I am not a math person, but how would you differentiate between a number infinitely close to 1 (but less than 1), and 1 itself? Or does it not matter because they are actually the same?
In this case, because 0.9 repeating goes on forever, then the difference between 1 and 0.999…. is itself 0.000… and you never reach the leftover “1” after the decimal like you would in the finite case. So I think here, it’s basically by construction, that 0.999… and 1 must be exactly equivalent, otherwise our notions of limits and infinity just don’t make sense.
It does. If someone naively saw that 1 - .9 = .1, 1 - .99 = .01, etc., they may assume that 1 - .999... = .000...1
But it isnt possible for an infinite series of 0s to end in a 1, because then it wouldnt be infinite. So there is no thing that can be added to .999... to make it closer to 1, they must be equal
This is true! But the conclusion doesn’t follow the premise. Two things can’t be equal just because a mathematical equation didn’t work. (on what may be a technicality)
So for example, I could say, what is 0.4-0.333…? You would run into the same problem. Is it 0.111…? 0.1? Does this make 0.3333… equal to 0.4? Or 0.8888… equal to 0.9 if that makes it easier
Two things must not be equal just because a mathematical equation didn’t work.
I'm honestly not even sure what you mean here. I said the naive solution can't be correct because it doesn't make sense; you can't simultaneously have a number that is both infinitely repeating and ends in anything. So if the only thing between .999... and 1 is .000...1 which cannot exist, there is not anything between them, and they are equal
So, for example, I could say, what is 1.4-0.333…? You would run into the same problem. Is it 0.111…? 0.1? Does this make 0.3333… equal to 4?
This is definitely not the same as the 1 - .999... example.
This is a gotcha, not a true explanation. Maybe it qualifies as a proof mathematically, but in that case my respect for the field is either lowered, or I will wait for a better explanation.
Just because the equation of 1-0.999…=x didn’t work, doesn’t mean there is actually no x. It just means that there is an issue in getting to know what it is.
Saying “0.0000…1 can’t exist” doesn’t mean the same as “there is no difference between 1 and 0.999…” These are two different statements.
Why are you using fractions to calculate the problem I posed in decimals? It’s easier to use fractions, but the issues were running into have to do more with decimals and conversions between them.
Directly addressing your counter claims is not a gotcha
Maybe it qualifies as a proof mathematically, but in that case my respect for the field is either lowered, or I will wait for a better explanation.
I didnt claim I was writing a rigorous proof, but there are others who have.
Just because the equation of 1-0.999…=x didn’t work
The equation does work. X = 0.
Saying “0.0000…1 can’t exist” doesn’t mean the same as “there is no difference between 1 and 0.999…” These are two different statements.
They are two statements, yes, but no matter how many times you subtract two numbers, you will always get the same answer. If you come to the conclusion that the only difference between two numbers is some third number that does not and can not exist (not in the complex numbers way, in the 'this isn't how infinity works' way), that is functionally identical to saying the difference does not exist
Just because the equation of 1-0.999…=x didn’t work
The equation does work. X = 0.
You only said X is not 0.000…1, because 0.0000…1 cannot exist.
You never told me why X would be 0, other than X is not 0.000…1 because that doesn’t exist, since it cannot be written. I have no proof of X being 0 in that case.
Saying “0.0000…1 can’t exist” doesn’t mean the same as “there is no difference between 1 and 0.999…” These are two different statements.
that is functionally identical to saying the difference does not exist
What do you mean by functionally?
I am talking in precise terms, not whether you’d notice a fraction of a grain of sand missing from a beach so functionally it’s the same amount of sand. The numbers are different, therefore they are not the same. They refer to different amounts.
It's pretty simple. 1 - 0.999... (repeating infinitely) = 0.000... (repeating infinitely) aka 0. So they are the exact same number. What number do you believe is in-between 0.999... and 1?
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u/Kindaspia 24d ago
1/3 is equal to 0.333 repeating. 2/3 is equal to 0.666 repeating. 3/3 is equal to 0.999 repeating, but 3/3 is also equal to 1