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.