if 0.999... = 1 does that mean that the set of all real numbers

>>8889589
>if 0.999... = 1 does that mean that the set of all real numbers less than one but greater than zero contains one?
What? No.

>how arbitrarily close are we allowed to get to one under this set?
As close as you like, so long as the distance isn't zero.

>Is there a specific amount of nines after 0.9 that we're allowed to have before that number is no longer a member of the set?
No; any number of nines would still be less than one.

>>8889602
>any number of nines would still be less than one.

So 0.999 is not 1?

>>8889611
0.999...*

sup{0.9, 0.99, 0.999, ...}=1

>>8889611
>>8889613
How many nines are in 0.999... ?

>>8889613
>>8889611
Infinite nines is not a number of nines.
>>8889616
Basically this. The supremum of this set is 1, and the limit of the sequence is 0.999..., and the real numbers are complete so 0.999...=1

>>8889589
>if 0.999... = 1 does that mean that the set of all real numbers less than one but greater than zero contains one?
This very statement implies 0.999...<1 but you have just (correctly) claimed 0.999...=1, a contradiction.

>>8889659
>Infinite nines is not a number of nines.

It is though, if i said that a number has 5 nines after it you would know it doesn't have less than 5 nines or more than five nines after it.

if a number has infinity nines after it we know it doesn't have five nines after it, that means infinity can be compared to other numbers and be treated as a number.

if infinity is not a number then you can't say 4 is greater than infinity, since you can't compare a value to something that you claim doesn't represent a value and claim it's greater than 4.

>>8889685
>if infinity is not a number then you can't say 4 is greater than infinity,
Strictly speaking, you can't say that.

>>8889685
if infinity is a number, would you mind rounding it to the nearest integer and giving me it's prime factorization?

>>8889685
This man is right, >>8889724
You'd actually be wrong if you said "4 is less than infinity" or "the harmonic series equals infinity", since infinity is not a number and it would make as much sense as saying "the harmonic series equals green". You would actually say "the harmonic series tends to infinity" and "4 is finite".

>>8889589

[math]0.\bar{9} = 1[/math] because there is no number, however precise, between the two numbers that is not equal to the two.
>how arbitrarily close are we allowed to get to one under this set?
if there is no possible number by any plausible definition within the set of [math]\mathbb{R}[/math] between any two numbers a and b, then a = b. [math]\lim_{x \to 0+} x = \lim_{x \to 0-} x = 0 \to (0) = 0 \neq 0.1[/math].

>Is there a specific amount of nines after 0.9 that we're allowed to have before that number is no longer a member of the set?

If by "the set" you mean [0.9, 1), then there must be a finite number of nines in the set, and explicitly finite:

[equ]0.\bar{9} = \displaystyle \lim_{m \to \infty}\Sigma_{n = 1}^{m} \frac{9}{10^n} = 0.9 + 0.09 + 0.009 + ...[/equ]
Outside of an infinite limit, however:
[math]\displaystyle \forall m \in \mathbb{P} \Sigma_{n = 1}^m \frac{9}{10^n} < 1[/math] by the definition of what makes [math]0.\bar{9}[/math] equal to 1.

>>8889816
edit: that mess that didn't post right is

[math]0.\bar{9} = \displaystyle \lim_{m \to \infty}\Sigma_{n = 1}^{m} \frac{9}{10^n} = 0.9 + 0.09 + 0.009 + ...[/math]

File: 4L_MYFJBQua.jpg (110KB, 914x892px) Image search: [Google]

110KB, 914x892px

Can I replace all integers with their .999... counterparts?
e.g. 315=314.999...

>>8889589
>if 0.999... = 1 does that mean that the set of all real numbers less than one but greater than zero contains one?
No, but the supremum of that set is 1.

>how arbitrarily close are we allowed to get to one under this set?
arbitrarily close

>Is there a specific amount of nines after 0.9 that we're allowed to have before that number is no longer a member of the set?

>>8890477
of course

>>8889771
>>8889777
If infinity is not a number then how can you claim that 0.999... = 1?

if infinity is not a number so we can't possibly know what 0.999... equals.

>>8889616
This is the correct answers.

>>8890477
Yes.

By the way, every infinitely repeating sequence can be written as a rational number, for example 0.145614561456....

Proof:
x = 0.1456....
10000x = 1456.1456...
10000x - x = 1456.1456... - 0.1456...
9999x = 1456
x = 1456/9999

>>8889589
>if 0.999... = 1 does that mean that the set of all real numbers less than one but greater than zero contains one?
>if 0.999... = 1 does that mean that 1 is less than 1?
No. It means 0.999... is not less than 1.

I fuckin' hate these threads.

Why the fuck do people still reply to these threads? We were explained this in class when we were 13 or something, and nobody thought it was weird. It's obvious that there can't be that many people here who that seriously cling to a belief for some entirely arbitrary reason, so there have to be trolls in these threads.

>>8891023
but that's wrong you nutjob. Infinity is a concept, not a number.

That's like saying "how can you know 1 is positive if "positive" is not a number?".