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

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).

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).

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

That one doesn’t make any sense either.

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).