This

sites.math.rutgers.edu/~zeilberg/Opinion146.html

it should read (0.999...)(T) = 1
where T is a transformation

Or
(0.999...) symbol 1
where symbol indicates that sides of the equation are different, but very similar

>where symbol indicates that sides of the equation are different, but very similar
should we also use 1/2 symbol 2/4 instead of 1/2 = 2/4 because both sides are "different, but very similar" ?

1/2 is exactly equal and same to 2/4 so no

>different, but very similar
How are they different aside from having different symbolic representations? In terms of value they're not just similar, they're completely identical.

>1/2 is exactly equal and same to 2/4
So are 0.999... and 1.
Show me one specific example of a way that 0.999... and 1 behave differently under identical circumstances.

0.999... is 1 - 0.000...1

in other words, 0.000...1 separates 0.999... and 1

nothing separates 1/2 and 2/4

0.999...^infinite =/= 1
1^infinite = 1

>0.999... is 1 - 0.000...1
0.000... 1 is 0.
1/2 - 0 = 1/2
1/2 - 0 = 2/4
1/2 - 0.000... 1 = 1/2
1/2 - 0.000... 1 = 2/4
1 - 0 = 1
1 - 0.000... 1 - 0 = 1
1 - 0 = 0.999...
1 - 0.000... 1 - 0 = 0.999...

sure you can take it as an axiom that 0.000...1 = 0 and 0.999... = 1 if that is what you want

i find it hard to believe you dont intuitively understand the logic behind the construction of numbers 0.999... and 0.000...1 and similar numbers

you take a number and change it by the least possible amount so that you can no longer consider them the same number

1^infinity is an indeterminate form. You can't just set it to 1, it won't behave normally if you try that. e.g.
logb(M^n) = n * logb(M)
If b = infinity, M = 1, and n = infinity, then
loginf(1^inf) = inf * loginf(1)
loginf(1) = inf * loginf(1)
0 = inf * 0
-inf = 0
-inf + inf = 0 + inf
inf-inf = 0 + inf
inf = inf + inf
inf = 2inf
1 = 2

1^infinity is an indeterminate form. You can't just set it to 1, it won't behave normally if you try that. e.g.
logb(M^n) = n * logb(M)
If b = infinity, M = 1, and n = infinity, then
loginf(1^inf) = inf * loginf(1)
loginf(1) = inf * loginf(1)
0 = inf * 0
0 = inf * 1/inf
0 = 1

Plus or minus 0.0…1

0.0…1 = 0

0.999...^infinite=0.999...

Hence it's true.

>least possible amount

0.000...1 > 0.000...01

Brainlet here.
What does this mean in regards to asymptotes, where an equation like 1/(x-1) reaches infinitely close to 1, but never actually touching? Would this mean it does actually touch 1 at one point, or am I understanding this wrong?

What do you get if you take 0.999... and subtract 0.000...1 again ? How many times do you need to do that to reach 0 ?

0.000...01 > 0.000...(0.000...01)

The function never reaches 1, but it's limit is 1. 0.999... is defined as a limit.

Wouldn't it mean that no matter how close to 1 it gets there's always a difference?

that's the same as 1-0
so you will never get down from 1

I'm asking about the case when you don't take the axiom 0.000...1 = 0

0.0…1 is not a number. How many times do you have to subtract ¥ from 1 to get to 0?

What ? I'm trying to understand the definition you gave:
>you take a number and change it by the least possible amount so that you can no longer consider them the same number

think of it this way:

1 = 9/10 + 1/10 = 0.9 + 10/100
= 0.9 + (9/100 + 1/100) = 0.99 + 10/1000
= 0.99 + (9/1000 + 1/1000) = 0.999 + 10/10000
= 0.999 + (9/10000 + 1/10000) = 0.9999 + 10/100000

and so on

so what's the big deal?

The difference here is that each line is exactly one, not approaching it.

line #1 is exactly 1
line #10 is exactly 1
line #98327498236483689 is exactly 1

even at infinity, it still is exactly 1

[math] \displaystyle
0.000...1 = \frac{1}{10^ \inf} = 0
[/math]

those three dots destroy anything to the right of it

So you were saying it's an axiom but now you say it's a theorem ? I'm confused. Are you even the person I was replying to ?

go to reddit if you care about that
this place just cares about the ideas themselves

Seems meaningless

I know about the classical mathematical setting where 0.000...1 = 0 and 0.999... = 1.
But someone was claiming that it is actually just an axiom and implying that you can work without it.
I'm asking how that would work and what kind of calculations you can make in that framework.
If you don't know the answer to that, please don't answer with unrelated stuff.

>you take a number and change it by the least possible amount so that you can no longer consider them the same number
There is no such number.

0.1 = 10^-1
0.01 = 10^-2
0.00...1=10^ -inf

yeah, totally unrelated

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