This is a discussion on Concept of Quantity within the General Discussions forums, part of the Community Boards category; Originally Posted by Mario F. 10 is a number. No, 10 is a representation of the number we call ten ...
No, 10 is a representation of the number we call ten in base ten.Originally Posted by Mario F.
That is true, but 10 = 1 * 10^1 + 0 * 10^0, where ^ denotes exponentation. Effectively, this is what we are saying by the symbol 10, in the context of base ten.
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That's really what this whole thing is driving at, isn't it? The question here is what 0.999... means. The problem we're having has to do with notation, not numbers.
Because there's no fixed, rigorous mathematical definition of what 0.999... actually means in the first place, I'm within my rights to claim that whatever it actually means, it's some quantity which equals 1, and you can claim it doesn't.
I think this whole debate reveals more about the problems in communicating mathematical concepts in a consistent way than it does about math itself.
Code://try //{ if (a) do { f( b); } while(1); else do { f(!b); } while(1); //}
+1
(my message is at least four characters)
Actually, that doesn't seem to be his problem at all. Because in that case his issue with my proof would clearly be that:
That's why I said that was the only debateable part, it's a question on the definition of the "...". Rather he went on to question the following:Code:0.9999... = lim[x -> infinity] 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x
So rather, he seems to question the definition of limits. Which I consider strange because the definition of limits are fairly strongly laid out and supported. And I can't really see how one would decide not to agree with the meaning, because that would CAUSE what you say he's arguing: communicating mathematical concepts.Code:0 = lim[x -> infinity] 10^-x
I'm not sure if there's a well put definition for "...". But I do know there's a fine one for limits.
No. I don't have a problem with:
0.9999... = lim[x -> infinity] (9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x)
And no. I don't have a problem with:
0 = lim[x -> infinity] (10^-x)
I have a problem with this:
It's your leap of faith. It's the conclusion you draw from a limit that I have a problem with; The fact that you treat the equality in the limit the same as you treat the equality in the conclusion of your proof. You don't bring the result of a limit into plain algebra without the equal sign changing meaning also. I've said this numerous, countless times. And here I am saying it again. I think that's enough of me saying this. Agree or disagree, but please do not again misinterpret my beef with your proof. That's bothersome.lim(x -> inf) ( 1*10^-x ) = 0
So:
1 - 0.9999... = 0
1 = 0.9999...
Indeed. I think that makes an excellent wrap up. I have my own idea of why that is so; I defend the current axioms and mathematical language cannot handle these type of numbers and hence the inconsistency in proofs, and hence why new axioms and languages have been devised to treat these Real numbers in the context of infinitesimals and non-terminating decimals.
The current axioms however hold true enough for most mundane purposes. And they don't need to be revised. So, for almost all purposes 0.999... can be 1. I cannot dispute that. Unless there's a specific need for an infinitesimal to have a meaning other than 0 in real numbers (e.g. 1, or 2), why should we bother?
Again, thanks you all for the debate. Loved it. Cheers.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
Ahh, so then your actual problem is not with this:
As that's the logical and easy consequence of the following two:Code:lim(x -> inf) ( 1*10^-x ) = 0 1 - 0.9999... = 0
That's undeniably a consequence of two numbers being equal (and yes, a limit produces a number). If you say you can't use a limit in a function then your problem is, I guess, with this line:Code:0.9999... = lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x ) and 1 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x ) = 0
As the last bit is simply the result of using that line. But as per *definition* of the limits, a limit produces a single real number. So obviously that can be used perfectly well in calculations. So again, if that is your problem you should re-read the definition of limits.Code:1 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
Or is it where I move the 1 inside the limit like this:
So your objection is one of the following if I understand you correctly:Code:lim(x -> inf) ( 1 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
1. A limit "operator" doesn't produce a real value and hence can't be used in calculations. This is plain wrong.
2. (1 - lim[x -> inf] f(x)) is not equal to (lim[x -> inf] ( 1 - f(x) ) )
I have to admit I've never proven (2) for myself, but I am quite sure it's true (and I might bother proving it if that's a point you fail to believe).
EDIT: actually, nevermind...
Last edited by Mario F.; 03-04-2011 at 06:06 PM.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
Is it? Limit theory makes sense when you use it, but if you use it precisely with all the rules. So lets see this expression:
this is equal toCode:lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
end of limit theory/algebra.Code:9*10^-1 + 9*10^-2 + 9*10^-3 + ...
Which means that this is a bad example. If you have
that means that limits are not useful everywhere, they are actually useful only when you have infinity. The point being is that if you are going to use "..." and "->" for me it is just pointless. You have to make everything into a limit and use the specific rules for it in order to get somewhere.Code:lim(x->1)(x) = 1
Concluding, Mario F. is absolutely right to say that 1 = 0.99999.... makes no sense if you are using limits or anything else. Because what would you do is say that you have 1 = lim(x->1)(x). Same thing different words? Duh, that is a reason limits were created weren't they
Well lets say that 10 is the symbol of "ten". Though, if I type 10 in a forum you don't know if I mean the symbol or the meaning, but in any case lets make the difference.No, 10 is a representation of the number we call ten in base ten.
So what is the meaning of 0.9999...? You can say it is something going very close to 1. Or you can say it is a decimal number with its decimal part having infinity 9.
Is there a difference? As day and night.
The first is
Symbol = 0.99999.....
Meaning = something close to 1
The second is
Symbol = 0.99999.....
Meaning = symbol that has infinite number of 9
So you translate a symbol to another symbol. What is the meaning is still not clear!
Think of this
0.99999 = 1
You are saying that the symbol 0.999.... has the meaning of the symbol 1. Which has the meaning of "one". Excellent, you use two symbols for the same thing.
But that doesn't stop you of doing algebra with 0.999...
So if you have
that is fine. In the end you have a symbol "1.999...." which you have to put a meaning. As simple as that. That meaning is NOT a number.Code:0.999999.... + 1 = 1.9999.....
To clarify even more, x is NOT a number in the context of lim(x->1)(x). It is "something that goes close to the number 1". Exactly why you have limits. To differentiate that this is something that contains more than numbers. When lim part is resolved you have again variables that are numbers. So
Code:lim(x->1)(x * y) = y x: no a number y: is a number
I'm very surprised people are saying things aren't numbers, but they are quantities, and quantities are numbers.
As far as I can tell, you can use limits on a sequence and the sequence need not reach its end for there to be a limit. If a sequence had to reach its end, you wouldn't be able to use limits to deal with infinity.Is it? Limit theory makes sense when you use it, but if you use it precisely with all the rules.
What you can disagree on is if you can add various things and get 0.999...
0.9 + 0.09 + 0.009 + 0.0009 + ... = 0.999...
You can, and apparently do disagree with that construction for some reason, which is fine.
Actually, ten is another symbol for the number that we call ten.
This is what I stated about the meaning of 10:
whiteflags' post #129 gives an idea of what I consider to be the meaning of 0.999... Of course, if you accept that 0.999... = 1, then 0.999... also means the number that we call one.
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The first thing you said is, I believe, in contrast with what you wrote here. See, let me repeat this:
Assume the following two are true:
You said that "y" (the result of a limit) is a number. So the above equations are perfectly good equations, agreed? Ignoring whether the proof is wrong or right; but the logic behind those equations is correct, according to what you said, agreed? And do you agree that from those two we can say:Code:0.9999... = lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x ) and 1 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x ) = 0
Assuming those two lines are correct. That's from what you said in your last sentence.Code:1 - 0.9999... = 0
I'm not saying that there's nothing you can disagree with, but I'm curious as to what it is.
Now there are two points you can question in the proof, which are the following two:
1:
2:Code:0.9999... = lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
If you disagree with 1, then we can agree that it is simply a differ on interpretation of "...". If you disagree with 2, I'll do my best to see if I can proof it (which I expect is quite easy).Code:1 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x ) = lim(x -> inf) ( 1 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
Or, of course, if there's something else you disagree with let me know. But I believe the proof is "foolproof" except for those two.
Indeed. The sequence is convergent. It is said the sequence approaches L. Or L is the limit of the sequence. Or more appropriately, the sequence converges to L.
Nowhere, though, does this realization establish the true identity of the sequence. Merely, its L.
For convenience we can safely use L in our algorithms (certainly for almost any use of Reals). But as a means to prove the true identity of a sequence, limits do nothing for you. L is not the indentity of a sequence anywhere else but within the domain of limits.
As a means to prove the identity of 0.999... this is just not good enough. You don't prove that there's a town hidden behind a mountain by increasing the magnification of your binoculars.
EDIT: In fact, if we consider the definition of 0.999... as that of a number that converges to 1 at infinity, it becomes evident that 0.999... is a number that will never be 1. Limits propose to give this number an identity still, for convenience sake. And that's fine. But they are not rigorous proofs of a the identity of a sequence. And this is why one must look at the context of the problem being proposed, before hastily mixing the equality signs of a limit and that of algebra. For this particular purpose of proving that 0.999... equals 1, that mixing is unacceptable.
EDIT2: And before you say "but it is for Reals!", no it is not! Exactly because we had problems with this limitation of Reals, extensions were proposed to include the notion of an infinitesimal into R. There's two solutions: A new number system (Hyperreals) which we ignore since it's not R, although it behaves just like R in ZFC. But alternatively, an extension to the language and the axioms (Internal Set Theory) which introduces new axioms and language in R. You can call it R 2.0. A new and revised R. A better and more complete R. One that stops this nonsense of trying to prove that a number that can never be 1 is 1.
Last edited by Mario F.; 03-05-2011 at 12:50 PM.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
Well, additionally -- and this is where you start agreeing -- it is nice to hold true that 1 = 0.999... because if you don't you have to break all sorts of arithmetic. We go back to other simple proofs involving thirds and such.
I did read the other paper you linked Mario. Trouble is I didn't see much point in bringing up that you couldn't do precise arithmetic on nonterminating decimals. We do anyway, to various degrees of approximation. I guess I felt like the author was throwing the baby out with the bath water, because you could do a simple proof like that in another base where numbers like 1/3 do terminate, and it's a tautology. It's boring. You can't write 0.9 in base 9, but then, bases only change how you express a number, not its quantity.
I suspect where he starts talking about Dedekind cuts is the meat of his problem with the simple proofs. I'd talk about that in detail if I knew, but I don't. He does start constructing new number sets though. That's where I start shrugging my shoulders and agreeing because 0.999... can't be 1 in all number sets.
This isn't important at all.But we don't get to argue about it enough in our own lives.
I originally wanted to turn the discussion in another direction because I found something on purplemath about how such proofs depend on the Axiom of Choice, and whether you accept that or not. But in looking up the Axiom of Choice, I find myself confused as to how it relates. People go out their depth when they argue these things anyway.
whiteflags: "I'm very surprised people are saying things aren't numbers, but they are quantities, and quantities are numbers."
They are not only quantities, that is what I mean. They are quantities with a condition of "being close to another quantity". They are not exact quantities if you like that term. Which makes calling them numbers kind of wrong at my point of view. Isn't exactly the point of calling a quantity a number to give it a specific value? If you have ten apples you can give that quantity a number. If you have "a lot of apples", yes there is a quantity of apples, but you wouldn't give them a number.
If you argue "true, but you have a quantity of apples close to 1000" then still you are not giving them a number, you are comparing them with an imaginary quantity of apples that you could give a number (1000).
But I am not correct with the above unless we clarify the question: is infinity a number? I would say it is fair to call it a quantity but not a number, something like this
So if infinity is not a number, I would say that 0.999... is certainly not a number as it is a sum of infinite numbers.
EVOEx: Despite if I am wrong or right you have to fully finish your equations, because for example:
is true, but you imply that it is equal to 0.Code:1 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x ) = lim(x -> inf) ( 1 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
Or using
is fine ifCode:1 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x ) = 0
is equal to 1, which is the exact point of our disagreement.Code:lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
As far as I can tell both are wrong assumptions. So lets start again more thoroughly
1.
We first solve the "lim" part asCode:0.9999... = lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
which is correct.Code:0.9999... = 9*10^-1 + 9*10^-2 + 9*10^-3 + ...
2.
again solve the lim partCode:1 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
then using the aboveCode:1 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ...
all of this are simple and clear. But I fail to see how the above actually equals to zero.Code:1 - 0.9999..
The whole point is why does 0.99999.... or 1 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... or any other form to point it using either "..." or "Σ" or any other symbol meaning a sum of infinite parts is equal to 1.
The code
is again perfectly correct.Code:0.9 + 0.09 + 0.009 + 0.0009 + ... = 0.999...
For me
but not equal. And I believe that limits exactly point that out. CompareCode:0.999.... -> 1
the whole point is that the first doesn't make sense in algebra that is why you have to use a limit to make it more clear. What you are doing withCode:x = oo limx(x->oo)(x)
defines that purpose**. If that is correct why would you have the symbol "->" at all???Code:0.999.... = 1
My point is always that using "lim" or "->" to prove the classical "0.999.... = 1" is not the correct method. There can sure be other methods, but using limits is more to solve the problem by adding the symbols "lim" and "->" exactly because the symbol "=" is not enough....
Lasersight: "Ten is another symbol for the number we call ten"
And calling ten is again another symbol, no difference if your write or say it. In mathematics there is not necessarily a practical or physical meaning on symbols...
** I use a lot of that term, but I am now doubting it is correct. I think I am actually saying the opposite of what I mean so please give me the correct phrase
Last edited by C_ntua; 03-05-2011 at 02:09 PM.
I wouldn't make this an English argument. You try to make a distinction for quantity and number and I see those words as synonyms in most contexts, and in all the ways you mention, so it's not really productive.They are not only quantities, that is what I mean. They are quantities with a condition of "being close to another quantity". They are not exact quantities if you like that term. [...] So if infinity is not a number, I would say that 0.999... is certainly not a number as it is a sum of infinite numbers.
I could also argue that it is completely wrong to say what you said at all since Euler's number, for example, is a real number, a transcendental number. We don't know what e precisely is, but we use it as a number and it is cancellable. Even if Euler's number is not a number for whatever reason, if we didn't use it in some formulas, we would be lost to other real world ideas, like compound interest.
You later say 0.999... is a sum of other numbers, but is the answer (the sum) not a number just because you would be constructing it forever in such a way? What else would it be? Mario puts it best that 0.999... could be a number always <1, but in proofs the difference is always so small as to be indistinguishable from 1.
We accept the concept of an infinitesimal as a number so small it is immeasurable. So there really is no problem with "what makes a number". In large part, arguments come from whether we rigorously address the concept of infinity in any proof that 0.999... = 1.
I'm done trying to prove whatever in the thread, because I'm not about to change my mind. (I want three thirds to be one no matter how I express it.) At the same time, I don't think you want to make a linguistic argument out of this, either.
Last edited by whiteflags; 03-05-2011 at 03:19 PM.
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