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Sir Col
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The Real Projective Line
« on: Sep 23rd, 2007, 9:37am » |
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Does anyone know much about the real projective line extension of the real number system?
http://en.wikipedia.org/wiki/Real_projective_line
I'm particularly interested in the "definition", a/0 = inf, where a is non-zero and belongs to the set of reals.
In which case, wouldn't inf*0 be undefined? (It could be equal to any real quantity.)
However, as inf*a = inf, then multiplying any non-zero real by infinity is properly defined. Hence it is multiplying by zero that becomes the new "undefined". It seems that they're just replacing one undefined calculation: a/0, with another: 0*inf.
In fact, is a*0 properly defined?
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towr
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Re: The Real Projective Line
« Reply #1 on: Sep 23rd, 2007, 10:25am » |
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on Sep 23rd, 2007, 9:37am, Sir Col wrote:I think Icarus explained it somewhere once.
Quote: I'm particularly interested in the "definition", a/0 = inf, where a is non-zero and belongs to the set of reals.
In which case, wouldn't inf*0 be undefined? (It could be equal to any real quantity.) |
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According to that wiki-page it is undefined.
Quote:| However, as inf*a = inf, then multiplying any non-zero real by infinity is properly defined. Hence it is multiplying by zero that becomes the new "undefined". It seems that they're just replacing one undefined calculation: a/0, with another: 0*inf. |
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Both are undefined normally, so I don't see how it's a replacement.
Actually, 0*inf isn't even a valid expression in normal arithmetic, because inf isn't a number there.
Quote:| In fact, is a*0 properly defined? |
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If a is not infinity, sure.
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| « Last Edit: Sep 23rd, 2007, 10:27am by towr » |
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Sir Col
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Re: The Real Projective Line
« Reply #2 on: Sep 23rd, 2007, 12:17pm » |
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on Sep 23rd, 2007, 10:25am, towr wrote:| Actually, 0*inf isn't even a valid expression in normal arithmetic, because inf isn't a number there. |
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Duh! I don't know what I was thinking. 
Which reminds me of three interesting "arguments"...
(i) inf * 0 = 0 + 0 + 0 + ... = 0
(ii) As y tends towards infinity x/y tends towards zero. Therefore x / inf = 0. Hence inf * 0 = x; that is, any finite value you want.
(iii) Consider the algebraic identity, x * 1/x = 1, which is defined for finite x. As x tends towards infinity, 1/x tends towards zero, so the limit of x * 1/x = inf * 0 = 1.
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Sir Col
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Re: The Real Projective Line
« Reply #3 on: Sep 23rd, 2007, 12:24pm » |
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And whilst I'm at it I may as well post another old favourite...
As x / y = 1, it follows that inf / inf = 1.
Therefore (inf + inf) / inf = inf / inf = 1
And (inf + inf) / inf = inf / inf + inf / inf = 1 + 1 = 2.
Hence 1 = 2. 
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towr
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Re: The Real Projective Line
« Reply #4 on: Sep 23rd, 2007, 12:28pm » |
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How about:
inf = card( )
0 =card( )
inf * 0 = card( x ) = card() = 0
And of course with these kinds of zeros and this kind of multiplication, unless I'm missing something, 0*a=0 for all a (even if a is infinite).
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Obob
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Re: The Real Projective Line
« Reply #5 on: Sep 23rd, 2007, 3:01pm » |
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Well we do want to be able to do arithmetic with more than just integers, so defining multiplication in terms of cardinalities can't be done in general.
Really, though, either you can define a/0=inf or inf.0=0 without having any serious arithmetic issues. Once you define both, the arithmetic no longer behaves as nicely as you want it to.
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Barukh
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Re: The Real Projective Line
« Reply #6 on: Sep 26th, 2007, 12:23am » |
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I like the geometrical apsect of this. Consider the following statement from Euclidian geometry:
Two lines intersect in exactly one point except when they are parallel.
In projective geometry (where lines are projective) the "except" part can be removed!
This is also true for some nice transformations (like inversion or reciprocation w.r.t. circle).
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Sameer
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Re: The Real Projective Line
« Reply #7 on: Sep 26th, 2007, 9:01am » |
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on Sep 26th, 2007, 12:23am, Barukh wrote:I like the geometrical apsect of this. Consider the following statement from Euclidian geometry:
Two lines intersect in exactly one point except when they are parallel.
In projective geometry (where lines are projective) the "except" part can be removed!
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This is new to me. So does this mean, in projective geometry, the parabola y=x2 and the line y=x have two points of intersection?
Edit: oops I meant 3. (0,0), (1,1), ( , )
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| « Last Edit: Sep 26th, 2007, 9:20am by Sameer » |
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
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Grimbal
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Re: The Real Projective Line
« Reply #8 on: Sep 26th, 2007, 9:16am » |
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yes
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