(in)exactness of complex numbers

[David C. Ullrich]

On Thu, 02 Aug 2001 14:28:45 -0700, Michael Ackerman <ack at nethere.com>
wrote:

>
>
>"David C. Ullrich" wrote:
>
>> Exactly what definition of
>> "complex number" do you have in mind here?
>> The _standard_ definition _is_ "pair of
>> real numbers".)
>
>If you define the complex numbers C as pairs of reals, you have given at
>most its structure of real vector space. Then you must define
>multiplication and prove that what you have is actually a ring.

Thanks for pointing this out yet one more time. Actually I was
aware of that... you omit the opening parentheses from the bit
you quote here: The _point_ to my post was not to insist that
one definition was correct, it was to say that it seems silly
to suggest that the mathematical definition has anything to
do with the way complexes should work in Python.

>But if
>you define C as the quotient ring R[X]/(X^2+1), then you needn't do
>anything more.

Only because you've _already_ proved what needs to be proved
about quotient rings.

> This definition is often used, e.g. in MacLane &
>Birhkoff's "Algebra", the most categorically oriented basic algebra
>text.  

I take back what I've been saying about how the definitions are
irrelevant to Python. Python should not represent complexes
as pairs of floats. Instead a complex should be an equivalence
class of real polynomials...

Perhaps "standard" was putting it too strongly; my point
of view is coming from complex analysis, not algebra. I've
seen the complexes defined many times in a complex-analysis
setting, never as anything other than a pair of reals.

Or rather as a pair of reals plus suitable definitions for
the operations.

>-- Michael Ackerman
>
>


David C. Ullrich