Re: What does "infinty as number" mean?
Ralph Baker Kearfott schrieb:
Dear Members,
What does the expression "infinity as number" mean to this 1788 group?
For better or worse, I had been under the impression it means the
following three things:
1. Infinities are not members of the interval, although they may appear
as endpoints of an interval to indicate an open (unbounded) endpoint.
This is infinity-not-as-number, or intervals-as-sets-of-reals.
2. To evaluate the difference (Inf-Inf) or ratio (Inf/Inf) of two
infinities, the infinities are replaced with a real number x and then
the arithmetic operation is considered in the limit as the magnitude x
tends towards infinity. The same is true for 0*Inf, which leads to 0*Inf=0.
3. Because of 2), Inf-Inf=0, Inf/Inf=1, and 0*Inf=0, but other
arithmetic operations involving infinities are as usual, such as
Inf+a=Inf for any real number a.
In my usage, infinity-as-number means that elements of R^* = R union
+-Inf are called numbers, and that these numbers may be elements of
intervals. Thus an interval [Inf,Inf] is nonempty and contains Inf,
and [0,Inf] also contains Inf.
I strongly recommend against this, since there is no natural
arithmetic on R^*.
In particular, while Inf+a=Inf for finite a, Inf-Inf _must_ be
undefined in an infinity-as-number mode and not zero since
otherwise (Inf+1)-Inf=0 against any intuition.
The recommendations in Part 7 of the Vienna Proposal do not concern
infinity-as-number, but only the directed rounding requirements to
be able to optimally work with infinite bounds, roughly according to
your 2. above. This is only in directed rounding mode, and _only_ to
get the correct results for intervals-as-sets-of-reals in the most
transparent way.
Nothing in the Vienna Proposal is dependent on the recommendations in
Part 7 of the Vienna Proposal.
Arnold Neumaier