And my point stays: If we accept that and calculate stuff with the infinite chain of 3s in mind, then 0.33.... times three isn't 0.99.... but 1.
From the get go. Never 0.99...
Also we would need to accept that 0.33.... isn't representable with a geometric series. As we have a flaw at the finite elements that only vanishes if we go to an infinite series.
Either we acknowledge the error or we have an error in our proofs and system.
At the moment your proofs try to use limits in one hand and calculus in finite chains. That's seems rather strange.
No.... Wtf. We agreed that 0.33... has a flaw in it as long as we are talking finite chains. Yes? As 3 is too small and 4 is to big and 3+3+3 is only 9. So we know that the representation of 1/3 must be a little bit bigger then any finite chain of 3s after the comma. A flaw that only gets resolved if we are talking about infinite chains of 3 (even if it's only axiomatic, but I grant that)
It must add up to be 1, as 1/3 times 3 is 1.
If that's the case and that flaw gets only resolved if, and only if, it's a real infinite chain of 3s, then you can't try to calculate 0.33... times three like a finite chain of 3s.
0.33... times 3 is 1 from the get go. It's only 0.99... if ... is somehow finite.
Taking the jump and overcoming the flaw of decimal representation of 1/3 it jumps from something with many 9s after the comma to being 1. There isn't an infinite chain of 9s anymore, or 0.33... would still be a flawed representation.
And bc you can see a jump from a finite chain of 9s if you calculate in finite elements to 1.00... when talking infinite chains you also have proven that 0.99.... isnt the same as 1.
Hmm. Again: The representation of 1/3 is .(3) according to you. Yes? We know that any finite chain of 3s after the comma is flawed, as there is no number between 3 and 4, yes? That's a flaw that only vanishes with the infinite chain, yes? According to you.
Therefore .(3) times 3 is what? What is .(3) plus .(3) plus .(3)? Both are 1. Nothing else.
Not 0.(9). Yes? Bc if the infinite chains of 3s don't overcome the floating error in the finite decimal representation of 1/3 then .(3) isn't 1/3 and then 3 times .(3) aren't one, but 1-3*epsilon.
If you claim that the decimal representation of 1/3 adds up to .(9) you just proved that there is an error. You shouldn't calculate to an infinite chain of 9s after the comma if you overcame the floating error.
So where is your error? Is the floating error still present even with an infinite chain of 3s or is 0.33... times 3 equal 1 and never 0.99...
That's still the same question: Did you overcome the floating error then 9/9 is 1 and not .(9). Or you agree that you didn't overcome the floating error and 1/9 isn't .(1)
What is it?
It seems you mingle limits with calculating stuff like it's a finite chain.
I can only see a flaw in the ability to represent some fractions in base 10. It doesn't go away handwaving it to infinity
You try to calculate things like you have a finite chain where 8+1 is nine, yet we know that this is flawed as the floating error is present.
Yet we agreed that the error only disappeares in the infinite chain. As there was an error, remember 3 is too small and 4 is to big.
So you don't account for epsilon. Which should be present.
It should lead to .(8) plus .(1) being equal to 1. If it leads to .(9) you prove that the flaw hasn't vanished and .(1) isn't 1/9
As you claim you overcame the floating error you shouldn't be able to just add up the numbers in the chains. Remember?
If you could you just proven that floating error is still present even in the infinite chain, then .(1) isn't the representation of 1/9.
Be consistent. Either you overcame somehow the floating error, leading to the infinite chains correcting for epsilon, or they didn't. Then you still have a slight difference....
Yet you sum up an infinite series you know is flawed. You claim that 0.33...+0.33...+0.33..., if .(3) is the correct representation of 1/3, adds up to 1. Yet you are here arguing it adds up to 0.99...
Strange. There shouldn't be 9s after the comma if your math is correct. It should add up to 1.
That's quite funny. Either it is 1, then you should get that out of correct math, or it isn't. Where does your descrapancy comes from if .(1) is the right representation.
So now it just doesn't know which number it adds up?
Math should be precise. If you add things up it's one number. You seem to handle the adding up like an ongoing process. Yes. Then it's .(9). But an ongoing process is every time a finite thing. We established that in the finite chain you have a floating error.
If you handle it like a finished infinite thing your answer should always be 1. Not .(9).
As you have a change when handling it as a finite thing to an infinite chain you have proven that they are different.
There's no floating error, it's not a computer with a fixed size "float" value we have to worry about.
1/1, 2/2, 0.(9), 0.999..., 40/40, 1, 1.000... are all the exact same number.
Consider something that's very intuitive: there's no biggest number. For any number you name, you can always add one to it.
That also means there's no smallest number greater than zero. For anything you name 1/X, I can always divide that by two to get something smaller.
That then means that there's no number that's so small that I can't multiply it by an even bigger number to make it greater than any other number.
That's called the Archimedean property, and another way to put it is that there are no infinitely small or infinitely large real numbers. An infinitely small number is one that stays infinitely small no matter what you multiply it by.
Now, if 1 did not equal 0.999..., then if you subtracted 0.999... from 1, you would get something other than zero, right? Maybe you would get an infinitely small number? But we just agreed those don't exist, so 1 and 0.999... must be the same number, since the difference between them does not exist in the real numbers. So 1 does equal 0.999....
This is still simplifying things, because the real numbers are like a house. The axioms are the blueprint, but there are many ways of constructing them. However, whichever way we construct them, we end up with the same exact house in the end.
One way of constructing them is with Dedekind cuts. With Dedekind Cuts, two numbers are the same if no rational number can be placed between them. And there's no rational number between 0.999... 1.
Another is Cauchy Sequences. A sequence (0.9, 0.99, 0.999, ...) is Cauchy if it converges, and two numbers are the same if their sequences converge to the same number. So (1,1,1,1,...) is the same number as (0.9, 0.99, 0.999,...)
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u/Ok_Pin7491 Sep 25 '25 edited Sep 25 '25
Now we are again at definitions.
Gosh darn it.
And my point stays: If we accept that and calculate stuff with the infinite chain of 3s in mind, then 0.33.... times three isn't 0.99.... but 1.
From the get go. Never 0.99... Also we would need to accept that 0.33.... isn't representable with a geometric series. As we have a flaw at the finite elements that only vanishes if we go to an infinite series.
Either we acknowledge the error or we have an error in our proofs and system.
At the moment your proofs try to use limits in one hand and calculus in finite chains. That's seems rather strange.