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You are not logged in. #1 20090206 11:25:38
0^0 equals 1 or undefined?I'd like to raise this question and see if there is an explicit answer or if this is just one of those topics that is not well established and yields different answers depending on who you speak to. He mentioned he also had another argument from a logic point of view but didnt show it. So now yesterday in our Abstract Algebra, our Professor who is a Algebraic Geometer mentioned during the lecture for some reason that 0^0 is undefined to which we all told him the discussion we had in Axiomatic Set Theory. Well, he couldnt believe that our other Professor had said it was equal to 1 since he says that it is obviously undefined. So in this thread I basically ask, which one of my Professors is right? Or is this one of topic in which no one is right and you assume whatever you want? Regardless, I would like to see your thoughts Last edited by LuisRodg (20090206 11:39:00) #2 20090206 11:30:31
Re: 0^0 equals 1 or undefined?I have always been of the opinion that 0^{0} should be defined to be 1, but some people (like mathsyperson) insist that that 0^{0} is undefined. #3 20090206 11:46:14
Re: 0^0 equals 1 or undefined?I just don't see how any value it could take would be logical. Why did the vector cross the road? It wanted to be normal. #4 20090206 12:10:45
Re: 0^0 equals 1 or undefined?To ease power series notation, 0^0 is taken to be 1 in complex analysis. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #5 20090206 14:03:39
Re: 0^0 equals 1 or undefined?For what it's worth, both the UNIX 'bc' program, and the basic on the C64 interpret 0^0 as 1. #6 20090206 21:11:29
Re: 0^0 equals 1 or undefined?
I’m not convinced by Mathsy’s argument. He claims there’s a “pattern” but I don’t see any. He is also claiming that for – which is false. I would need a much better argument to be convinced that 0^{0} should be left undefined. Last edited by JaneFairfax (20090206 23:31:03) #7 20090206 21:14:34
Re: 0^0 equals 1 or undefined?I've seen a nice combinatorial 'proof' that 0^0 = 1. Last edited by Daniel123 (20090206 21:29:54) #8 20090206 23:25:35
Re: 0^0 equals 1 or undefined?Oof... Why did the vector cross the road? It wanted to be normal. #9 20090207 00:26:49
Re: 0^0 equals 1 or undefined?No, Mathsy, it’s not a question of intuition. It’s a question of usefulness (as Ricky has said). Hence there is no way to define so that both and are continuous from the right at . Well, rather than being cynical and leaving to be undefined at this point, why not choose which of the two functions to make continuous (from the right) at 0? Now is already undefined for half of the real line. Choosing to make it continous from the right at 0 by defining to be 0 would serve very little purpose indeed. On the other hand, defining would make continuous over the entire real line, which might be potentially very useful. And indeed, it turns out that in many mathematical applications, having is very useful – even to the point of seeming “natural”. That is why defining has advantages over leaving it undefined. #10 20090207 02:05:48
Re: 0^0 equals 1 or undefined?Doesn't 0^0 = 0/0? I suspect that this is one of the proofs with a fundamental flaw (like the proof 1=2) but where is the flaw? Is it as simple as dividing by 0? To me this says: 0^0 is undefined because 0/0 is undefined. What is complex analysis? Is it simply analyzing complex functions the Reinman Hypothesis? Or does it mean something else? There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. #11 20090207 02:25:28
Re: 0^0 equals 1 or undefined?
Then to you is also undefined because is undefined. But jolly well defined!! Who says it isn’t? So do you now see the flaw (in both “proofs”)? #12 20090207 02:32:57
Re: 0^0 equals 1 or undefined?
I see that there is a flaw, but I'm not sure what. If I had to guess, I'd say it's the divide by zero flaw, but which step specifically is the nono? Is it 0^0 = 0^(11) step? I'm not sure why, but I think it is. There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. #13 20090207 02:35:47
Re: 0^0 equals 1 or undefined?Not that one. Here is the flawed step:
Compare that with this: #14 20090207 06:11:45
Re: 0^0 equals 1 or undefined?
Looks like we got a Platonist
That's pretty much it. The property of being differentiable as a complex function is much stronger than being differentiable in the real sense, and complex analysis is figuring out how much stronger it is and precisely what it gives us to play with. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #15 20090208 14:33:39
Re: 0^0 equals 1 or undefined?Google says 0^0 is 1. Linux FTW #16 20090209 03:41:35
Re: 0^0 equals 1 or undefined?
If that's the case, I have a used car to sell you. I have a bunch of people that will tell you it's a great deal. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #17 20090209 05:16:09
Re: 0^0 equals 1 or undefined?
Excel says it's undefined. There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. #18 20090209 06:22:07
Re: 0^0 equals 1 or undefined?
Sorry, I'm too young to drive. My feet don't even reach the pedals. Last edited by simron (20090209 06:22:57) Linux FTW #19 20090209 08:18:19
Re: 0^0 equals 1 or undefined?
In my opinion, this is AliceinWonderland nonsense. Sorry. Moreover, you appear to be confusing “empty sum” with “empty product” . When you multiply no numbers together, you get the empty product, not the empty sum. The empty sum is 0, but the empty product is 1, not 0. Last edited by JaneFairfax (20090209 09:56:05) #20 20090209 10:17:10
Re: 0^0 equals 1 or undefined?Good point Jane... Linux FTW #21 20120701 11:43:53
Re: 0^0 equals 1 or undefined?Here's an argument that 0^0 could be 0 or 1. #22 20120722 09:24:00
Re: 0^0 equals 1 or undefined?0^0 can be defined nicely immediately after introducing the field axioms which give us multiplication and the multiplicative identity 1. Last edited by noelevans (20120722 09:29:51) Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional). LaTex is like painting on many strips of paper and then stacking them to see what picture they make. #23 20120722 10:24:11
Re: 0^0 equals 1 or undefined?Hi noelevans The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #24 20120722 12:55:17
Re: 0^0 equals 1 or undefined?I'm afraid there is a fallacy in the argument that 0^0 = 0/0. Last edited by noelevans (20120723 10:05:31) Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional). LaTex is like painting on many strips of paper and then stacking them to see what picture they make. #25 20120722 13:03:25
Re: 0^0 equals 1 or undefined?Hi The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment 