Re: What does "infinty as number" mean?
Dear all,
Let me see if I can take a stab at this.
> Date: Sun, 01 Mar 2009 18:07:35 -0600
> From: Ralph Baker Kearfott <rbk@xxxxxxxxxxxxx>
> To: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
> Subject: What does "infinty as number" mean?
>
> Dear Members,
>
> What does the expression "infinity as number" mean to this 1788 group?
I'm fairly sure it means different things to different people.
But let's not get hung up on that & rather see what dealing
with infinity implies about intervals.
>
> 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 so for standard intervals.
Of course a non-standard interval such as [1,-1] have what
appears to be a projective infinity as an internal member.
But let's leave them out of the discussion for the moment.
For most people there is a further restriction that -inf
may only appear as a lower bound & +inf may only appear
as an upper bound. This completes a representation of
contiguous sets of real numbers by providing endpoints
for sets which are unbounded in that direction.
This includes [-inf,+inf] but leaves us with no natural
representation for the empty set which, again, let's leave
out of the discussion.
An exception is [+inf,+inf] & [-inf,-inf] which may be
included in some people's definition of 'infinity as number'.
>
> 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.
I disagree with this interpretation & the interpretation
that falls out naturally from 754 involving NaNs.
I feel the proper interpretation of inf OP inf should be
lim inf x OP y as x & y approach the desired
infinity INDEPENDENTLY
if the result is to appear as a lower bound &
lim sup x OP y as x & y approach the desired
infinity INDEPENDENTLY
if the result is to appear as an upper bound.
No other interpretation gives strict containment of the
associated real expression.
In most cases this will result in -inf for lim infs &
+inf for lim sups.
>
> 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.
Therefore,
lim inf inf-inf = -inf
lim sup inf-inf = +inf
lim inf inf/inf = -inf
lim sup inf/inf = +inf
lim inf 0/inf = -inf
lim sup 0/inf = +inf
but this last could be +/-0 in some cases in a definition in
which strict control of the infinities & the signs of zero
are maintained. I believe this must be part of the price we
must pay for 'infinity as number'.
Similarly,
lim inf x OP a as x approaches the desired infinity &
lim sup x OP a as x approaches the desired infinity
but only of such a limit exists. An exception would be x/0.
Divide by zeros could be considered exactly infinity with
the sign a question of inf or sup & how you are controlling
the sign of zero.
It is this last which forms one of the justifications of
'infinity as number' as I understand it.
>
> Did I abuse the terminology too much?
>
> Sincerely,
>
> Nate Hayes
> Sunfish Studio, LLC
At this point I don't think you can be accused of 'abuse'
of that which we have yet to clarify ourselves.
Let us take on that task & see if we can agree on a correct
interpretation & a terminology to go with it.
Don't hold your breath. This may take some time.
Yours,
Dan