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You only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude

You only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude that it has to be equivalent to 1.

CogitoCogito 6y ago
> You only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude that it has to be equivalent to 1. Sure you can provide a hand-wavy argument and try to give some intuition if you'd like. That doesn't make it any sort of logical proof though. I guess it depends on what you're after.
- marcosdumay 6y ago
  The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means. You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.
  - CogitoCogito 6y ago
    > The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means. > You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way. I _know_ that. I think I maybe should just bow out of this conversation. I'm apparently incapable of explaining myself in a way that is understandable to people here. I'll consider this my fault. I'll just summarize: I don't think any "proof" that 0.9... = 1 that is only expressed in terms of arithmetic operations and does not make use of limits is legitimate. In other words I claim that a proof like "0.9... = x" means "9.9... = 10x" means "9 = 9x" is illegitimate. Instead of taking "0.9... = 1" on faith it takes "10 x 0.9... = 9.9..." and "9.9... - 0.9... = 1.0... = 1" on faith. There's no proof here. It's just shifting around symbols. Of course there are logical proofs, but they make use of limits/completeness/properties of real numbers explicitly. Feel free to disagree...
    - dmitshur 6y ago
      FWIW, I believe I understand, and agree with the point you’re trying to make. In most circumstances I wouldn’t have a left comment because there’s nothing useful for me to add, just a thumbs up. But I wanted to make an exception this time. It’s worth remembering that most people who understand and agree rarely leave a reply.
    - marcosdumay 6y ago
      Oh, ok. If I understand correctly, you mean that there isn't any standard algorithm for handling a sum with infinite terms if you don't include limits. Well, I do disagree, not with the above statement, but the meaning of "0.9..." itself requires limits, so the discussion can never go anywhere if your assumptions do not include limits.

You only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude that it has to be equivalent to 1.

CogitoCogito 6y ago

> You only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude that it has to be equivalent to 1. Sure you can provide a hand-wavy argument and try to give some intuition if you'd like. That doesn't make it any sort of logical proof though. I guess it depends on what you're after.

marcosdumay 6y ago
The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means. You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.
- CogitoCogito 6y ago
  > The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means. > You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way. I _know_ that. I think I maybe should just bow out of this conversation. I'm apparently incapable of explaining myself in a way that is understandable to people here. I'll consider this my fault. I'll just summarize: I don't think any "proof" that 0.9... = 1 that is only expressed in terms of arithmetic operations and does not make use of limits is legitimate. In other words I claim that a proof like "0.9... = x" means "9.9... = 10x" means "9 = 9x" is illegitimate. Instead of taking "0.9... = 1" on faith it takes "10 x 0.9... = 9.9..." and "9.9... - 0.9... = 1.0... = 1" on faith. There's no proof here. It's just shifting around symbols. Of course there are logical proofs, but they make use of limits/completeness/properties of real numbers explicitly. Feel free to disagree...
  - dmitshur 6y ago
    FWIW, I believe I understand, and agree with the point you’re trying to make. In most circumstances I wouldn’t have a left comment because there’s nothing useful for me to add, just a thumbs up. But I wanted to make an exception this time. It’s worth remembering that most people who understand and agree rarely leave a reply.
  - marcosdumay 6y ago
    Oh, ok. If I understand correctly, you mean that there isn't any standard algorithm for handling a sum with infinite terms if you don't include limits. Well, I do disagree, not with the above statement, but the meaning of "0.9..." itself requires limits, so the discussion can never go anywhere if your assumptions do not include limits.