(in)exactness of complex numbers
David C. Ullrich
ullrich at math.okstate.edu
Wed Aug 8 15:39:48 CEST 2001
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
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