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January 24th, 2013, 01:55 PM  #21  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  Re: Complex numbers and ordering Quote:
(3,4) > (2,7) (3,4)  (3,4) > (2,7)  (3,4) (0,0) > (1,3) And this to me is undesirable because the vector on the left has magnitude 0 while the other vector has positive magnitude sqrt(10), so i agree with [color=#0000BF]ZardoZ[/color] that any attempt to define order consistent with ARITHMETIC for the complex numbers is probably doomed.  
January 24th, 2013, 10:41 PM  #22 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Complex numbers and ordering
I think you mixed up dictionary ordering with absolute value ordering. According to dictionary ordering, (0,0) > (1,3) but that doesn't imply  (0,0)  >  (1,3)  is obvious. I am on HallsOfIvy's side since complex numbers can be ordered and such definitions of ordering can be used in practice. 
January 25th, 2013, 01:53 AM  #23 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  Re: Complex numbers and ordering
You're right, my argument is flawed, i realized as soon as i posted, but [color=#0000BF]Weiler's[/color] argument above is sound IMHO. 

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