Bill Page wrote:
> >> But I worry if the proposal really is that
> >>
> >> x/0 = 0
> >>
> >> since that could do a lot of damage to many other desirable properties
> that
> >>
> >> Float has Field
> >>
> >> should imply, e.g. as Bertfried pointed out concerning it's
> >> topological properties.
> >
> > Float is bad example because strictly speaking it is not a field
> > (operations are nonassociative).
>
> True. I assumed that we were already glossing over that issue.
>
> > However defining 'x/0' does not
> > really change properties of field: normal field axioms only
> > say what happens when you divide by nonzero element.
>
> If the inverse of 0 is 0 then the domain should not even be a Group
> let alone a Field.
>
Let us recall field axioms:
1) field is abelian group with respect to addition. Defining 1/0 = 0
does not affect this at all.
2) _nonzero_ elements of the field form an abelian group with
respect to multiplication. Again defining 1/0 = 0, because
here we consider only nonzero elements (using 1/0 = x with
nonzero x would be more tricky)
3) distributive law:
\forall_{a, b, c} a*(b+c) = a*b + a*c
Since distributive law does not involve division it is satisfied
if 1/0 = 0.
The axioms imply \forall_a 0*a = 0. This is fundamental property
of fields and means that it is impossible to define 1/0 in such a
way that _all_ elements of field form multiplicative group.
> But I think your example of how to define such a domain in FriCAS is
> still interesting. I am curious why it seemed to be more difficult to
> do this than one might have expected. Are there some hidden
> assumptions somewhere in FriCAS about Float?
Nothing special about Float. Spad compiler signals type error
in simpler code -- ATM I do not know if there is some problem
with simpler code or it is a compiler bug.
--
Waldek Hebisch
[email protected]
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