On Wed, Jul 9, 2008 at 3:38 PM, Bill Page <[EMAIL PROTECTED]>
wrote:
> >>>
> >>>> *yixin.cao wrote:* -- In the light of polynomial, (2d) is a
> >>>> non-zero polynomial, so that it's always safe to write (1/(2d))
> >>>>
> >> ...
>
> > On Wed, Jul 9, 2008 at 10:38 AM, Bill Page wrote:
> >> But is computing in the field Q(d) "safe" if we eventually intend to
> >> replace d with some non-symbolic value?
> >
>
> On Wed, Jul 9, 2008 at 3:56 PM, Gabriel Dos Reis wrote:
> > That is a question separate from whether the polynomial 2*d is
> > nonzero or not. There is no doubt it is nonzero.
> >
>
> Call it whatever you like. My question is: "Is yixin.cao's original claim
> true?"
>
Which claim?
In the light of polynomial, (2d) is a non-zero polynomial, so that it's
always safe to write (1/(2d))
If this, see
http://en.wikipedia.org/wiki/Field_of_fractions
In mathematics <http://en.wikipedia.org/wiki/Mathematics>, every integral
domain <http://en.wikipedia.org/wiki/Integral_domain> can be embedded in a
field <http://en.wikipedia.org/wiki/Field_%28mathematics%29>; the smallest
field which can be used is the *field of fractions* or *field of
quotients*of the integral domain. The elements of the field of
fractions of the
integral domain *R* have the form *a/b* with *a* and *b* in *R* and *b* ≠ 0.
The field of fractions of the
ring<http://en.wikipedia.org/wiki/Ring_%28mathematics%29>
*R* is sometimes denoted by Quot(*R*) or Frac(*R*).
The result of evaluation of a polynomial doesn't restrict the properties of
this polynomial. Or else a/b for any Polynomial Float b is wrong, for that
there is always at least one complex number to evaluate b to 0. But I'm
saying in the light of polynomial, it's safe.
>> Isn't there a possibility that
>> such computations could lead to incorrect results due to (for example)
>> implicit divisions by 0?
>
> This is a classic problem of interpretation, discussed by classic volumes
> of algebra books, and classics of symbolic computation.
>
True. What is your point?
> And this is precisely one of the fundamental reasons why I prefer
> a distinct domain Symbolic T, instead of reusing Polynomial or
> similar thing. From my point of view, this is also a good reason
> to reject Union(Variable d, Float).
>
How will 'Symbolic T' deal with this issue? How will it be different
from Polynomial?
> ...
Regards,
Bill Page.
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