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November 17th, 2011, 10:44 AM  #1 
Member Joined: Aug 2011 Posts: 85 Thanks: 1  Complex numbers and ordering
It is thought that complex numbers unlike real numbers don't have ordering. But it seems that even though individual complex numbers don't have ordering, sets of complex numbers do. On the basis of their magnitude. So, instead of having a complex line, like the real line, there is complex circle that determines the magnitudes of complex numbers. Of course as, by definition, a circle ,unlike a point, has dimensions, the complex circle differentiates between infinite sets of complex numbers. Is this idea valid? 
November 17th, 2011, 01:14 PM  #2  
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Complex numbers and ordering Quote:
It is not thought, it is so. Take the positive half of the complex plane let us say , meaning , then , but there exists an element with which is impossible for a "positive" element.[/color]  
November 17th, 2011, 01:17 PM  #3 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Complex numbers and ordering
This is trivially valid if you define magnitude in the ordinary way, since real numbers are wellordered.

November 17th, 2011, 01:30 PM  #4 
Member Joined: Aug 2011 Posts: 85 Thanks: 1  Re: Complex numbers and ordering
Is there any other way to define complex magnitude other than using the pythagorean theorem on the complex plane? I mean any nonequivalent way?

November 18th, 2011, 11:38 AM  #5  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Complex numbers and ordering Quote:
 
November 19th, 2011, 03:46 PM  #6 
Member Joined: Aug 2011 Posts: 85 Thanks: 1  Re: Complex numbers and ordering
Yes, but that's not my point. My point is that complex numbers provide ordering in 2D, that of a plane, that could more simply be done through circles, a thing that they do.

November 20th, 2011, 11:25 AM  #7  
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Complex numbers and ordering Quote:
 
November 21st, 2011, 05:36 AM  #8 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Complex numbers and ordering
First, it is NOT true that the complex number cannot be ordered. We can say that a+ bi< c+ di if a< c or, when a= c, b< d. What is true is that the complex numbers are not an "ordered field". That is, there is no way to define an order such that if a< b then a+ c< b+ c and if 0< c, then ac< bc. As for your ordering sets of complex numbers, which of A= {a+ bi 1< a< 2, 1< b< 2} and C= {a+ bi 2< a< 1, 2< b< 2} is the smaller? 
November 21st, 2011, 05:47 AM  #9  
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Complex numbers and ordering Quote:
where did you see that, can you name the book where it is written? Let for example z=a+bi for a,b>0, according to what you are writting you can say that ? Where did you see it?[/color]  
November 21st, 2011, 10:31 PM  #10  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond  Re: Complex numbers and ordering Quote:
 

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