(in)exactness of complex numbers

[Michael Abbott]

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.

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.