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Other people have pointed out that induction never makes the jump from a finite number of 9s to an infinite number of 9s. I feel the easiest "proof" is a proof

Other people have pointed out that induction never makes the jump from a finite number of 9s to an infinite number of 9s. I feel the easiest "proof" is a proof by contradiction. 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. If 0.999...!= 1 then there must exist a number A, such that 0.999... < A < 1. Now since 0 < A < 1 (as 0 < 0.999...) it should be easy to see that A's decimal expansion is of the form 0.abcdef... . Since, A != 0.999... one of the digits in the decimal expansion of A has to be something other than a 9. For instance, we might have A = 0.99998999... . However, this would mean A < 0.999... as all digits other than 9 are smaller than 9 [1], but this contradicts our initial assumption that 0.999... < A and since we're dealing with strict inequalities both can't be true at the same time! Thus no such A should exists and thus 0.999... = 1. Now this isn't a rigorous proof, the thing that makes me most uncomfortable is the bit that I state A < 0.999..., but I'm uncomfortable because A might have multiple decimal expansions and I don't know how the algorithm in [1] interacts with that, however, if someone quibbles about that bit of the proof for that reason I feel they should have already accepted that 0.999... = 1 via another rigorous proof. [1] If this is not clear think about how you would compare two decimal expansions to see if one is smaller than the other. You go through every digit until you find one that is different between the two numbers and then you compare those.

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.

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.