Re: Sign conventions

From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 03/03/05 

Date: Thu, 3 Mar 2005 16:58:42 +0000 (UTC)

In article <20050303012643.023b621f.kevin@hotmail.com>,
Kevin <kevin@hotmail.com> wrote:

>I learned that the complex numbers aren't ordered (or well
>ordered, I don't know the difference actually), but anyway, I
>thought I learned that you can't order complex numbers, so i > 0
>is nonsense.

What you learned, presumably, is that it is impossible to define an
ordering on the complex numbers ->that is compatible with the
operations<-.

What this means is that it is impossible to define an order relation >
on the complex numbers such that:

(i) For each complex number a, one and only one of a<0, a=0, and a>0
     holds.

(ii) a<0 if and only if (-1)*a > 0

(iii) If a > 0 and b > 0 then a+b >0 and ab>0.

To see why, you first note that -1<0; for if (-1) > 0, then by (ii)
you woul dhave -(-1) = 1 < 0, but by (iii) you would also have
(-1)*(-1) >0. So -1 < 0.

Now take i; if i>0, then i*i = -1 >0, but that is impossible. If i<0,
then -i>0, and again we have that (-i)*(-i)>0, which contradicts that
-1<0.

HOWEVER, it is quite possible to define an ordering on the complex
numbers. For example, a very easy and natural ordering on the complex
numbers is to define

  (a + bi) < (c + di) if and only if a<c or (a=c and b<d).

This is the "lexicographic order", and on the complex plane it is the
same as saying that being to the left of, or on the same vertical line
and below, is the same as being "smaller than". In this definition, i
is indeed bigger than 0.

This is a perfectly fine ordering that has a lot of good uses. It is
not compatible with the operations, however, so it also fails to be
any good for a number of (algebraic) applications. That does not mean
you cannot "define an ordering on the complex numbers."

(Oh, and if you accept the Axiom of Choice, then of course the complex
numbers ->can<- be well ordered, though we cannot write down a rule
for this ordering). 

-- 
======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu
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