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Author Topic:   0.99999~ = 1 ?
Dr Adequate
Member (Idle past 284 days)
Posts: 16113
Joined: 07-20-2006


Message 61 of 237 (543454)
01-18-2010 7:58 AM
Reply to: Message 57 by Straggler
01-18-2010 6:17 AM


Re: Is 0.999R a Whole Number?
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: 09-30-2006


Message 62 of 237 (543460)
01-18-2010 9:13 AM
Reply to: Message 59 by PaulK
01-18-2010 6:38 AM


Presentation and NOT Presentation
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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 Message 59 by PaulK, posted 01-18-2010 6:38 AM PaulK has replied

Replies to this message:
 Message 63 by PaulK, posted 01-18-2010 9:21 AM Straggler has replied

  
PaulK
Member
Posts: 17822
Joined: 01-10-2003
Member Rating: 2.2


Message 63 of 237 (543462)
01-18-2010 9:21 AM
Reply to: Message 62 by Straggler
01-18-2010 9:13 AM


Re: Presentation and NOT Presentation
quote:
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.
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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 Message 62 by Straggler, posted 01-18-2010 9:13 AM Straggler has replied

Replies to this message:
 Message 65 by Straggler, posted 01-18-2010 9:32 AM PaulK has replied

  
Straggler
Member
Posts: 10333
From: London England
Joined: 09-30-2006


Message 64 of 237 (543463)
01-18-2010 9:26 AM
Reply to: Message 60 by Son Goku
01-18-2010 7:41 AM


Re: Infinite sums.
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: 09-30-2006


Message 65 of 237 (543464)
01-18-2010 9:32 AM
Reply to: Message 63 by PaulK
01-18-2010 9:21 AM


0 and Infinitesimally Small - Something and Nothing
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.

This message is a reply to:
 Message 63 by PaulK, posted 01-18-2010 9:21 AM PaulK has replied

Replies to this message:
 Message 66 by PaulK, posted 01-18-2010 9:40 AM Straggler has not replied
 Message 67 by Dr Jack, posted 01-18-2010 10:01 AM Straggler has replied
 Message 68 by Briterican, posted 01-18-2010 10:57 AM Straggler has replied

  
PaulK
Member
Posts: 17822
Joined: 01-10-2003
Member Rating: 2.2


Message 66 of 237 (543467)
01-18-2010 9:40 AM
Reply to: Message 65 by Straggler
01-18-2010 9:32 AM


Re: 0 and Infinitesimally Small - Something and Nothing
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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 Message 65 by Straggler, posted 01-18-2010 9:32 AM Straggler has not replied

  
Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 67 of 237 (543470)
01-18-2010 10:01 AM
Reply to: Message 65 by Straggler
01-18-2010 9:32 AM


Re: 0 and Infinitesimally Small - Something and Nothing
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.

This message is a reply to:
 Message 65 by Straggler, posted 01-18-2010 9:32 AM Straggler has replied

Replies to this message:
 Message 70 by Straggler, posted 01-18-2010 1:31 PM Dr Jack has replied

  
Briterican
Member (Idle past 3948 days)
Posts: 340
Joined: 05-29-2008


Message 68 of 237 (543471)
01-18-2010 10:57 AM
Reply to: Message 65 by Straggler
01-18-2010 9:32 AM


Re: 0 and Infinitesimally Small - Something and Nothing
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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 Message 65 by Straggler, posted 01-18-2010 9:32 AM Straggler has replied

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 Message 69 by Straggler, posted 01-18-2010 1:08 PM Briterican has seen this message but not replied

  
Straggler
Member
Posts: 10333
From: London England
Joined: 09-30-2006


Message 69 of 237 (543476)
01-18-2010 1:08 PM
Reply to: Message 68 by Briterican
01-18-2010 10:57 AM


Re: 0 and Infinitesimally Small - Something and Nothing
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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 Message 68 by Briterican, posted 01-18-2010 10:57 AM Briterican has seen this message but not replied

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Straggler
Member
Posts: 10333
From: London England
Joined: 09-30-2006


Message 70 of 237 (543477)
01-18-2010 1:31 PM
Reply to: Message 67 by Dr Jack
01-18-2010 10:01 AM


Re: 0 and Infinitesimally Small - Something and Nothing
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.

This message is a reply to:
 Message 67 by Dr Jack, posted 01-18-2010 10:01 AM Dr Jack has replied

Replies to this message:
 Message 72 by Dr Jack, posted 01-18-2010 1:49 PM Straggler has replied

  
Straggler
Member
Posts: 10333
From: London England
Joined: 09-30-2006


Message 71 of 237 (543478)
01-18-2010 1:33 PM


Headbanging Lunacy
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.

  
Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 72 of 237 (543479)
01-18-2010 1:49 PM
Reply to: Message 70 by Straggler
01-18-2010 1:31 PM


Re: 0 and Infinitesimally Small - Something and Nothing
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.

This message is a reply to:
 Message 70 by Straggler, posted 01-18-2010 1:31 PM Straggler has replied

Replies to this message:
 Message 73 by Straggler, posted 01-19-2010 5:20 AM Dr Jack has replied
 Message 84 by Apothecus, posted 01-23-2010 6:03 PM Dr Jack has not replied

  
Straggler
Member
Posts: 10333
From: London England
Joined: 09-30-2006


Message 73 of 237 (543550)
01-19-2010 5:20 AM
Reply to: Message 72 by Dr Jack
01-18-2010 1:49 PM


Re: 0 and Infinitesimally Small - Something and Nothing
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?

This message is a reply to:
 Message 72 by Dr Jack, posted 01-18-2010 1:49 PM Dr Jack has replied

Replies to this message:
 Message 75 by Dr Jack, posted 01-19-2010 10:01 AM Straggler has not replied

  
Jumped Up Chimpanzee
Member (Idle past 4942 days)
Posts: 572
From: UK
Joined: 10-22-2009


Message 74 of 237 (543581)
01-19-2010 9:39 AM
Reply to: Message 69 by Straggler
01-18-2010 1:08 PM


Re: 0 and Infinitesimally Small - Something and Nothing
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: 07-14-2003
Member Rating: 8.7


Message 75 of 237 (543582)
01-19-2010 10:01 AM
Reply to: Message 73 by Straggler
01-19-2010 5:20 AM


Re: 0 and Infinitesimally Small - Something and Nothing
Which is exactly why I am not disagreeing with you?
Aye, it was a comment, not an argument

This message is a reply to:
 Message 73 by Straggler, posted 01-19-2010 5:20 AM Straggler has not replied

  
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