- #1
Nano-Passion
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[tex]\frac{0}{0}[/tex]
Is it 0 or undefined? I thought this was interesting, it seems a paradox in its own.
Is it 0 or undefined? I thought this was interesting, it seems a paradox in its own.
nucl34rgg said:Well, assuming you are working with the field of real numbers, 0 doesn't have an inverse multiplicative element. There is no real number x such that 0*x=1. So the expression 0/0 which really means 0 times the multiplicative inverse of 0 has no meaning, since the multiplicative inverse of 0 isn't in the field of reals and doesn't actually exist. Thus 0/0 is undefined.
Intuitively, dividing zero by zero makes no sense because you are asking 0=0*x for what number x? Well, x could be any real number and it would satisfy that equation. In other words, the expression 0/0 isn't defined to be a particular number, whereas when we define division as a/b for real numbers a and nonzero real numbers b, we mean a/b=x where x is the unique solution to the equation x*b=a. The expression a/b is supposed to spit out a single real number. In our 0/0 case, it sort of gives us literally every real number as an output, which means it is useless if we are trying to describe a specific number with it.
You might want to hide your post before the mathematicians see this and die of shock and mad rage! lol
HallsofIvy said:We might note that many text refer to "0/0" as "undetermined" rather than "undefined" because if you have a limit of a fraction where the numerator and denominator both go to 0, the actual limit itself can exist and can be anything.
edgepflow said:What about looking at:
[itex]\frac{lim}{x -> 0}[/itex][itex]\frac{x}{x}[/itex]
and apply l'Hôpital's rule to obtain:
[itex]\frac{1}{1}[/itex] = 1
micromass said:Also see the FAQ on this topic: https://www.physicsforums.com/showthread.php?t=530207
edgepflow said:What about looking at:
[itex]\frac{lim}{x -> 0}[/itex][itex]\frac{x}{x}[/itex]
and apply l'Hôpital's rule to obtain:
[itex]\frac{1}{1}[/itex] = 1
Nano-Passion said:That was interesting, thank you. You said that [itex] \frac{1}{0}[/itex] isn't ∞ because as you approach 0 in a rational function then it can either be positive or negative infinity.
So then would you be able to state:
[itex]\frac{0}{0}= -∞ < x < +∞ [/itex], where x exists anywhere on the extended real number line.
Then the probability of x being a particular value on the one of the real numbers would be [itex]\frac{1}{∞}[/itex] would be undefined. Therefore, that might imply that [itex]\frac{1}{0}[/itex] is undefined.
jgens said:As a slightly unrelated note on probability, consider the following problem: If you select an integer at random from Z, what is the probability that the integer you chose is 0? It turns out the probability is zero. Therefore, there are events with probability 0 that can still occur. Likewise, there are events with probability 1 that do not occur. These are just some neat things that happen when you consider probability on infinite sample spaces.
Nano-Passion said:Okay I agree with your post. Something with probability one does not have to occur. But how can something with probability 0 occur? It could be an infinitesimal and not occur that much I agree, [itex]lim_{Δx\ to 0}[/itex] isn't necessarily 0. But 0 is a bit of a different number.
jgens said:In the real number system, there are no infinitesimal elements. The same is true in the extended reals and projective reals as well. In fact, most mathematicians rarely (if ever) do any work that uses formal infinitesimals. There are number systems that have infinitesimal elements (like the hyperreal numbers), but most of these have roots in model theory and are fairly difficult to define formally. If you are interested, non-standard analysis is the subject that deals with the calculus of these infinitesimal numbers, but non-standard analysis is far from one of the more active areas of research in analysis.
Therefore, it is often best not to resort with reasoning using infinitesimals. Without using their formal properties, it is easy for your intuition to deceive you. It turns out most people have terrible intuition when it comes to infinitesimals.
Now, it is important to note that [itex]lim_{h \to 0} h = 0[/itex]; that is, the value of the limit is 0. The limit is not infinitesimally close to 0, but actually is 0. This is an extremely important point to understand.
Finally, keeping what I've said above in mind, something with probability 0 can occur in just the same manner as something with probability 1 not occurring.
Nano-Passion said:Hey, thanks for your patience. You haven't really argued on how something with probability 0 can occur. I'm not completely convinced at the moment. I'll try to put it in words for the sake of argument; to me 0 is absolutely nothing, so for absolutely nothing to happen is a paradox. Can you throw in a bit of mathematics, I'm interested.
Nano-Passion said:[tex]\frac{0}{0}[/tex]
Is it 0 or undefined? I thought this was interesting, it seems a paradox in its own.
sankalpmittal said: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.
0/0 = x , where x can be any number. So this is kinda indeterminable.
sankalpmittal said: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 : http://mathforum.org/library/drmath/view/55722.html
Read it , it is very interesting.
jgens said:Here are some simple examples:
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.
- 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)?
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?
micromass said: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.
Anti-Crackpot said:Please...
A classic example of how mathematics can "straightjacket" common sense. An equation with infinite solutions is NOT undefined, nor is it "indeterminate" excepting to the mind that requires finite solutions to questions with infinite answers.
Anti-Crackpot said:The seeming human inability to accept that the equation 0*x = 0 is satisfied by any value one could possibly imagine for x, to the point where one must suggest the algebraic manipulation of that formula via division to be "undefined," or "indeterminate" is just beyond (IMHO) "retarded." It is perfectly "well-defined."
X = "any value" one can possibly imagine.
Therefore, 0/0 = anything you may wish it to equal, including 1, which typically is the value one obtains when dividing some positive or negative quantity by itself.
-1/-1 = 1
1/1 = 1
But (-1 + 1)/(-1 + 1) is somehow not equal (at least) to 1?
Please...
A classic example of how mathematics can "straightjacket" common sense. An equation with infinite solutions is NOT undefined, nor is it "indeterminate" excepting to the mind that requires finite solutions to questions with infinite answers.
shriomtiwari said:0/0
one of many ways of writing all the no.s in one go.
None of the above.wilsonb said:Combining the both, should it be 0, 1 (since X amount of apples divided by X person) or undefined??
D H said:None of the above.
1/0 is undefined using the standard definition of the real numbers. 0/0, on the other hand, is indeterminate.
Division is the inverse of multiplication. a/b=c means c is the unique number such that a=b*c. In the case of 1/0, there is no such number c∈ℝ such that 1=0*c. So 1/0 is "undefined".
In the case of 0/0, every number c∈ℝ satisfies 0=0*c. This makes it appear that one could assign any number whatsoever as the value of 0/0. This has two problems, both of them killer. One obvious problem is the lack of uniqueness. An even bigger problem is that assigning anyone specific value opens the door to all kinds of contradictions. Mathematical systems must be contradiction-free.
I think you completely missed the point of my post, which was that the correct term for 0/0 is that it is indeterminate rather than undefined. 1/0 is undefined, but 0/0 is indeterminate.wilsonb said:very simple.