(in)exactness of complex numbers

[Mikael Olofsson]

On 01-Aug-2001 David C. Ullrich wrote:
 >  (Not that it makes any difference here, but
 >  since you undertstand the math so much better
 >  than I do: Exactly what definition of
 >  "complex number" do you have in mind here?
 >  The _standard_ definition _is_ "pair of
 >  real numbers".)

I'd say it's not the standard definition, but rather the standard
representation. The complex field C is an extension of degree 2 of the 
reals R (by the way, the only possible extension of R). To get a 
representation of the complex field you need a root of an irreducible
polynomial over R. The standard representation uses the polynomial 
x**2 + 1, and we usually call that root i or j. This means that we have
i**2 = -1. This choice is natural, since it gives the easiest rules for
arithmetic operations. However, you could use any other irreducible 
polynomial over R, e.g. x**2 + x + 1. You still would use a pair of 
reals to represent a complex number, but another pair than you would
have in the standard representation. And the arithmetic rules would
be different as well.

What I believe Greg is aiming at is that the complex numbers are more
than a pair of reals. A pair of reals is not a complex number unless
you specify the arithmetic rules of the pair.

Now *I* would not mind to allow complex numbers with integers as 
coefficients. That way I would get Gaussian integers without specifying
my own class for the purpose. _But_, I understand that very few would
have a need for that. I mean, how many acually uses complex numbers
at all in their programs. Most people still regard complex numbers as
a very obscure corner in math. About only a 150 years ago, when complex
numbers were new, many _mathematicians_ thought of them as something 
the devil had produced.

/Mikael

-----------------------------------------------------------------------
E-Mail:  Mikael Olofsson <mikael at isy.liu.se>
WWW:     http://www.dtr.isy.liu.se/dtr/staff/mikael               
Phone:   +46 - (0)13 - 28 1343
Telefax: +46 - (0)13 - 28 1339
Date:    02-Aug-2001
Time:    08:48:18

         /"\
         \ /     ASCII Ribbon Campaign
          X      Against HTML Mail
         / \

This message was sent by XF-Mail.
-----------------------------------------------------------------------