/sci/ - Science & Math

newfag here, what is the sum of all natural numbers?

0.999... < r < 1
Does r exist?

Look at it in a variety of ways.
Are there infinitely many divisions between 0 and 1? If yes, there is a z value where z×infinity = 1, and 1-z = 0.999...
This requires infinity be an arbatrily large real number. If it isn't, and is instead a vector, z cant exist because there is no end to the counting of possible division parts between 0 and 1, yet if there is no end to the counting of 9's in 0.999..., it can not ever rationally reach or get close enough to 1.
>infinity is a value: 0.999... =/= 1 because [math]z_1=\frac{1}{\infty}, 1-z_1 = 0.\bar{9}[/math]
>infinity is a vector: 0.999... =/= 1 because there exists only more and more 9's never getting closer to 1
>infinity is a value without a specific definition: 0.999... =/= 1 because you can not increment or count to infinity, but any attempt to do so will reveal greater and greater local maximal real number amounts of possible 9's, where [math]\frac{1}{LMRN} = z_2 \in \mathbb{R}[/math], a value similar to math]z_1[/math] to satisfy the same evaluation of [math]1-z_2 = 0.\bar{9}[/math]

>"[math]\frac{1}{3} = 0.\bar{3}, 0.\bar{3} × 3 = 0.\bar{9} = \frac{3}{3} = 1[/math]": no possible amount of 3's will ever rationally and succinctly finitely equate the decimal value of 1/3, so [math]\frac{1}{3} > 0.\bar{3}, \frac{2}{3} > 0.\bar{6}, \frac{3}{3} > 0.\bar{9}[/math]
>"x= 0.9... ; 10x= 9.9... ; 10x-x = 9 ; x=1": y=0.9, 10y=9.0 || y=0.999, 10y= 9.990 || y=0.(n nines), 10y=9.(n-1 nines) ^ 10y-y = 8.999...1; 9y/9 = y = 0.999...

Question:

How do you figure out if there is an infinite amount of something?
Do you count each thing that makes up something?
How long does it take to count to infinity?
How big is infinity × 2?

Only PROFESSIONAL mathemeticians and serious replies are welcome.

Isn't infinity a shitty concept in math?
I mean, you get retarded shit like
1+2+3+4+... = -1/12
or
1-1+1-1+1-1+... = 1/2
which is OBVIOUSLY wrong, but then they get pretty accepted within the community because it is valid to the current rules of the game and the rules don't get changed no matter how retarded the results that it produces are