My Math Forum  

Go Back   My Math Forum > College Math Forum > Complex Analysis

Complex Analysis Complex Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
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:
Isn't that like saying (3, 4) is greater than (2, 7)?
Yes, it is. And that is a perfectly valid order on .
HallsofIvy is offline  
 
November 27th, 2011, 09:29 AM   #12
Global Moderator
 
greg1313's Avatar
 
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.
greg1313 is offline  
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:
Originally Posted by greg1313
Are you not clear in what an "order" is?
Since you are a specialist in "ordering", you must be familiar with the fact that in the complex plane there is no natural ordering for complex numbers.

Quote:
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.
According to your saying, we can claim that a complex number is greater or lesser than another one, still haven't replied to me, in which book did you read this? I am curious to know, unless you invent these things by your own. If it is written in a book of mathematics, I guess that the author is a kind of "magician", reminds me of the very famous magician Harry Houdini.[/color]
ZardoZ is offline  
November 27th, 2011, 11:36 AM   #14
Global Moderator
 
greg1313's Avatar
 
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:
Originally Posted by ZardoZ
Quote:
Originally Posted by greg1313
Are you not clear in what an "order" is?
I didn't say that.
greg1313 is offline  
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.
HallsofIvy is offline  
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?
Eureka is offline  
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.
Weiler is offline  
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.
Sawyier is offline  
January 20th, 2013, 09:28 AM   #19
Senior Member
 
Joined: Aug 2012

Posts: 2,414
Thanks: 755

Re: Complex numbers and ordering

Quote:
Originally Posted by ZardoZ
According to your saying, we can claim that a complex number is greater or lesser than another one, still haven't replied to me, in which book did you read this? I am curious to know, unless you invent these things by your own. If it is written in a book of mathematics, I guess that the author is a kind of "magician", reminds me of the very famous magician Harry Houdini.
This is called the lexicographic ordering.

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.
Maschke is offline  
January 24th, 2013, 07:59 AM   #20
Math Team
 
Joined: Sep 2007

Posts: 2,409
Thanks: 6

Re: Complex numbers and ordering

Quote:
Originally Posted by The Chaz
This is trivially valid if you define magnitude in the ordinary way, since real numbers are well-ordered.
Strictly speaking you should say "linearly ordered". A set is said to be "well ordered" if every subset contains a smallest member.
HallsofIvy is offline  
Reply

  My Math Forum > College Math Forum > Complex Analysis

Tags
complex, numbers, ordering



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Can complex numbers (and properties of complex numbers) be.. jonas Complex Analysis 2 October 13th, 2014 03:03 PM
Complex complex numbers Tutu Algebra 11 June 26th, 2012 01:36 PM
complex numbers Tutu Algebra 12 June 25th, 2012 02:29 PM
complex numbers fe phi fo Complex Analysis 4 June 10th, 2012 04:50 AM
Am I doing this right ? complex numbers mathslog Algebra 2 May 6th, 2012 08:32 AM





Copyright © 2019 My Math Forum. All rights reserved.