Is 1 = 0.999... ? Really?

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!

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.

No I don’t. “0.333…” is not a quantized number, a “quantity”. But 1/3 is a quantity.

I agree that math operators do not work on non-quantity items (anything ending with “…”).

So, obsrvr,

So, This is an interesting theory of numbers!

9/3 = 3
10/3 = 3

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

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?

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?

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

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

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

This infinite product never attains (0).

Actually, the only way it can NEVER equal a zero is if it adds 1/10th sequentially. Otherwise, it’s a zero.

What does it mean to add 1/10th sequentially?

1/10th. STOP * 1/10th STOP. * 1/10th STOP etc…

I don’t understand.

It’s means that 1/10th MOVES for EVERY 9 instead of the 1 never occurring (the “the end”)

Wikipedia proofs treat these symbols as representing infinite sums.

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

Remember that (\infty) has an infinite sum equal to it which is (1 + 1 + 1 + \cdots). This means that if you can’t do arithmetic with infinities (because they are qualities, as you say, and not quantities) then you can’t do arithmetic with infinite sums either. (Which would invalidate Wikipedia proofs.)

Unless, for some strange reason, you don’t think that infinity can be represented as an infinite sum. In that case, I could simply stop talking about infinities and start talking about. . . infinite sums. There would be no difference with regard to my argument.

Do you agree that infinite sums come in different sizes?

Do you agree that ( (1 + 1 + 1 + \cdots) + 1 > 1 + 1 + 1 + \cdots)?

If you do, thank you very much.

But if you don’t, this means that:

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

Do you agree that (1 + 1 + 1 + \cdots = 1 + 1 + 1 + \cdots)?

Remember that one of the Wikipedia proofs claims that (0.9 + 0.09 + 0.009 + \cdots = 0.9 + 0.09 + 0.009 + \cdots).

If you don’t agree with this, you also don’t agree with Wikipedia proofs.

If you do agree, let’s subtract (1 + 1 + 1 + \cdots) from the above equation.

What do we get?

We get (1 = 0).

Do you agree with the conclusion?