An infinite question: Why doesn't .999~ = 1?

[watchwolf49]

I don’t understand grammering so it’s all bullshit to me …

However, I do understand your username … shame on you … [giggle]

But, yes, I’m not using people as a singular entity … so it should have the plural form of the verb are

[netzweltler]

@Budget_Player_Cadet

Um… What?

You’ve confused two totally different things - a set, which is a defined group of things, and a range, which is all the numbers between a high and a low value. What’s more, it doesn’t map to a set unless you do more to define it, such as telling us what set of numbers within that range you want.

And in this case, if the set you’re talking about is {x ∈ ℝ, 0 ≥ x ≥ 1} (all real numbers between 0 and 1), then we run into entirely different problems, because the real numbers are uncountably infinite - meaning you literally cannot get from point A to point B by counting. If you remove 1 from that set, changing it to {x ∈ ℝ, 0 ≥ x > 1}, then you literally cannot define a “greatest number” for the set, making the situation quite different.

I agree with that. We cannot define a “greatest number”. That’s why we cannot define an endpoint.

[netzweltler]

@leahcim

Without some definition, it is impossible to say anything about distance between sets and points. That is how math works – you define things and then use those definitions to prove things. There is no, “I don’t want to define this thing, but I am sure I know something about it”.

Is there definition missing?

Wouldn’t we rather have to define elements as close as zero distance to point 1, if we want the set [0, 1) to be as close as zero distance to point 1? I mean, shouldn’t we be able to name such elements?

@leahcim

Again, if you posit a definition for “endpoint” as “the greatest lower bound or the least upper bound of the set” then it is obvious. { 0, 0.9, 0.99, 0.999 } has endpoints 0 and 0.999. [0, 1) has endpoints 0 and 1. Start with the definition, derive the theorem.

If you don’t like that definition, propose another, but you can’t get away without one.

This “work from analogy” thing is great for English common law, but it is not mathematics.

[watchwolf49]

@netzweltler

I agree with that. We cannot define a “greatest number”. That’s why we cannot define an endpoint.

So, the line segment [0, 1) with the defined endpoints at 0 and 1 has no defined endpoints … interesting … that’s running loose with the definition of definition … that’s for sure …

[RealityChuck]

We’re still arguing this? It’s easily proven with basic math.

for .999999…

x = .99999…

Multiply by 10

10x = 9.9999999…

Subtract .999999… from both sides

10x - .9999999 = 9

From the first equation, it’s given that: x = .999999…

So, substituting that in, you get 10x - x = 9

Simply

9x = 9

Divide both sides by 9

x = 1

QED

This works for any repeating decimal as a way to find a whole number value for it. Try .333333…

x = .333333…
10x = 3.33333…
10x - .333333… = 3
10x - x = 3
9x - 3
x = 3/9 = 1/3 Thus .333333… = 1/3

If the proof is wrong for .99999…, it’s also wrong for .3333333… But you can easily prove that 1/3 = .333333… by simple division. Therefore, the proof is correct.

Any other argument ignores math.

[watchwolf49]

Yes, this is taking longer than we thought.

[The_Great_Unwashed]

@RealityChuck

If the proof is wrong for .99999…, it’s also wrong for .3333333… But you can easily prove that 1/3 = .333333… by simple division. Therefore, the proof is correct.

Honestly, that’s a horrible argument, a “proof” might well be wrong for both Proposition A and Proposition B but even if we could prove Proposition B by independent other means that would have no bearing on the soundness of said previous argument for the truth of Proposition A.

[The_Great_Unwashed]

@netzweltler

Wouldn’t we rather have to define elements as close as zero distance to point 1, if we want the set [0, 1) to be as close as zero distance to point 1? I mean, shouldn’t we be able to name such elements?

I said it sometime before, easily missed: 0.999… is not an element of [0,1) so I don’t know why you think it of the least importance that we cannot name its “closest to 1” element.

[John_W.Kennedy]

@The_Great_Unwashed

I said it sometime before, easily missed: 0.999… is not an element of [0,1) so I don’t know why you think it of the least importance that we cannot name its “closest to 1” element.

I may be wrong, but I think that, from his viewpoint, you’re begging the question.