(in)exactness of complex numbers

[David C. Ullrich]

On Wed, 8 Aug 2001 08:46:17 +0000 (UTC), Michael Abbott
<michael at rcp.co.uk> wrote:

>ullrich at math.okstate.edu (David C. Ullrich) wrote in
>news:3b6ea347.838398 at nntp.sprynet.com: 
>
>> On Mon, 6 Aug 2001 07:18:56 +0000 (UTC), Michael Abbott
>> <michael at rcp.co.uk> wrote:
>> 
>> Why "probably": Of course it's impossible to use the mathematical
>> R[X]/(X^2+1) literally; the elements are equivalence classes of
>> polynomials, in particular a complex number is an infinite set
>> of polynomials, and hence would take too much storage space.
>
>No, no, it's not impossible at all.  You simply use one representative at a 
>time to represent a value, but use the relation when you need to decide 
>whether or not two values are equal.

Of course. I _said_ that. You _quote_ part of where I said that
below.

(Doesn't matter, though - we have a special diss-pensation for
off-topic posts. Um, <wink>...)

>So, if we want to represent S/R where S is a set and R is an equivalence 
>relation on S then we simply use elements of S, but use R to implement the 
>equality test.
>
>Obviously, if there's a canonical choice of a representative, then we can 
>just use that and not take time to compute R.
>
>> Here there _is_ a natural choice of one polynomial to use
>> from each equivalence class, but if you use that one your
>> complex numbers have become precisely pairs of reals.
>
>Quite so.
>


David C. Ullrich