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alofatti
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It is true, 0.9999.... is, in the real system of numbers, equal to 1. There are a lots of proofs of this, some of them using analytic concepts of series, other working directly with some basic properties of the real number.
Frogger: decimal representeations and the real numbers are actually bijective (i.e., they have the same cardinal) even though one of them is complete(the real numbers) and the other not.
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alofatti
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Frogger:
when I talk about the decimal representation I am talking about an element which has this form:
xxxxxxxx.xxxxxxx...........
where x is a digit between 0 and 9 and which could extend to the infinite, I am not talking about rational numbers (which are not biyective to the real numbers).
This decimal representation number system has the same cardinal than the real numbers, there is a mapping between them that is both injective and suryective.
The inyection is easy, it is clearly seen that the real numbers are included in this set.
The surjection is somewhat trickier. It is possible to show that this set includes both this sets:
- the set of numbers which do not end with a repeating 9 (which are easily shown to be bijective to the real numbers)
- the set of numbers which do end with a repeating 9 (which are numerable).
Since c (the cardinal of R) + Aleph_O (the cardinal of a numberable set) equals c, then they are both coordinable between themselves.
A complete set is a set which, in an informal way, does not have successions which converge to "ghost" (i.e., inexistent) points. More formally, every Cauchy sequence does have a limit.
I have said in a post before this that the set of decimal representable numbers are not complete, which is not right. First is necesary to define a distance between points to talk about completeness.
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KrazyHorse
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Macedonia
May 2001
time: 00:22
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quote:
Originally posted by Asher
The reason why there's disagreement is because math is such a stupid field. 
Like it seems the most common approach these days is to use the logic that something like 1/3 is 0.3 repeating, so 3/3 must be 0.9 repeating, but it's also 1, therefore they're equal... |
Asher, have you ever learnt what a limit is?
Take the limit of the sequence 0.9, 0.99, 0.999, ...
By inspection 0.9999999.... is a limit
By going over the definition of the requirements of being a limit, 1 is also a limit.
By a fairly standard proof, if a sequence has a and b as limits, then a=b
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Asher
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Calgary, Alberta
Nov 1999
time: 22:22
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quote:
Originally posted by Frogger
Asher, have you ever learnt what a limit is?
Take the limit of the sequence 0.9, 0.99, 0.999, ...
By inspection 0.9999999.... is a limit
By going over the definition of the requirements of being a limit, 1 is also a limit.
By a fairly standard proof, if a sequence has a and b as limits, then a=b |
Yes yes, I know that you can formally prove it now.
And yes, I've taken Calculus I and II and courses with combinatorial proofs.
I just don't like assuming repeating numbers are anything else other than repeating numbers. It would have helped if I read his post other than just the title, but it's still a silly math rule to get around how math geeks still haven't come up with a better representation of numbers.
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KrazyHorse
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Macedonia
May 2001
time: 00:22
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quote:
Originally posted by alofatti
Frogger:
when I talk about the decimal representation I am talking about an element which has this form:
xxxxxxxx.xxxxxxx...........
where x is a digit between 0 and 9 and which could extend to the infinite, I am not talking about rational numbers (which are not biyective to the real numbers).
This decimal representation number system has the same cardinal than the real numbers, there is a mapping between them that is both injective and suryective.
The inyection is easy, it is clearly seen that the real numbers are included in this set.
The surjection is somewhat trickier. It is possible to show that this set includes both this sets:
- the set of numbers which do not end with a repeating 9 (which are easily shown to be bijective to the real numbers)
- the set of numbers which do end with a repeating 9 (which are numerable).
Since c (the cardinal of R) + Aleph_O (the cardinal of a numberable set) equals c, then they are both coordinable between themselves.
A complete set is a set which, in an informal way, does not have successions which converge to "ghost" (i.e., inexistent) points. More formally, every Cauchy sequence does have a limit.
I have said in a post before this that the set of decimal representable numbers are not complete, which is not right. First is necesary to define a distance between points to talk about completeness. |
You're absolutely and fundamentally wrong here.
Under the standard decimal representation of the real numbers, every real number has a decimal representation (i.e. Decimal->Real is onto). Unfortunately, the representation is not a bijection, as 0.999999999.... and 1 are both mapped to the same real number.
BTW, when you use the word "complete", most people use the word "closed" to represent a set which contains all of its limit points. And the set of reals and the set of decimal reps are both closed. 
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alofatti
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You are not obliged to have o.9999..... equal to 1.
You could very well define your own number system where 0.9999... is not 1 (as well it is correctly defined), though perhaps it would share very little resemblance to the real system.
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alofatti
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Actually, it is interesting to notice that lots of things are studied around the number 1.
For example, the fact that 0 < 1 requieres a formal proof, which is not obvious (but also not that hard, though).
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alofatti
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Quoted from Frogger:
"Under the standard decimal representation of the real numbers, every real number has a decimal representation (i.e. Decimal->Real is onto). Unfortunately, the representation is not a bijection, as 0.999999999.... and 1 are both mapped to the same real number. "
Not necessarily. One could easily define a map between two sets as he wants to. I am not obliged to send 0.9999... to 1. In terms of cardinality, both sets have the same cardinal. There is a bijection between these sets (which does not send 0.999... and 1 to 1 because that would not be inyective).
Quoted from Frogger:
"BTW, when you use the word "complete", most people use the word "closed" to represent a set which contains all of its limit points. And the set of reals and the set of decimal reps are both closed"
Well, in metrics spaces "closed" is weaker than "complete" (i.e., a space which is complete is closed), but you are right: both sets are closed (in fact, they are complete).
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