[Fsf-friends] [OT]Notations for representing infinity

[Ramanraj K]

On 6/8/05, V. Sasi Kumar <sasi.fsf@gmail.com> wrote:
> On Wed, 2005-06-08 at 08:32 +0530, Ramanraj K wrote:
> > (BTW division by zero is assumed to be infinity and afaik there is no proof
> > for that).
> 
> If I remember my math, division by zero is properly said to be
> indeterminate, and not infinity. Infinity is not a number, but a concept
> -- if that does make sense.

Commonly, division by  zero is treated as an  illegal operation.  But,
Bhaskara in his Lilavati gives khahara (infinity) as the answer to 1/0
and it  does seem  to make sense,  though there  is no proof  for this
hypothesis or the claim that division by zero is an illegal operation.
Interestingly, 1/0  is also a notation  for khahara -  infinity in the
works  of our ancients.   Now, we  may follow  the same  notation n/0,
where the numerator  n may stand for the initial  value of an infinite
range of values. 

> > There should be a way to represent ... or infinity in the above
> > equation on computers.  If there are no standard ways of doing it
> > then, we should devise a way to do it.  It is fairly important to be
> > able to represent infinity on computers just as easily as we represent
> > numbers, because it has many practical uses as well.  We may have to
> > define max and min values for variables, and sometimes it has to be
> > set at infinity.
> >
> > If there are no standards for this, then:
> > [1] A special character could represent infinity (lemniscate :
> > sleeping 8 :) or three dots ...
> >
> > AND/OR
> >
> > [2] The last bit could be used to represent infinity.  If a n bit word
> > is used to represent integers, then the allowed integers have values
> > between -2^(n-1) and (2^(n-1)) - 1.  The maximum integer value could
> > be reserved to represent infinity
> 
> I am a little confused. Are you talking of writing infinity? In that
> case, the infinity character is available. But when you want to do
> actual computation, you cannot obviously do an iteration an infinite
> number of times. So there we are forced to select an appropriately large
> number that could stand for infinity.

Both issues have to be dealt with almost simultaneously: [1] How could
we represent infinity within the  byte words used for numbers; and [2]
Explore the means for doing actual computation once its representation
and notation can be taken for granted.

There  is  a  discussion  at   ieee  on  whether  there  should  be  a
representation for -0.  After all, 0-0  = 0 and there is nothing wrong
in having a representation for -0 for the sake of balance, and further
is would  also help  to represent infinity  in a simpler  manner.  The
integer values allowed are in the range -2^(n-1) and (2^(n-1)) - 1 and
there is  a slight imbalance there.   If -0 is  also represented, then
the  imbalance would be  corrected and  then the  range would  be just
-2^(n-1) and 2^(n-1). The values -2^(n-1) and 2^(n-1) themselves could
represent negative and positive infinity respectively.

> It may be interesting to know that there are different kinds of
> infinity. For instance, there are an infinite number of integers. But
> there are an infinite number of rational numbers between any two
> integers. So these two infinities cannot be considered to be of the same
> kind. If I remember right, they are represented by the Hebrew alphabet
> aleph. So we have aleph1, aleph2 and so on.

Thanks for the  info about aleph. I am using  both types of infinities
to define legal data in  the calpp project.  Any given procedure could
have a  number of  steps, and by  using rational numbers  to represent
steps, any future new steps  created, could always be inserted between
any  two  given steps  taking  advantage  of  the infinite  number  of
rational numbers between two integers.

If we need to use infinity in our computations, we can implement them
on our own, making our own rules, but standardisation would be better.

The max and min values possible for integers and floats could represent 
-∞  ∞ and  -א  א  respectively.  Assignment could be done more simply
using the notation of our ancients in the form n/0

Thanks,
Ramanraj.