Is 1 = 0.999... ? Really?

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!

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

So the “proof” using n/o is invalid.

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.

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…