Is 1 = 0.999... ? Really?

But you don’t need a number. All you need is a symbol that is different from every other symbol used in the sum.

That’s called “adding dimensionality”

So first we agree that you can’t add an infinite amount of “new ones”, only a finite amount.

Problem is, that means the ones weren’t infinite (endless) in the first place… you push the “last” one up one more place, which is the same as adding a 1 at the end.

Okay, let’s say for the sake of argument that you can have more symbols than your number system can count. As if that makes any sense in itself.
You can manipulate those symbols using the rules of math but at the end you are forced to convert the result into a number.

If the result has more symbols than your number system can count, then you end up with an infinity.

You still don’t have a quantified number of 1s.

Try this example :

Let’s say that you have a counting system (1,2,3,4,5). Anything over 5 is infinity.

1+1+1+… is infinity : over 5.

(1+1+1+…)+1 is over 5. Therefore it is also equal to infinity in your number system.

(1+1+1+…)+2 is over 5. Therefore equals infinity.

infinity+infinity is over 5. Therefore equals infinity.

I need to be absolutely clear about what you’re saying. Are you actually saying that:

(0.9 + 0.09 + 0.009 + \cdots) = (0.\dot9) = (\infty)

You can do arithmetic with finite terms in order to create an infinite sum. But once it’s created, you can’t do arithmetic with that.

I’ll explain (like that will have any effect):

You can do arithmetic with a series of 1s:

1 + 1 + 1 = 3

You can do arithmetic with a larger series of 1s:

1 + 1 + 1 + 1 + 1 + 1 = 6

There is no limit to the number of 1s you can do arithmetic with. You can do arithmetic with an infinite number of 1s:

1 + 1 + 1 + … = (\infty)

But now you can’t take that and do further arthemtic with it:

(1 + 1 + 1 + …) + n ← Can’t do.

(\frac{(1 + 1 + 1 + …)}{n}) ← Can’t do.

((1 + 1 + 1 + …)^{n}) ← Can’t do.

^ Note that not all infinite sums give you infinity. So we can do the same thing with the 9s:

0.9 + 0.09 + 0.009 = 0.999

0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + 0.000009 = 0.999999

0.9 + 0.09 + 0.009 + … = (\dot9)

^ Unless you think (\dot9) = (\infty), you have not derived (\infty) here. Therefore, you can do further arithmetic with it.

Ooooor… I can just avoiding doing arithmetic with it.

I didn’t say you can’t do arithmetic with infinite sums, I said if the infinite sum gives you (\infty), THEN you can’t do arithmetic with it. (0.\dot9 \neq \infty). (<-- Is this seriously lost on you?)

I agree with step 1. The mistake is not there. It’s not in any of the steps. It’s the fact that you took steps to begin with. You should have stopped at (\infty + 1 = \infty). Why? Take a guess, Magnus. Because you can’t do what with infinity?

I will say this: Another rendition of what I’m trying to say is that you can do arithmetic with infinity, but then different rules apply. So the rule that adding 1 to something gives you a greater number no longer applies. Instead, adding 1 to (\infty) still gives you (\infty).

So don’t do arithmetic with (\infty) or accept that different rules apply. ← Take your pick.

Magnus,

I’m going to explain a very sophisticated concept to you and I hope you can take it in.

Your fallback position is that planes are infinity larger than lines. This is a mistake James made.

Have you ever seen a “2D” optical illusion that moves when you look at it, or gives you a three dimensional image if you look at it correctly.

The same thing can be done in unary with 2D images.

What I’m trying to explain to you here is that planes aren’t a greater infinity than lines.

I hope you understood all that.

Not really.

I’m merely asking why do you think that you can do arithmetic with some infinite sums (such as (0.9 + 0.09 + 0.009 + \cdots)) but not all (such as with (1 + 1 + 1 + \cdots)). Is it because the latter evaluates to (\infty) or because you think the former equals to a finite number or because of something else? And how did you come to your conclusion?

Why is it okay to say that (10 \times (0.9 + 0.09 + 0.009 + \cdots) = 9 + 0.9 + 0.09 + \cdots) but not okay to say that (10 \times (1 + 1 + 1 + \cdots) = 10 + 10 + 10 + \cdots)?

If I understand you correctly, we can do arithmetic with (0.9 + 0.09 + 0.009 + \cdots) because it’s not equal to (\infty).

Why is that so?

But you didn’t explain why.

The mistake must be in one of the steps.

You say that (\infty + 1 = \infty) is true. You have no problem with that expression. Even though what you’re doing here is what you say you can’t do with (\infty), which is arithmetic, you have no problem evaluating its truth-value. But for some strange reason, you have a problem with (\infty + 1 - \infty = \infty - \infty)? You can’t evaluate the truth-value of that expression?

Very curious.

The mistake is in the assumption that (\infty - \infty) equals to (0). It does not. Even though (\infty = \infty), (\infty - \infty) is not necessarily (0).

An infinite line plus a single dot yields more locations than the number system provides. Two infinite lines provides an infinity more than the number system provides. In neither case is the total quantity a “number”.

Relevance? The decimal system doesn’t have a top or final number.

Incoherent.

i got your PM, ecmandu, but i don’t yet feel comfortable with publishing. there still seems to be much disagreement among our mathematicians and i’d like to wait until i see unanimous agreement.

maybe you guys’ll figure out if 1 = 0.999 THIS FUCKING YEAR, and i can proceed with the project.

More to the point, any number of them here already have.

Go ahead, ask them. :wink:

Ok I see you all have moved on, but check it:

The a and b thing is true, but in the infinite sum operation you misidentified a and b. Because the sum is infinite, any added number to a is still a. In that case, there is no a +1. Because a is already all the 1s.

You think there aren’t more symbols than there are natural numbers? The set of natural numbers is the set of all symbols?

How about ({a_1, b_1, c_1, a_2, b_2, c_2, a_3, b_3, c_3, \dotso})? That’s (3) times more than ({1, 2, 3, \dotso}).

But I understand. You don’t think that (3 \times \infty > \infty).

There you go again!

What’s an infinite orange?

What’s an infinite orange + an infinite pear??

What are two infinite oranges?

Do you really believe that any of these guys will change their mind?
I don’t.

Obsrvr! I could say the same about you. Think before you post!

Psh.

Promethean has no understanding of the finer points of arythmetic.

At the end of the day, you can’t count them with the natural numbers. Which was important in context because you had a 1 for each symbol. Your 1+1+1+… is infinite whether you use symbols or not.

You have to make up your mind whether you will consider all infinities the same, in which case 3*infinity=infinity. Or you will consider them to be distinct, in which case you need a way to track the size of each infinity.

That’s the problem with the math here:

The first line says that all infinities are equal. “If I add a grain of sand to a heap of sand, then I still have a heap of sand.”

The second line says that each infinity is distinct and the distinct quantity can be subtracted on both sides of the equal sign. “I know how many grains of sand are in a heap.”

Both lines cannot be true. If you think that line one is true then line two is false. If you think that line two is true then line one is false.

One has to make up one’s mind in order to be consistent. One can swing either way but standard math says all infinities are the same.

So you believe that two infinite lines depict the same amount of locations as one? An infinite plane has the same amount of locations depicted as an infinite line? There is not a one-to-one correspondence.

Yes. It’s called 1:1 correspondence.

No. there is not a one-to-one correspondence between a single infinite line and two infinite lines.