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Member (Idle past 2295 days) Posts: 2870 From: Limburg, The Netherlands Joined: |
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Author | Topic: 0.99999~ = 1 ? | |||||||||||||||||||||||||||
Dr Adequate Member (Idle past 284 days) Posts: 16113 Joined: |
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? Yes. Specifically, it's the number 1. Edited by Dr Adequate, : No reason given.
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Straggler Member 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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PaulK Member Posts: 17822 Joined: Member Rating: 2.2 |
quote: I would say that human intuition can't handle it (with maybe a few rare exceptions). But your unease is the product of your intuition. Which is why when you see 0.999R you think of it as something different from and less than 1. But if you accept that 0.999R is just another way of writing 1 it IS a matter of presentation. Your intuition wouldn't rebel against "1 = 1" or "1 is a whole number", you'd just regard them as trivial and obvious truths.
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Straggler Member 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 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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PaulK Member Posts: 17822 Joined: Member Rating: 2.2 |
That's because infinity is weird.
Remember that we had to redefine equality to do integration ? That's because integration is - effectively - summing up an infinite number of infinitesimals. But if we use the definition that lets us work with infinitesimals, then infinitesimals - or the sum of any finite number of infinitesimals equal zero. And in that case we are still left 0.999R = 1.
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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.7 |
The difference isn't infinitesimally small, it's 0.
0.9999999~ is the limit of 0 + 9/10 + 9/100 + 9/1000 + 9/10000 + .... The limit. And the limit of that series is 1, not something infinitesimally different from one.
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Briterican Member (Idle past 3948 days) Posts: 340 Joined: |
straggler writes: 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. That's how my brain wanted to approach the problem, but several members posted this very useful way of looking at it... 1/3 + 1/3 + 1/3 = 1 1/3 = .333R .333R + .333R + .333R = .999R therefore .999R = 1 Precisely and exactly equal. Edited by Briterican, : No reason given.
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Straggler Member 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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Straggler Member Posts: 10333 From: London England Joined: |
The difference isn't infinitesimally small, it's 0. Indeed. It approaches 1 and is 1 at the limit. No disagreement there.
Strag writes: 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. 0.9999999~ is the limit of 0 + 9/10 + 9/100 + 9/1000 + 9/10000 + .... The limit. And the limit of that series is 1, not something infinitesimally different from one. Isn't that what I said? That at the limit the difference between 1 and 0.999R is 0.
Strag writes: 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. I am not arguing with your maths here. I am trying to explain why it still feels intuitively wrong.
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Straggler Member Posts: 10333 From: London England Joined: |
In my time at EvC there have been some truly head banging conversations. Ones that have gone down in forum folklore as examples of brazen lunacy. One about the center of the surface of a sphere springs to mind. As does one by Buz about a non-bending steel bar in curved space-time.
At this point I feel that this one might be on the verge of going down in history as the Straggler says 1 doesn’t equal 1 thread.
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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.7 |
Isn't that what I said? That at the limit the difference between 1 and 0.999R is 0. Yes, but the point is that 0.99999~ is a limit. There is no 0.9999~ that isn't a limit. All real numbers are limits, that's how they're constructed.
I am trying to explain why it still feels intuitively wrong. Oh, sure, it feels intuitively wrong, no doubt. But as in higher level physics, intuition is a shoddy guide to higher maths.
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Straggler Member Posts: 10333 From: London England Joined: |
Oh, sure, it feels intuitively wrong, no doubt. But as in higher level physics, intuition is a shoddy guide to higher maths. Which is exactly why I am not disagreeing with you?
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Jumped Up Chimpanzee Member (Idle past 4942 days) Posts: 572 From: UK Joined: |
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? As a non-mathematician, the above explanation is interesting as it seems to indicate something I'd never considered: that the decimal system is not always adequate (or, indeed, 100% accurate). Expressing 3 thirds as equalling one is perfectly expressed by the use of fractions (1/3 + 1/3 + 1/3) but is not perfectly expressed by the recurring decimals. I guess that's why it feels intuitively wrong. The fact that you have to state that the decimal numbers are recurring is an indication in itself that it's an inadequate system because you can never perfectly express a third in decimals, unless I suppose you are using base 3, or a multiple thereof, but then you'll have problems expressing other fractions.
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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.7 |
Which is exactly why I am not disagreeing with you? Aye, it was a comment, not an argument
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