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January 24th, 2013, 01:55 PM   #21
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Re: Complex numbers and ordering

Quote:
Originally Posted by greg1313
Quote:
Originally Posted by HallsofIvy
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.
Isn't that like saying (3, 4) is greater than (2, 7)?
I'm not so sure you can do valid ARITHMETIC with a lexicographical order, for example if

(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.

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January 24th, 2013, 10:41 PM   #22
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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.
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January 25th, 2013, 01:53 AM   #23
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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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