Actually, ten is another symbol for the number that we call ten.Originally Posted by C_ntua
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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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.
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 04:19 PM.
No, in such a long topics everything that becomes a "linguistic argument" can be fairly dropped and we can focus on the more important part.
So I would stick on "0.9999.... = 1" issue.
The forum ate the first attempt to edit and the first attempt at posting.
Probably. Possibly from the standpoint or formalism versus intuition, the notion or limits, and the nature of infinitesimals?I believe there is the confusion?
Meh. Anyway...
Soma
Indeed. The issue isn't on the formal validity of the so-called proof. There is really no much point arguing over that. The problem on the conclusion that is being drawn from that limit. This is valid in calculus, is often also accepted that the result of a limit can transit into algebra, but for this particular case in which the identity of a sequence is trying to be established, this simply cannot be used as proof. It's circular reasoning. It's of no use as a means to form proof. For that matter, I'd take a fraction-based proof any day before I can accept the result of a limit that is then transported into algebra.
As for infinitesimals... it's a mess. Real numbers cannot have them because the language wasn't properly constructed (or wasn't intended?) to accept them. It lacks resolution. Yet, Cauchy sequences which lie deeply ingrained in the concept of limits and sequences, depend on the concept of an infinitesimal. There's inconsistencies in the language and the axioms. It's no wonder these debates can then emerge and pretty much go on forever.
My real beef isn't so much about the idea that 0.999... is 1. I can live with that; Have been saying so for 4 pages in this thread. The world has live with that! My real beef is with the fact this notion actually developed a group of partisans who pretend to prove what cannot be proven in R. No proof can exist in R that establishes the identity of a rational number with a non terminating sequence, no more than the impossibility of establishing the identity of an irrational number. Still wikipedia articles are written on the matter in condescending tones, as if any of the so-called proofs had any mathematical validity whatsoever. Pretty amazing.
I'd rather prefer some humility. Something like "0.999... is 1 in R. It cannot be proven. But this is why it must be [insert formulaic explanation the likes of which we have seen on this thread]"
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.
First: Σ(x = 0, x->infinity) 9*10^-x = 9.999...
Second: .999... = Σ(x = 0, x->infinity) .9*10^-x = .9/(1-.1) = 1
Again, sticking only on the mathematical part, I wouldn't agree on the second. Though of course all my posts are completely pointless, would have been easier if I actually opened a math book rather than thinking and "arguing" but I will skip this part....
So if the second is believed to be true you can elaborate a bit...
I also can live with "0.999.... = 1", but nevertheless if it can be proven and how it can be proven are interesting parts for me.
All limit theory says, in this case, is that as the number of 9's increases then the value on the left hand side of the expression continually gets closer to 1. Or, to put it another way, we can get as close as we like to 1 by adding more 9's.
Conventionally mathematicians state that that the values will be equal in the limiting case where the number of 9's is truly infinite.
This thread appears to be taking the rather novel stance that conclusions derived using the theory of limits, and associated conventions, are invalid. According to that stance, the result is unprovable in the context of this thread.
Yes, but you are adding every time a smaller amount. So in the end you are adding something very close to 0. Which in the case of a truly infinite amount of 9s that will imply that you are actually adding 0 thus you will never get to 1. In other words the amount you are adding can get closer and closer to 0 and in the limiting case it will equal to 0.
When it is not specified otherwise, R is implied. In *R, .999... may not equal 1, but who considers *R when it is not specified? No one.Something like "0.999... is 1 in R. It cannot be proven. But this is why it must be [insert formulaic explanation the likes of which we have seen on this thread]"
It is true that lim (x->infinity) of 10^-x is zero. And so is 9 * {lim (x->infinity) of 10^-x} ..... 9 times zero is still zero, whether that zero is a limiting case or not.
Or, each time a digit 9 is added, it is adding less than was added by the preceding digit 9, because it is being divided by a larger power of 10.
Very close, yes, but never equal. When dealing with limits, the difference (epsilon) approaches but never equals 0. If it does, the limit is undefined.
There is no "limiting case," you simply add increasingly smaller amounts forever.
This was an interesting thread, but I have to say there is quite a bit of confusion involving limits and their definition and rationale, especially about the "epsilon," or what some have been calling the "infintesimal." E > 0 always. NOT 0.
Some would be better served reading a calculus textbook (limits and infinite series) than attempting to disprove the calculus. I guess the problem is one of viewpoint; whether or not a (finite) limit produces a single, unique real number or not, and whether this number cannot be used like any other real number for some strange reason.
In other words, its a debate about the validity of the definition of a (finite) limit whether or not 0.999... = 1.
Overall, I have to say that reading this thread has not convinced me that the definition of a limit is flawed.
Last edited by MacNilly; 03-08-2011 at 02:43 PM.
I was talking about the formalism of the implied limit used in the proof.The issue isn't on the formal validity of the so-called proof.
*shrug*
Same difference I guess.
Soma
I do not agree since the claim that this holds "all the way towards infinity" is suspect as you are extrapolating from examples that do not actually consider this.
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