish $$0.\dot9$$ from $$1$$, nor $$0.\dot0{1}$$ from $$0$$

Saying there's always a smaller gap, therefore there always is "a" gap that never fully vanishes, is no different from saying there's never any point at which the gap can be said to come into existence.
"Undefined" $$\to$$ "a gap exists" is erroneous. But it is valid to say that any such gap can never be defined therefore it's undefinable and cannot be said to exist at all.
Any reversal of something like "it can't be said to not exist either" does not mean "It can be said to exist" - this would commit the formal fallacy of "affirming a disjunct".

Fortunately for me $$0.\dot0{1}$$ isn't a valid quantity in the first place as it's bounding either end of an endlessness, which is a contradiction. So it's unsound to use it in any of the further reasoning that you'd need to define any gap never vanishing.
Combine that with "A gap can be said to exist" being a formal fallacy and bingo: "no gap can be said to ever exist" is valid and the equality cannot be disproven.

But as long as you just simply "think" that's exactly wrong, does that make you exactly right?

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

Where you're misleading is in allowing this subjective display positionings of the blues and the yellow to affect the objective position of the purples, which do not overlap or have anything to do with the operation of summing the blues to get the yellow. Likewise for the reds, you could subjectively display them anywhere you like and they'd still objectively sum to the purples. It's only subjectively convenient to put them next to the blues so the visual grouping of the greens is clearer, but even if you displayed them anywhere else that you liked: the first element of the greens would still be the first element of the blues, the second element of the greens would still be the first element of the reds and so on.

If you're saying that my calculations are influenced by my image, you're wrong. The image is nothing but a reflection of my calculations. It's supposed to show how I'm doing my calculations (and it is implied that the way I'm doing them is the way they should be done.)

The first element of the purple rectangle is $$0.9$$ and it's constructed from the second element of the green rectangle which is $$0.09$$. It is not constructed from the first element of the green rectangle. That's why the purple rectangle does not start at the same place as the green rectangle.

There is no "product itself" of a divergent infinite product - you never get there.

There is. The fact that the product never ends does not mean it does not exist. (Indeed, I can use Gib's argument against you: infinite sums have no temporal dimension, so they can be considered complete.)

What's true is that there may be no finite number equal to an infinite product. In the case of $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots$$, there is no finite number equal to it. It has a limit, which is $$0$$, but that's not the same thing as its result (its result being greater than its limit.)

The only thing you can do is identify some limit that it's tending towards because of the very fact that it never gets there.

Not really.

$$1+\infty$$ is almost literally "more boundless" (by the finite quantity of "one").
Something can't be more boundless than boundless, so it's boundless whatever finite thing you do to it along its boundlessness.

Quantities are either finite or they are infinite. There are no degrees. That's where we agree. Where we disagree is that $$\infty + 1$$ means "more infinite". It does not. What it means is "larger infinite quantity".

That's why adding one results in boundlessness both before and after you do it

The truth value of the statement $$\infty + 1 = \infty$$ depends on its meaning.

If what we mean is "Some infinite quantity X + 1 = Some infinite quantity that is not necessarily equal to X" then the statement is true.

If what we mean is "Some infinite quantity X + 1 = The same infinite quantity X" then it is false.

It's literally impossible to test whether you added, took away or did whatever bounded thing to a boundless length after you've done it and are therefore able to make an equation about it. You can only validly comment on what you did with the finite quantity of 1 apple before it was absorbed into the boundless non-finite mass, you cannot validly comment on that resulting infinity that stays infinite in the same and only way that infinity can be infinite. So with no change in the result, there is therefore no valid equation or statement to make about the result as changed.

It seems to me that like Gib you're not able to distinguish between conceptual and empirical matters.

Gib keeps asking questions such as "How can we empirically determine (specifically, through direct observation) whether any two infinitely long physical objects are equal in length or not?"

The question is irrelevant and it is so precisely because it's empirical and not conceptual.

We're talking about concepts here, and to concepts we should stick. A conceptual matter cannot be resolved empirically.

Your error is to say that since 1=0 is a contradiction, we ought to be able to treat infinites alongside finites as though they were compatible.

That's not what I'm saying.

The contradiction that you speak of arises as a result of not understanding the implications of the concept of infinity that is being employed.

I've previously said that $$\infty + 1 = \infty$$ is true. There is no doubt about it. But that's only the case if the symbol $$\infty$$ means "some infinite quantity, not necessarily the same as the one represented by the same symbol elsewhere". In such a case, you can't subtract $$\infty$$ from both sides and get $$1 = 0$$. This is because $$\infty - \infty$$ does not equal $$0$$ given that what that expression means is "Take some infinity quantity and subtract from it some other infinite quantity that is not necessarily equal to it". If the two symbols do not necessarily represent one and the same quantity, then the difference between them is not necessarily $$0$$.

But that's PRECISELY what mathematicians do when they try to prove that $$0.\dot9 = 1$$. There is literally no difference between people proving that $$0 = 1$$ by subtracting $$\infty$$ from both sides of the obviously true equation $$\infty + 1 = \infty$$ and various Wikipedia proofs that $$0.\dot9 = 1$$ except that it's much easier to see that the former conclusion is nonsense and that the proof must be invalid.

Sure, if saying what something "isn't" gives it a meaning about what it "is".
You don't seem to understand the extent to which "ends" apply when it comes to definition.
Full definitions that include what something "is" as well as what it "isn't" are separating what it "is" from what it "isn't" by a bound (i.e. an "end") that's as clear as possible.

To define some symbol S is to verbally (or non-verbally) describe the meaning of that symbol S. The usual aim is to communicate to others what meaning is assigned to the symbol.

In other words, the meaning of a symbol precedes its verbal (or non-verbal) representation (which is what definitions are.) They are superficial things, very much in the Freudian "tip of the iceberg" sense.

The meaning of a word does not have to be described in order for it to have a meaning. This means the word "infinity" is a meaningful word so as long there is some kind of meaning assigned to it regardless of what kind and how many descriptions of its meaning exist.

You can describe the meaning of words any way you want, so as long your descriptions do the job they are supposed to do (which is to help others understand what your words mean.) If you can define words using "is not" statements and get the job done, then your definitions are good.

The reason that expressions that include e.g. division by zero are undefined is because you can't clearly bound what it "is" from what it "isn't", since in this case any answer is no more valid than any other:

$$\frac{1}{0}$$ has no valid answer (not a single one) since there is no number that gives you $$1$$ when you multiply it by $$0$$.

As above, limits are at the core of infinite series.

Infinite series and limits is what they taught you at school (and thus what you're familiar with) but they are not relevant to the subject at hand.

See how you're only going for one side of the "undefined" and concluding that the other side therefore doesn't exist?
$$0.\dot01$$ never gets to $$0$$ also means no gap can ever come into existence between $$0.\dot01$$ and $$0$$.

That's not true. It simply means the gap cannot be represented using one of those numbers you can understand.

$$0.\dot01$$ must attain $$0$$ in order for there to be no gap.

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

Magnus,

Your graph is nice, but it’s actually not different from simply stating 0.333... * 3 = 0.999...

I want to stay on orders of infinity with you before really laying into silhouette
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:The undefinability of any "gap" means there's no grounds to distinguish $$0.\dot9$$ from $$1$$, nor $$0.\dot0{1}$$ from $$0$$

I suppose what you want to say is: no gap between numbers = no difference between numbers = equal numbers. If that's what you're trying to say, then I agree.

That's not the point of our disagreement. What we're disputing is your claim that there is no gap between $$0.\dot01$$ and $$0$$.

Saying there's always a smaller gap, therefore there always is "a" gap that never fully vanishes, is no different from saying there's never any point at which the gap can be said to come into existence.

This is the problematic statement. It's a non-sequitir.

A non-existent gap would be a gap that has zero size. The size of an infinitely small gap is greater than zero -- by definition. No partial product of $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$ is equal to $$0$$. Note that this is different from $$0 \times 0 \times 0 \times \cdots$$ where every partial product is equal to $$0$$. Both are infinite, never-ending, products but one evaluates to a finite number that is $$0$$ (is equal to it) whereas the other doesn't evaluate to any finite number and is provably greater than $$0$$.

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

Silhouette wrote:
obsrvr524 wrote:I think that Magnus is exactly right about that and that your claim is exactly wrong.

That's nice.

I mean, I just disproved it (again) in the post right at the end of the previous page:

Silhouette wrote:$$0.\dot0{1}$$ never gets to $$0$$ also means no gap can ever come into existence between $$0.\dot0{1}$$ and $$0$$.

The undefinability of any "gap" means there's no grounds to distinguish $$0.\dot9$$ from $$1$$, nor $$0.\dot0{1}$$ from $$0$$

Neither of those claims make any sense to me. So you haven't proven anything to me. Why would your first claim be true?

Silhouette wrote:Saying there's always a smaller gap, therefore there always is "a" gap that never fully vanishes, is no different from saying there's never any point at which the gap can be said to come into existence.

That one doesn't make any sense either.
You have been observed.
obsrvr524

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

Magnus Anderson wrote:
Silhouette wrote:The undefinability of any "gap" means there's no grounds to distinguish $$0.\dot9$$ from $$1$$, nor $$0.\dot0{1}$$ from $$0$$

I suppose what you want to say is: no gap between numbers = no difference between numbers = equal numbers. If that's what you're trying to say, then I agree.

That's not the point of our disagreement. What we're disputing is your claim that there is no gap between $$0.\dot01$$ and $$0$$.

Saying there's always a smaller gap, therefore there always is "a" gap that never fully vanishes, is no different from saying there's never any point at which the gap can be said to come into existence.

This is the problematic statement. It's a non-sequitir.

A non-existent gap would be a gap that has zero size. The size of an infinitely small gap is greater than zero -- by definition. No partial product of $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$ is equal to $$0$$. Note that this is different from $$0 \times 0 \times 0 \times \cdots$$ where every partial product is equal to $$0$$. Both are infinite, never-ending, products but one evaluates to a finite number that is $$0$$ (is equal to it) whereas the other doesn't evaluate to any finite number and is provably greater than $$0$$.

I’ll argue Silhouette for silhouette!

The reason Silhouette sees no difference between .000...1 and zero is because the 1 in 0.000...1 is never arrived at. It’s ALWAYS zero!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:"They" have a proof that if $$0\times{n}=0$$, "n" can be any number you want since all numbers multiplied by zero equal zero. Therefore dividing both sides by 0 to get $$\frac0{0}$$ it's not $$\infty$$ that this equals, but "undefined".

Except that ∞, by definition is NOT a number ("n").

So the "proof" using n/o is invalid.

Ecmandu wrote:The reason Silhouette sees no difference between .000...1 and zero is because the 1 in 0.000...1 is never arrived at. It’s ALWAYS zero!

So you are saying that you never get to the end??

And that would mean that there is always a "9", never a "0" in 0.999..., wouldn't it.
You have been observed.
obsrvr524

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

obsrvr524 wrote:
Silhouette wrote:"They" have a proof that if $$0\times{n}=0$$, "n" can be any number you want since all numbers multiplied by zero equal zero. Therefore dividing both sides by 0 to get $$\frac0{0}$$ it's not $$\infty$$ that this equals, but "undefined".

Except that ∞, by definition is NOT a number ("n").

So the "proof" using n/o is invalid.

Ecmandu wrote:The reason Silhouette sees no difference between .000...1 and zero is because the 1 in 0.000...1 is never arrived at. It’s ALWAYS zero!

So you are saying that you never get to the end??

And that would mean that there is always a "9", never a "0" in 0.999..., wouldn't it.

Absolutely. But that’s not my focus right now, I’m debating Magnus on orders of infinity, that’s my focus.

I’ll get to Silhouette later.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:
obsrvr524 wrote:And that would mean that there is always a "9", never a "0" in 0.999..., wouldn't it.

Absolutely. But that’s not my focus right now, I’m debating Magnus on orders of infinity, that’s my focus.

So if it never has a zero at the end, because it doesn't have an end, then it cannot be equal to 1.000..., which has zero throughout.
You have been observed.
obsrvr524

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

obsrvr524 wrote:
Ecmandu wrote:
obsrvr524 wrote:And that would mean that there is always a "9", never a "0" in 0.999..., wouldn't it.

Absolutely. But that’s not my focus right now, I’m debating Magnus on orders of infinity, that’s my focus.

So if it never has a zero at the end, because it doesn't have an end, then it cannot be equal to 1.000..., which has zero throughout.

Yes. Correct. Actually, it can be best seen as a problem with operators ... some think that problem makes an equality, and some (me) don’t.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:
obsrvr524 wrote:So if it never has a zero at the end, because it doesn't have an end, then it cannot be equal to 1.000..., which has zero throughout.

Yes. Correct. Actually, it can be best seen as a problem with operators ... some think that problem makes an equality, and some (me) don’t.

I wasn't using any operators. I was talking about the static situation of an already infinite string.
You have been observed.
obsrvr524

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

obsrvr524 wrote:
Ecmandu wrote:
obsrvr524 wrote:So if it never has a zero at the end, because it doesn't have an end, then it cannot be equal to 1.000..., which has zero throughout.

Yes. Correct. Actually, it can be best seen as a problem with operators ... some think that problem makes an equality, and some (me) don’t.

I wasn't using any operators. I was talking about the static situation of an already infinite string.

You don’t understand what you’re saying!

If 1/3 = 0.333...

And 0.333... *3 = 0.999... not 1/3!

That’s an operator problem!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:If 1/3 = 0.333...

And 0.333... = 0.999...

That’s an operator problem!

That might be so but I wasn't multiplying anything. People who tried that as a proof have that issue, not me.
You have been observed.
obsrvr524

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

obsrvr524 wrote:
Ecmandu wrote:If 1/3 = 0.333...

And 0.333... = 0.999...

That’s an operator problem!

That might be so but I wasn't multiplying anything. People who tried that as a proof have that issue, not me.

Do you agree that 1 whole number (1) divided by 3 equal 0.333... ?

Do you agree that 0.333... times 3 equals 0.999... ?

If all that is true, then operators don’t work. At least for base-1.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:
obsrvr524 wrote:
Ecmandu wrote:If 1/3 = 0.333...

And 0.333... = 0.999...

That’s an operator problem!

That might be so but I wasn't multiplying anything. People who tried that as a proof have that issue, not me.

Do you agree that 1 whole number (1) divided by 3 equal 0.333... ?

Do you agree that 0.333... times 3 equals 0.999... ?

No I don't. "0.333..." is not a quantized number, a "quantity". But 1/3 is a quantity.

Ecmandu wrote:If all that is true, then operators don’t work. At least for base-1.

I agree that math operators do not work on non-quantity items (anything ending with "...").
You have been observed.
obsrvr524

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

So, obsrvr,

So, This is an interesting theory of numbers!

9/3 = 3
10/3 = 3
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

I'm not seeing where you are getting that.
Why would 10/3 = 3?
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obsrvr524

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

Here's a proof that $$1 = 0$$.

$$(1 + 1 + 1 + \cdots) + 1 = 1 + 1 + 1 + \cdots$$

Agree?

If the answer is yes, subtract $$1 + 1 + 1 + \cdots$$ from both sides.

What do we get?

$$1 = 0$$

But if the answer is no, it appears to me that it follows that one of the two sides of the expression is greater than or less than the other -- and that means that infinities come in different sizes.

Assuming that I'm wrong, can you help me understand what I'm doing wrong?

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

Let me see if I understand you.

You have an infinite line and under it you have a dot.

Then you subtract the infinite line away and are left with a dot.

And that confuses you?
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obsrvr524

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

And if that confuses you...

When you have 3 parallel lines and subtract 1 parallel line, how many are left?
2

If you then subtract another parallel line, how many are left?
0

2 - 1 = 0
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obsrvr524

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

obsrvr524 wrote:I'm not seeing where you are getting that.
Why would 10/3 = 3?

I’m using shorthand before the expansion...

The expansion is .333...

The shorthand works just as well.

9/3 = 3

10/3 = 3

The latter is what Silhouette is arguing
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:The reason Silhouette sees no difference between .000...1 and zero is because the 1 in 0.000...1 is never arrived at. It’s ALWAYS zero!

$$0.000\dotso1$$ represents $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$.

This infinite product never attains $$0$$.

Magnus Anderson
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