Thread: 0 divided by 0
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Nov30-11, 01:26 AM
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Quote Quote by sankalpmittal View Post
Hii , Nano-Passion !!
This is a very interesting question.

0/0 is neither ∞ nor 0. It is what is "Indeterminate".

For example we can say that 1/0 = ∞ because 0x1 = 0 , 0x10100000000000000000 = 0. So we assume that somehow at an undefined place that is ∞ 0 will become 1.

But in case of 0/0 , every equation is satisfied !

0/0 = x , where x can be any number. So this is kinda indeterminable.

Here is the best explanation of 0/0 by Doctor Math :

Read it , it is very interesting.
Hey , thanks for sharing.

Quote Quote by jgens View Post
Here are some simple examples:
  • If you choose an integer at random from Z, what is the probability that the integer chosen is 0?
  • If you choose an integer at random from Z, what is the probability that the integer lies between -N and N?
  • If you choose a real number at random from R, what is the probability that the real number chosen is rational (or algebraic)?
In each case, the probability in question is 0. The third statement is a little more complicated, but it has a nice proof once you have measure-theoretic concepts.

I will prove that the probability of the second statement is 0:
Let [(2N)m] = {-(2N)m, ... , (2N)m}. Then for a fixed m, the probability of choosing an integer between -N and N is (2N)1-m. By letting m → ∞, we see that the probability goes to 0. In particular, in the limiting case (when we are choosing elements from Z), the probability is 0.

I should probably write this more formally and nicely, but it captures the point. So there's your example. If you don't think that the limit actually is 0, but rather is something else, what do you propose that something else should be?
The third question is very interesting actually haha.

I agree with you, when you are dealing with an infinite amount of numbers then it would be 0. But what about a finite amount of numbers? Can a number occur with 0 probability, such that n is a finite number [in this case let us limit n to a world consisting only of 50 digits].

Quote Quote by micromass View Post
Hmm, probability 0 is indeed a silly concept. Most people think of probability as throwing dice, and indeed: throwing a 5.5 with a dice has probability 0 and thus never happens. But it is important not to generalize this situations. There are some probability 0 situations which can happen.

As an example: choosing an arbitrary number in the interval [0,1]. It is clear that all numbers have the same probability p of being chosen. However, saying that a number has probability p>0 is wrong, since [itex]\sum_{x\in [0,1]}{p}\neq 1[/itex]. So we NEED to choose p=0. So choosing probability 0 for this is actually quite unfortunate and caused by a limitation of mathematics.

However, there is another way of seeing this. Probability can be seen as some "average" value. For example, if I throw dices n times (with n big), then I can count how many times I throw 6. Let [itex]a_n[/itex] be the number of 6's I throw. Then it is true that

[tex]\frac{a_n}{n}\rightarrow \frac{1}{6}[/tex]

So a probability is actually better seen as some kind of average.

Now it becomes easier to deal with probability 0. Saying that an event has probability 0 is now actually a limiting average. So let [itex]a_n[/itex] be the number of times that the event holds, then we have

[tex]\frac{a_n}{n}\rightarrow 0[/tex]

It becomes obvious now that the event CAN become true. For example, if the event happens 1 or 2 times, then we the probability is indeed 0. It can even happen an infinite number of times.
Probability 0 should not be seen as a impossibility, rather it should be seen as "if I take a large number of experiments, then the event will become more and more unlikely". This is what probability 0 means.
Hey, thanks for the reply.

To me probability rings to my neurons as a tendency to become a value over a period of time or over n times. But that is just my definition of course. If we take this definition in that context, then perhaps a probability of 0 would imply that it has 0 tendency to become any value over a period of time or n times. But then I guess this doesn't hold true in the mathematical context.

I wonder, if something has 0 probability in Quantum Mechanics, can it happen? I suppose it can, which would support your statement.