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November 27th, 2011, 08:03 AM  #11  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Complex numbers and ordering
Are you not clear in what an "order" is? An order for a set is a relation on the set such that if a< b and b< c, then a< c. That is all that is required. If we want a linear order, then we require that, for any two members of the set, a and b, one and only one must be true: a= b, a< b, or b< a. Any set can be given an order. My point was that there is no order for the complex numbers consistent with its field properties. greg1313: Quote:
 
November 27th, 2011, 09:29 AM  #12 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  Re: Complex numbers and ordering
Thanks, I didn't know that.

November 27th, 2011, 11:22 AM  #13  
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 [color=#000000] Quote:
Quote:
 
November 27th, 2011, 11:36 AM  #14  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  Re: Complex numbers and ordering Quote:
 
November 28th, 2011, 10:52 AM  #15 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Complex numbers and ordering
Any book on set theory and some texts on Complex numbers will discuss this. Again, any set can be given a linear order. What is not possible in the complex numbers is to give them an order which is consistent with the field properties.

January 16th, 2012, 12:44 PM  #16 
Member Joined: Aug 2011 Posts: 85 Thanks: 1  Re: Complex numbers and ordering
Is there then a justification for the visual ordering of negative square roots that is the imaginary axis?

January 16th, 2012, 01:39 PM  #17 
Senior Member Joined: Dec 2011 From: Argentina Posts: 216 Thanks: 0  Re: Complex numbers and ordering
I'd contribute with the following. The imaginiary unit is the basis of the complex number. So lets consider the following by definition: Then we have, multiplying by that which can't be true. Take, again by definition: Then multiplying by makes the inequality sign changes and we get Again we get that under the ordinary definition of ordering of numbers, the complex numbers are not ordered. 
January 15th, 2013, 04:18 AM  #18 
Newbie Joined: Jan 2012 Posts: 4 Thanks: 0  Re: Complex numbers and ordering
In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). A totally ordered set is one that rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy* holds. The natural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitude whereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via x+iy < u+iv if and only if x < u or (x = u and y < v), but this is not ordering by magnitude (as HillsofIvy stated) in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute magnitude but are not of "equal nature". *Trichotomy: generally, it can be defined as the property of an order relation on a set X that for any x and y, exactly one of the following holds: x < y, x = y, x > y. 
January 20th, 2013, 09:28 AM  #19  
Senior Member Joined: Aug 2012 Posts: 2,414 Thanks: 755  Re: Complex numbers and ordering Quote:
http://en.wikipedia.org/wiki/Lexicographical_order The point (which has already been made perfectly well by others in this thread) is that the complex numbers can not be made into an ordered field. But they can certainly be ordered. That's because "ordered field" is one thing and order is another. Two different definitions.  
January 24th, 2013, 07:59 AM  #20  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Complex numbers and ordering Quote:
 

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