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 [split] 0.999... equals 1
2nd February 2012, 09:30
Post: #321
 CliveStaples Senior Member Religious Views: Protestant Christian Posts: 532 Joined: 30th January 2012 Reputation: 5
RE: [split] 0.999... equals 1
(12th December 2011 06:22)houseofcantor Wrote:

But it's mathematical chicanery. The mind always wants to make numbers real. Here we go: 5.13 x 10^61. The number of the universe; well, age in Planck time. But the difference between finite and infinite is not a number, it is a perspective. One cannot write 9/10^(n) on to every n quark in the universe to add to to one, just ain't done; so whether or not it really sums to one is a metaphysical concern.

If you want to deal only with finite quantities, then just use some real analysis.

Take x = .999...

If we agree that x is indeed a real number, then by trichotomy either x > 1, x < 1, or x = 1. Clearly x > 1 is false; thus x < 1 or x = 1.

Now, suppose x < 1. Then |1 - x| must be positive; call it c. But for any positive c you give, I can show that c must be smaller. Thus c cannot be positive. Thus x < 1 is false.

Hence x = 1.
“The truth of our faith becomes a matter of ridicule among the infidels if any Catholic, not gifted with the necessary scientific learning, presents as dogma what scientific scrutiny shows to be false.”
2nd February 2012, 13:57 (This post was last modified: 2nd February 2012 15:24 by Categories+Sheaves.)
Post: #322
 Categories+Sheaves Subgame Perfect Pony Religious Views: ZFC Posts: 164 Joined: 25th January 2012 Reputation: 4
RE: [split] 0.999... equals 1

See: Wikipeida or Ed Nelson's introduction to nonstandard analysis for systems of analysis where this sort of thing need not happen.
There systems do treat infinitesimals as genuine, nonzero quantities. Here we have that .999... ~= 1 i.e. they are infinitely close, but we do not insist that .999... = 1.

Now, I'm a big fan of the standard real numbers, and I strongly prefer using them to the other systems out there. But the other systems are still out there. We do get this equivalence of .999... and 1 once we accept all that business with cauchy sequences and epsilon-delta. But definitely not before then.
So these philosophers were all like, "That Kant apply universally!" And then these mathematicians were all like, "Oh yes it Kan!"
2nd February 2012, 14:58
Post: #323
 CliveStaples Senior Member Religious Views: Protestant Christian Posts: 532 Joined: 30th January 2012 Reputation: 5
RE: [split] 0.999... equals 1
(2nd February 2012 13:57)Categories+Sheaves Wrote:  I'mma drop some links

See: Wikipeida or [url=http://www.math.princeton.edu/~nelson/books/1.pdf]Ed Nelson's introduction to nonstandard analysis[/i] for systems of analysis where this sort of thing need not happen.
There systems do treat infinitesimals as genuine, nonzero quantities. Here we have that .999... ~= 1 i.e. they are infinitely close, but we do not insist that .999... = 1.

Now, I'm a big fan of the standard real numbers, and I strongly prefer using them to the other systems out there. But the other systems are still out there. We do get this equivalence of .999... and 1 once we accept all that business with cauchy sequences and epsilon-delta. But definitely not before then.

Well, of course I assume we're talking about the standard presentation of the real numbers, which are an Archimedean field.

(But I suspect that even in the hyperreals, .999... = 1.)
“The truth of our faith becomes a matter of ridicule among the infidels if any Catholic, not gifted with the necessary scientific learning, presents as dogma what scientific scrutiny shows to be false.”
25th February 2012, 07:28 (This post was last modified: 25th February 2012 07:29 by Categories+Sheaves.)
Post: #324
 Categories+Sheaves Subgame Perfect Pony Religious Views: ZFC Posts: 164 Joined: 25th January 2012 Reputation: 4
RE: [split] 0.999... equals 1
(2nd February 2012 14:58)CliveStaples Wrote:  (But I suspect that even in the hyperreals, .999... = 1.)

Mmk. Goldblatt's Lectures on the Hyperreals came in the mail this week...

Without getting into any business with ultrafilters, it turns out that there are too many elements in *R that I would have wanted .999... to be.
The issue is that we don't know whether .999... is referring to the equivalence class of [.9, .99, .999, ...] or [.99, .9999, .999999, ...] etc. (yes, these are two distinct elements of *R, and the first is less than the second).

So if '.999...' is a well-defined symbol for a single thing, we'd have to treat it as an element of R and not *R - R...

So: Touché. If our writing '.999...' is to stand any chance of being intelligible, .999... = 1
So these philosophers were all like, "That Kant apply universally!" And then these mathematicians were all like, "Oh yes it Kan!"
25th February 2012, 09:11
Post: #325
 Tiberius Question Everything. Religious Views: Shell B Posts: 10873 Joined: 25th August 2008 Reputation: 91
RE: [split] 0.999... equals 1
Another win for mathematics. :-)
20th February 2013, 03:32
Post: #326
 john_gabriel Junior Member Religious Views: Agnostic Posts: 1 Joined: 20th February 2013 Reputation: 0
RE: [split] 0.999... equals 1
All the proofs in favour of the equality 0.999... = 1 are debunked in the following article:

<snip!>

Allow me to debunk the most common fallacious proofs:

1. 1/3 = 0.333....
Well, a little known fact is that 1/3 is NOT equal to 0.333...
See Pages 33-36 of article.

2. There is no number between 0.999... and 1.
True. This is due to the fact that 0.999... is not a well-defined number.
Article explains more.

3. x = 1(0.999...)
10x= 10 (0.999...)
9x = 9(0.999...)
x = 0.999...

Wha?! Yes. If you don't do anything stupid, like try to multiply a quasi-number object by 10, you can predict the output of the algorithm exactly.
Arithmetic is designed to work with well-defined mathematical objects called the rational numbers.

For more on this, see pages 12-18.

To learn much more, read the entire article. Do visit my New Calculus site for the first rigorous formulation of calculus in history!

 (20th February 2013 at 16:45) Kayenneh edited this post for the following reason: Please establish your presence here before advertising your own material.
20th February 2013, 09:18
Post: #327
 Aractus Who? Religious Views: Anglican Posts: 1616 Joined: 12th October 2012 Reputation: 17
RE: [split] 0.999... equals 1
Firstly welcome to the forums.

Secondly, you're full of shit.
"That's disgusting. There were clean athletes out there that have had their whole careers ruined by people like Lance Armstrong who just bended thoughts to fit their circumstances. He didn't look up cheating because he wanted to stop, he wanted to justify what he was doing and to keep that continuing on." - Nicole Cooke
20th February 2013, 09:30
Post: #328
 Tiberius Question Everything. Religious Views: Shell B Posts: 10873 Joined: 25th August 2008 Reputation: 91
RE: [split] 0.999... equals 1
I quickly skimmed over the first few pages of your article. I stopped when I came across a sentence blaming an apparent confusion in set theory on the fact that the father of set theory was a "bipolar Jew".
20th February 2013, 10:18
Post: #329
 Dee Dee Ramone Posting Freak Religious Views: Apes Posts: 805 Joined: 20th November 2012 Reputation: 11
RE: [split] 0.999... equals 1
Bi = 2... I don't see the problem :-)
20th February 2013, 10:52
Post: #330
 Aractus Who? Religious Views: Anglican Posts: 1616 Joined: 12th October 2012 Reputation: 17
RE: [split] 0.999... equals 1
(20th February 2013 09:30)Tiberius Wrote:  I quickly skimmed over the first few pages of your article. I stopped when I came across a sentence blaming an apparent confusion in set theory on the fact that the father of set theory was a "bipolar Jew".
ROFL! I stopped when john_gabriel couldn't even present an argument without referencing this mythical material in a PDF (in other words, I didn't bother with the PDF).
"That's disgusting. There were clean athletes out there that have had their whole careers ruined by people like Lance Armstrong who just bended thoughts to fit their circumstances. He didn't look up cheating because he wanted to stop, he wanted to justify what he was doing and to keep that continuing on." - Nicole Cooke
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