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Member (Idle past 2324 days) Posts: 2870 From: Limburg, The Netherlands Joined: |
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Author | Topic: 0.99999~ = 1 ? | |||||||||||||||||||||||||||
Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Surely the difference between 0.999R and 1 is practically non-existant and philosophically massive?
It is the difference between claiming complete certainty (for example) and always allowing for the possibility of that which is unexpected (no matter how likely or unlikely). It is the difference between an obtainable destination and that which can never exist or be obtained in reality. I am no mathematician. But surely the difference between 1 and NOT 1 is as significant as ever. No matter what the NOT 1 may be? Edited by Straggler, : No reason given.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Well It appears that the genaral consensus amongst those who are more mathematically literate than myself is that I am wrong.
In which case I stand corrected. And in which case I will need to find a new nomenclature for expressing all but certain without the philosophical possibility of complete and absolute certainty. But just to be clear is it false to say that 0.999R < 1?
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
This is a genuine question. As it appears that my understanding up until now has been incomplete (to put it politely - or just wrong if we are to be more blunt.
But is it wrong to say that 0.999R <1?
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Nope, they're completely the same, philosophically and otherwise. Well up until now I have (rightly or wrongly) been expressing the difference between probablistic certainty or otherwise as the differnce between 1 and NOT 1. Where (in my mind at least) 0.999R is not equal to 1. It appears I am need to change my nomenclature. But I am not sure what I should use instead?
Nope. There is no uncertainty, 0.9999~ is 1. In fact all decimal representations are infinite sequences, it's just some of them finish in an infinite number of 0s. This is not, as it may sound, a cute aphorism but actually fundamental to the construction of the real numbers. Real numbers are limits of inifnite sequences. Well I get what you are saying. I think. But it still seems counter-intuitive that 1/0.999R is entirely equal to 1/1. For example.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Double post. See below.
Edited by Straggler, : No reason given.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
I'm afraid that it is. Yeah I am starting to get that despite inbuilt resistance.
Unless you can show that 1 - 0.999R is greater than zero. Can it be shown mathematically that 1 - 0.999R is equal to zero?
(Hint: What is the decimal expansion of 1 - 0.999R. How far do you have to go to find a non-zero digit ?) One step further down the infinite chain than you need to go to make 0.999R the same as 1?
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Yes, since they are the same. Fair enough. Is there somewhere I can see the mathematical proof of that?
How many steps does an infinte chain contain then? And what is infinity + 1? Well is infinity + 1 > infinity. I guess not. But even in my fairly limited context of undergraduate level physics (as opposed to pure or higher level maths) the concept of more rapidly approaching infinity and thus "degrees of infinity" has arisen. At least in some sort of conceptual principle. So even if I am taking the role of the idiotic punchbag for the more mathematically literate here I do so on the basis of asking whether things are as obvious as you seem to be claiming?
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Yes we can show that there is no number between it and zero, for instance (you can get that from the fact that there is no number between 1 and 0.999R) Put like that - I guess so. But does that mean that infinity squared is the same as infinity to the power of 10? For example.
Nice attempt to turn it around. Well I had to try.
However, would you agree that 1 - 0.999R is zero to an infinite precision? Yes. But once we start invoking infinity all sorts of mad sounding stuff is mathematically true in ways that mathematicians I have spoken to never sound wholly convinced by themselves in ways that are well beyond me to know why they have such arguments. Maybe I am extrapoloting too far. Maybe every mathematician would say that 1=0.999R exactly and without any reservation of any sort. Without any debatable notion of infinity being involved. I am just not knowledgeable enough to say. But it still strikes me that once you need to invoke infinity nothing is as cut and dried as seems to be being asserted here. Edited by Straggler, : No reason given.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
More than infinity? How's that even possible? Screwed if I know. But then degrees of infinity is no less mad and that is a genuine mathematical concept.
Now, how many rooms does the hotel have? Still an infinite amount. Well once we invoke infinity we can make all sorts of stuff sound insane. Wich is kinda the problem with invoking infinities. They are not very practically helpful.
Ah yes, there are more then one infinity. So I am told.
Strag writes: So even if I am taking the role of the idiotic punchbag for the more mathematically literate here I do so on the basis of asking whether things are as obvious as you seem to be claiming? Apparently, to math guys, they are. In my limited experience maths guys can argue about the nature and application of infinity infinitely.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
DA writes: False as false can be. Well I accept what you say. And when you guys explain it does all kinda make sense. But there is still something that seems intuitively wrong about the whole thing. For example is it true to say that 0.999R is a whole number?
I shall give the proof of the statement above, because it's one of my favorite proofs in mathematics. Thanks for this. That does actually make sense. Even to me.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
I get all the arguments for 0.999R being entirely equalt to 1. But it still seems "wrong" that 0.999R is a whole number. Surely I am not alone in this intuitive feeling? Otherwise it wouldn't even be a topic worth highlighting.
Of course, this also means that 6.999R ≡ 7.0 Oh God. Please. Let's not go there. You know what I mean about the inherent impossibility of certainty in evidence based arguments just as well as I do.
Bertie Russel writes:
"To my mind the essential thing is that one should base one's arguments upon the kind of grounds that are accepted in science, and one should not regard anything that one accepts as quite certain, but only as probable in a greater or a less degree. Not to be absolutely certain is, I think, one of the essential things in rationality".
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Isn't that just a matter of presentation? I don't think so. I think it is because the human brian (well mine at least) cannot cope with infinity. And for this to make sense we need to think of 0.99999999999 - to infinity. If I were to say that the difference between 1 and 0.999R was 0.00R1 I would be talking mathematical nonsense. I don't dispute that at all. But I still think pretty much everyone here would have a vaguely instinctive understanding of the cencept I am trying to convey. So I think it is more than just presentation. I think it is our inabilioty to really conceptualise infinity that lies at the heart of my self proclaimed unease.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
I think the lack of intuitivity comes about because we need to think in terms of infinity.
The concept of infinity doesn't come naturally. What's more if we are going to accept infinity as a reasonable concept then it also intuitively seems no more or less reasonable to accept the concept of infinitesimal. In which case saying that: 1 > 0.999... by an infinitesimal amount sounds more intuitively reasonable than saying 1=0.999R
(b)Accept that the use of the infinite in its construction is valid and attempt to give a meaning to that construction. The only meaning that makes sense and agrees with the mathematics we already know is the one I've given above and under that meaning it is 1. Yep I accept that. I am simply arguing out of bloody minded obstinacy at this point. Not because I think I have a mathematical case for refuting anything being said here. But it is interesrting looking at ones own thought processes and trying to work out why something that is so logically provable seems intuitively so wrong.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
But if you accept that 0.999R is just another way of writing 1 it IS a matter of presentation. Well sure. Except that it still seems like 0.999R should be infinitesimally less than 1. Which I suppose it is. If by infinitesimally small we mean tending to 0. Which brings us to distinguishing between 0 and something that is infinitesimally small. Which brings us to distinguishing between something and nothing. Which is where I think the whole intuition things takes over and gets things wrong.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
That's how my brain wanted to approach the problem Well it's nice not to be the only lunatic in the asylum
1/3 + 1/3 + 1/3 = 1 1/3 = .333R .333R + .333R + .333R = .999R therefore .999R = 1 Precisely and exactly equal. Yeah I figured that one out for myself. But it still feels wrong don't it?
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