1 = 0,99........

[Ghieri]

Because we have no way of measuring numbers that small at this point in
time. The difference between 0.9~ and 1 is immeasurably small, but it
still exists. I don't consider it proven because the arithmetic
operators +, -, * and / are designed to work with measurable numbers. As soon as you start using numbers that have no definite value the math stops working.

I think you misunderstand, "infinitely small", "infinite", etc are not actual numbers. You can measure numbers. You cannot count to infinity, or measure infinite anything. There is no countable distance between .99~ and 1. It is not that we cannot measure infinity "Yet", it's that infinity is immeasurable by nature. It only exists because theoretically, there is no natural upper limit of numbers.

An infinitely small number is not an actual number, because you can continuously divide the number and it will only get smaller and smaller. It does not, by definition, have a value, because you cannot reach the value, or plug it into an equation. It's on the same level as "undefined". Something that exists in mathematics as a way to tell the mathematician they are doing something wrong. Does that make sense?

[Bridge]

Because we have no way of measuring numbers that small at this point in
time. The difference between 0.9~ and 1 is immeasurably small, but it
still exists. I don't consider it proven because the arithmetic
operators +, -, * and / are designed to work with measurable numbers. As soon as you start using numbers that have no definite value the math stops working.

I think you misunderstand, "infinitely small", "infinite", etc are not actual numbers. You can measure numbers. You cannot count to infinity, or measure infinite anything. There is no countable distance between .99~ and 1. It is not that we cannot measure infinity "Yet", it's that infinity is immeasurable by nature. It only exists because theoretically, there is no natural upper limit of numbers.

An infinitely small number is not an actual number, because you can continuously divide the number and it will only get smaller and smaller. It does not, by definition, have a value, because you cannot reach the value, or plug it into an equation. It's on the same level as "undefined". Something that exists in mathematics as a way to tell the mathematician they are doing something wrong. Does that make sense?

I know. But if a number is "nothing" it is 0.

0.00 ... 01 is not zero and yet we have no way of ascertaining its exact value. It's not necessarily infinite, just too small to calculate. Does that make sense? That's all I'm trying to say. Saying it is "nothing" is ignorant and lazy. We have not yet discovered the real "building block" of matter, and yet no reputable scientist would ever dream of saying it was "nothing". Just because it is too small to measure today does not mean it doesn't exist.

[Ghieri]

Saying it is "nothing" is ignorant and lazy.

Yet the worlds most prominent Mathematicians accept it as fact.

It's not necessarily infinite, just too small to calculate.

It's not "too small to calculate" it's "so small it doesn't have a value"

Let me explain it this way:

0.00 ... 01

The number of 0's in front of that one are infinite, meaning there are infinite number of zeros, meaning the value can only possibly be resolved as 0. It is not larger than zero, because when running it through calculations, we never reach that last digit.

[Your Computer]

The number of 0's in front of that one are infinite, meaning there are infinite number of zeros, meaning the value can only possibly be resolved as 0. It is not larger than zero, because when running it through calculations, we never reach that last digit.

Is that to say that prominent mathematicians fail at logic or that you aren't a mathematician? For how can you end an "infinite" with "01" and still call it "infinite"? And how can you end an "infinite" with "9" and still call it "infinite"? In order for the equation to work you have to assume that "infinite" or "..." is measurable. Thankfully, "..." doesn't necessarily mean "infinite," in which case 0.99...9 != 1.

[Ghieri]

For how can you end an "infinite" with "01" and still call it "infinite"?

We are talking about infinitely small numbers, which have to have their "last" digit as 1. My point is that the infinite number of zeroes never reach that point. Therefore, infinitely small.

Thankfully, "..." doesn't necessarily mean "infinite," in which case 0.99...9 != 1.

We are not talking about measurably small numbers, so "..." does not equal a finite number of places in this case. If "..." were a finite number of places then I would agree. But it isn't.

Is that to say that prominent mathematicians fail at logic or that you aren't a mathematician?

I'm not really sure what I did that put a stick up your ass, but whatever it is probably isn't relevant to the discussion. Anyways, there are several mathematical proofs on the topic:

n = .99~

10n = 9.99~

10n-n = 9n

9n = 9
n = 1

1/9 = .11111~
9(1/9) = 9(.11111~)
1 = .99999~

More here.

So while it seems counter intuitive. It is mathematically solid.

[Bridge]

So while it seems counter intuitive. It is mathematically solid.

Because you assume 0.00 … 01 is "infinitely small". I assure you, if you actually used the real number that is being represented, you would not get the solution "1=0.9~". It's like saying:

1/inf x + 1 = 1
0 + 1 = 1
1/inf x = 0

Nobody can say this isn't valid because the number is not calculable and therefore the solution unknowable. 1 divided an infinite amount of times is also an infinitesimal (perhaps even the same number), but it is most certainly not zero. This is an assumption that the solution needs in order to work. It's a loss of data, no matter how much you say it isn't because it "never reaches the last digit" (now there's an absurd mathematical concept for you).

[BAndrew]

All
the proof you have is based around the flawed assertion that 1-0.9~ is
0. All of your calculations depend on this, and you provided no proof
for this assertion. Occam's Razor dictates that the explanation that
makes the least amount of unnecessary assumptions is the most efficient
and most likely correct. I am not making any assumptions by saying
1-0.9~ > 0 because 0.9~ < 1. Prove without making any assumptions how it is not.

I didn't make any assumptions.



Tell me where the assumption is.

lim n->nf {0.00 … 01} (n-1) …

I already told that this is not an assumption. It is archimedean property.

[Your Computer]

We are talking about infinitely small numbers, which have to have their "last" digit as 1. My point is that the infinite number of zeroes never reach that point. Therefore, infinitely small.

Then any value with "..." is self-invalidated. If you can never reach 01 in 0.00...01 or 9 in 0.99...9, then you can never subtract from it, never add to it, etc. To say that i can is an assumption without evidence--or, more precisely, an assumption contrary to contrary evidence.

We are not talking about measurably small numbers, so "..." does not equal a finite number of places in this case. If "..." were a finite number of places then I would agree. But it isn't.

Would you say 1 is measurable? If so, then you are claiming that a measurable number equals a non-measurable number.

I'm not really sure what I did that put a stick up your ass, but whatever it is probably isn't relevant to the discussion.

Words on paper have no emotion, neither do words on a screen.

So while it seems counter intuitive. It is mathematically solid.

It's not just counter-intuitive, it's illogical.

Look at the following example:

Line 1: n = .99~
Line 2: 10n = 9.99~
Line 3: 10n-n = 9n
Line 4: 9n = 9
Line 5: n = 1

In order for line 3 to lead to line 4 i would have to assume that n is equal to 1. However, line 1 shows that it is equal to 0.99~. Likewise, evidence that n is equal to 1 has not yet been shown. In other words, i have to commit multiple logical fallacies to reach line 5. Of course, if we were to have divided 10 from 10n, therefore making n = .99~, we'd be back at line 1. Likewise, therefore, i would have proven that n cannot be anything but 0.99~, therefore 0.99~ can never equal 1.

[BAndrew]

It's not just counter-intuitive, it's illogical.

Look at the following example:

Line 1: n = .99~
Line 2: 10n = 9.99~
Line 3: 10n-n = 9n
Line 4: 9n = 9
Line 5: n = 1

In order for line 3 to lead to line 4 i would have to assume that n is equal to 1. However, line 1 shows that it is equal to 0.99~. Likewise, evidence that n is equal to 1 has not yet been shown. In other words, i have to commit multiple logical fallacies to reach line 5. Of course, if we were to have divided 10 from 10n, therefore making n = .99~, we'd be back at line 1. Likewise, therefore, i would have proven that n cannot be anything but 0.99~, therefore 0.99~ can never equal 1.

No it doesn't. You have 10x - x which is equal to 9x. You are probably refering to 9n = 9 which is indeed a mistake as we don't know n = 1(that's it if you don't accept that 9,99... - 0,9999 = 9).

See the proof with the limits which is mathematically accepted.

[Ghieri]

In order for line 3 to lead to line 4 i would have to assume that n is equal to 1.

We know 9n = 9 because 10(.99~)-(.99~) = 9.99~ - .99~ = 9
and 10-1 = 9.

Therefore n=9/9 = 1.

It's not just counter-intuitive, it's illogical.

...except I mathematically proved it. You're welcome to try counter proof but I held up my burden.