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"If 0.999...!= 1 then there must exist a number A, such that 0.999... < A < 1." Thanks for the reply, but I don't know how you can make the above claim, becaus

by vfinn 6y ago

"If 0.999...!= 1 then there must exist a number A, such that 0.999... < A < 1." Thanks for the reply, but I don't know how you can make the above claim, because to me the question of what we mean by 0.999... is intertwined with the claim itself. I mean to me it seems to be a matter of how you interpret the approach to infinity. I don't see why there has to be A in between, if you interpret 0.999... as the "biggest possible number below 1", as then there would be also a "difference of the smallest possible amount" between those numbers approaching 0, but not quite getting there. But then again, if it's by some fundamental definition (limit) that 0.999... = 1, then ok. I'm slightly embarrassed I don't know more mathematics, but I'm trying to learn some more...

kosievdmerwe 6y ago
That statement is based on: > First hopefully we can agree that if we have two real numbers x and y that are not equal then we have a number z, such that x < z < y. The easiest example is z = (x + y)/2. It's one of the properties of the reals and rationals that if two numbers aren't equal, then there are infinitely many numbers between them. It doesn't work with the integers, 2 and 3 aren't equal, but there's no integer x, such that 2 < x < 3. So if we say 0.999... != 1 then that means 0.999... < 1 and then that means (0.999... + 1) / 2 is a number different from 1 and different 0.999... but that lies between them. But this is modern mathematics. In the past mathematicians dealt with "infinitesimals", especially in the early days of calculus, but I think they were discarded because they were confusing and also not necessary in favor of limits. This is where I think where some of your confusion is coming from. Infinitesimals don't exist in the real (and therefore rational) number space, but the concept exists for other "weirder" number systems. According to the wikipedia page "This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...)" The clearest resolution to "if you interpret 0.999... as the "biggest possible number below 1"" is that in the reals and rational this concept doesn't exist. There's no biggest number smaller than x and ditto for smallest number bigger than y. (The distinction from the wikipedia definition is the difference between < and <=, <= exists, but < doesn't) It's similar to saying you can't divide by 0. Sure in some cases you can define it, but doing so causes many issues and costs you so much that it's not worth it.[1] Another example is 1 not being prime, there's no reason for it not to be prime, but it's just much more convenient to say arbitrarily that it's not prime.[2] The other thing that might confuse you is a proof by contradiction, I know it certainly confused me the first few times I saw it. I'm happy to help you if this is tripping you up too. > I'm slightly embarrassed I don't know more mathematics, but I'm trying to learn some more... No problem, we all start knowing nothing :) [1] https://www.youtube.com/watch?v=BRRolKTlF6Q https://www.youtube.com/watch?v=BRRolKTlF6Q [2] https://www.youtube.com/watch?v=IQofiPqhJ_s https://www.youtube.com/watch?v=IQofiPqhJ_s
- vfinn 6y ago
  Thanks :). I think the resolution to my problem is moving away from infinitesimals. I did learn some calculus, linear algebra, number theory etc. in university, but I did it quite superficially, since it didn't feel that relevant to me at that time. Now I feel different.

by vfinn 6y ago

kosievdmerwe 6y ago

That statement is based on: > First hopefully we can agree that if we have two real numbers x and y that are not equal then we have a number z, such that x < z < y. The easiest example is z = (x + y)/2. It's one of the properties of the reals and rationals that if two numbers aren't equal, then there are infinitely many numbers between them. It doesn't work with the integers, 2 and 3 aren't equal, but there's no integer x, such that 2 < x < 3. So if we say 0.999... != 1 then that means 0.999... < 1 and then that means (0.999... + 1) / 2 is a number different from 1 and different 0.999... but that lies between them. But this is modern mathematics. In the past mathematicians dealt with "infinitesimals", especially in the early days of calculus, but I think they were discarded because they were confusing and also not necessary in favor of limits. This is where I think where some of your confusion is coming from. Infinitesimals don't exist in the real (and therefore rational) number space, but the concept exists for other "weirder" number systems. According to the wikipedia page "This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...)" The clearest resolution to "if you interpret 0.999... as the "biggest possible number below 1"" is that in the reals and rational this concept doesn't exist. There's no biggest number smaller than x and ditto for smallest number bigger than y. (The distinction from the wikipedia definition is the difference between < and <=, <= exists, but < doesn't) It's similar to saying you can't divide by 0. Sure in some cases you can define it, but doing so causes many issues and costs you so much that it's not worth it.[1] Another example is 1 not being prime, there's no reason for it not to be prime, but it's just much more convenient to say arbitrarily that it's not prime.[2] The other thing that might confuse you is a proof by contradiction, I know it certainly confused me the first few times I saw it. I'm happy to help you if this is tripping you up too. > I'm slightly embarrassed I don't know more mathematics, but I'm trying to learn some more... No problem, we all start knowing nothing :) [1] https://www.youtube.com/watch?v=BRRolKTlF6Q https://www.youtube.com/watch?v=BRRolKTlF6Q [2] https://www.youtube.com/watch?v=IQofiPqhJ_s https://www.youtube.com/watch?v=IQofiPqhJ_s

vfinn 6y ago
Thanks :). I think the resolution to my problem is moving away from infinitesimals. I did learn some calculus, linear algebra, number theory etc. in university, but I did it quite superficially, since it didn't feel that relevant to me at that time. Now I feel different.