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Well... except that, if by equality you mean mathematical equality as real numbers, and by "0.9…" you mean the limit of the sequence (0.9, 0.99, 0.999, …), then

Well... except that, if by equality you mean mathematical equality as real numbers, and by "0.9…" you mean the limit of the sequence (0.9, 0.99, 0.999, …), then in fact "0.9…" does equal 1. I'd argue that this is almost certainly what you in fact meant when you wrote "0.9", "1", and "equals". In fact, "0.9999… = 1" is true, and an appreciation of mathematics beyond a purely-syntactic level will make this clear. It will also help clear up puzzles like the OP.

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- ColinWright 12y ago
  The comment to which this is a reply has been deleted. To give this comment (its siblings) context, there is the text: Yes I meant that. It makes no sense, because with that flawed logic you can formulate the following: 1/3 = 0.3... -> 0.3... * 3 = 0.9... -> 0.9.. = 3/3 = 1 If anything, 0.9... APPROACHES 1. Something like 0.9... ~ 1 is a better notation. Consider the expression 0.9999... It's easy to write syntactic nonsense, so it's not obvious that this has any meaning at all. However, if you are going to give that a meaning, then the most reasonable meaning to assign is that it's the limit of the sequence: 0.9, 0.99, 0.999, 0.9999, 0.99999, ... The limit is 1. The terms of the sequence approach 1, yes, but the limit does not approach anything because it doesn't move and it doesn't change. The limit is a value, and that value is 1. Your comment about 0.999... approaching 1 is genuinely wrong, and very, very misleading. It's even more wrong (whatever that means) to say that 0.9999... ~ 1. That's genuinely not a better notation. The interpretation of the meaning of the expression 0.9999... as given above does not allow that the expression 0.9999... should have any value other than 1.[0] [0] Assuming you're working in the standard reals, which form the Archimedean totally ordered complete infinite field. If you're in non-standard analysis then that's something completely different.
- judofyr 12y ago
  I'm not sure how much there is to discuss unless you want to challenge all of the proofs at http://en.wikipedia.org/wiki/0.999 http://en.wikipedia.org/wiki/0.999.
- Tmmrn 12y ago
  Notice that you write 0.9... with dots instead of just writing the number... because it isn't a "normal" number. Of course I agree, it approaches 1 for any number of digits. But infinitely many digits are not a number of digits. Please, define the exact meaning of the ... you use.
- euank 12y ago
  Fyi, "3/3 = 1" is true as well, which you seem to be indicating is not the case. 3/3 does not approach 1; it is exactly 1. In fact, the idea of "approaches" cannot happen with just rational numbers. You need a limit in order for the word "approaches" to be correct. I think you need to read up on math terminology a little (and just math in general) if you want to try and discuss this... though there's no point discussing this because that implies there's more than one side. In the case of .99.., that could be expressed using a limit and thus you could say it approaches 1, though it's much more accurate to say that it equals 1.
  - pdonis 12y ago
    > In fact, the idea of "approaches" cannot happen with just rational numbers. Yes, it can. The sequence 0.9, 0.99, 0.999, ... is entirely composed of rational numbers, and its limit is also a rational number. The issue with rational numbers is that there are sequences which are composed entirely of rational numbers but do not have limits that are rational numbers. An example is 1.4, 1.41, 1.414, ..., i.e., successive truncations of the decimal expansion of sqrt(2). The limit of the sequence is sqrt(2) itself, which is irrational. So in that particular case, the concept of "taking the limit of a sequence" can't be applied within the rational numbers. But that doesn't mean it can't be applied at all; you just need to restrict to sequences that have rational limits.
    - ColinWright 12y ago
      Just a thought ... euank> In fact, the idea of "approaches" cannot euank> happen with just rational numbers. pdonis> Yes, it can. The sequence 0.9, 0.99, 0.999, ... pdonis> is entirely composed of rational numbers, and its pdonis> limit is also a rational number. Everything you write is correct, but I think you're refuting something that wasn't intended. I think euank's comment meant that you can't have a concept of "approaches" when you're just dealing with single values. That's what the "just" is intended to mean, you're "just" dealing with numbers, not with sequences. With that reading your comment is misplaced, although I can completely understand the interpretation you put on it, and were that what had been intended, your comment would have been well stated.
- shawnz 12y ago
  In the off chance you're not trolling, I'll bite. > with that flawed logic you can formulate the following: [...] What you have written there is just the original statement in different words. You can't say "X can't be true, because then X would be true which doesn't make sense". You haven't said anything which conflicts with the original statement. In fact it is true that "0.999..." is exactly equal to 1, which is also exactly equal to 3/3. Those are three different representations of the exact same number, just like how 1/3 is exactly equal to "0.333...". There is no "approaching" involved.
  - baby 12y ago
    > In the off chance you're not trolling, I'll bite. If you want to know something on internet, just say something wrong and someone will come up to explain.

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ColinWright 12y ago
The comment to which this is a reply has been deleted. To give this comment (its siblings) context, there is the text: Yes I meant that. It makes no sense, because with that flawed logic you can formulate the following: 1/3 = 0.3... -> 0.3... * 3 = 0.9... -> 0.9.. = 3/3 = 1 If anything, 0.9... APPROACHES 1. Something like 0.9... ~ 1 is a better notation. Consider the expression 0.9999... It's easy to write syntactic nonsense, so it's not obvious that this has any meaning at all. However, if you are going to give that a meaning, then the most reasonable meaning to assign is that it's the limit of the sequence: 0.9, 0.99, 0.999, 0.9999, 0.99999, ... The limit is 1. The terms of the sequence approach 1, yes, but the limit does not approach anything because it doesn't move and it doesn't change. The limit is a value, and that value is 1. Your comment about 0.999... approaching 1 is genuinely wrong, and very, very misleading. It's even more wrong (whatever that means) to say that 0.9999... ~ 1. That's genuinely not a better notation. The interpretation of the meaning of the expression 0.9999... as given above does not allow that the expression 0.9999... should have any value other than 1.[0] [0] Assuming you're working in the standard reals, which form the Archimedean totally ordered complete infinite field. If you're in non-standard analysis then that's something completely different.
judofyr 12y ago
I'm not sure how much there is to discuss unless you want to challenge all of the proofs at http://en.wikipedia.org/wiki/0.999 http://en.wikipedia.org/wiki/0.999.
Tmmrn 12y ago
Notice that you write 0.9... with dots instead of just writing the number... because it isn't a "normal" number. Of course I agree, it approaches 1 for any number of digits. But infinitely many digits are not a number of digits. Please, define the exact meaning of the ... you use.
euank 12y ago
Fyi, "3/3 = 1" is true as well, which you seem to be indicating is not the case. 3/3 does not approach 1; it is exactly 1. In fact, the idea of "approaches" cannot happen with just rational numbers. You need a limit in order for the word "approaches" to be correct. I think you need to read up on math terminology a little (and just math in general) if you want to try and discuss this... though there's no point discussing this because that implies there's more than one side. In the case of .99.., that could be expressed using a limit and thus you could say it approaches 1, though it's much more accurate to say that it equals 1.
- pdonis 12y ago
  > In fact, the idea of "approaches" cannot happen with just rational numbers. Yes, it can. The sequence 0.9, 0.99, 0.999, ... is entirely composed of rational numbers, and its limit is also a rational number. The issue with rational numbers is that there are sequences which are composed entirely of rational numbers but do not have limits that are rational numbers. An example is 1.4, 1.41, 1.414, ..., i.e., successive truncations of the decimal expansion of sqrt(2). The limit of the sequence is sqrt(2) itself, which is irrational. So in that particular case, the concept of "taking the limit of a sequence" can't be applied within the rational numbers. But that doesn't mean it can't be applied at all; you just need to restrict to sequences that have rational limits.
  - ColinWright 12y ago
    Just a thought ... euank> In fact, the idea of "approaches" cannot euank> happen with just rational numbers. pdonis> Yes, it can. The sequence 0.9, 0.99, 0.999, ... pdonis> is entirely composed of rational numbers, and its pdonis> limit is also a rational number. Everything you write is correct, but I think you're refuting something that wasn't intended. I think euank's comment meant that you can't have a concept of "approaches" when you're just dealing with single values. That's what the "just" is intended to mean, you're "just" dealing with numbers, not with sequences. With that reading your comment is misplaced, although I can completely understand the interpretation you put on it, and were that what had been intended, your comment would have been well stated.
shawnz 12y ago
In the off chance you're not trolling, I'll bite. > with that flawed logic you can formulate the following: [...] What you have written there is just the original statement in different words. You can't say "X can't be true, because then X would be true which doesn't make sense". You haven't said anything which conflicts with the original statement. In fact it is true that "0.999..." is exactly equal to 1, which is also exactly equal to 3/3. Those are three different representations of the exact same number, just like how 1/3 is exactly equal to "0.333...". There is no "approaching" involved.
- baby 12y ago
  > In the off chance you're not trolling, I'll bite. If you want to know something on internet, just say something wrong and someone will come up to explain.