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        riddles >> general problem-solving / chatting / whatever >> The Real Projective Line
        
(Message started by: Sir Col on Sep 23rd, 2007, 9:37am)
Title: The Real Projective Line
Post by Sir Col on Sep 23rd, 2007, 9:37am
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?
Title: Re: The Real Projective Line
Post by towr on Sep 23rd, 2007, 10:25am

on 09/23/07 at 09:37:34, Sir Col wrote:
Does anyone know much about the real projective line extension of the real number system?
http://en.wikipedia.org/wiki/Real_projective_line
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.)
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.
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?
If a is not infinity, sure.
Title: Re: The Real Projective Line
Post by Sir Col on Sep 23rd, 2007, 12:17pm

on 09/23/07 at 10:25:35, towr wrote:
Actually, 0*inf isn't even a valid expression in normal arithmetic, because inf isn't a number there.

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.
Title: Re: The Real Projective Line
Post by Sir Col on Sep 23rd, 2007, 12:24pm
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.  ::)
Title: Re: The Real Projective Line
Post by towr on Sep 23rd, 2007, 12:28pm
How about:

inf = card(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbn.gif)
0 =card(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/emptyset.gif)
inf * 0 = card(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbn.gif x http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/emptyset.gif) = card(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/emptyset.gif) = 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).
Title: Re: The Real Projective Line
Post by Obob on Sep 23rd, 2007, 3:01pm
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.
Title: Re: The Real Projective Line
Post by Barukh on Sep 26th, 2007, 12:23am
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).
Title: Re: The Real Projective Line
Post by Sameer on Sep 26th, 2007, 9:01am

on 09/26/07 at 00:23:42, 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!


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), (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif, http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif)
Title: Re: The Real Projective Line
Post by Grimbal on Sep 26th, 2007, 9:16am
yes


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