Knuth's argument looks compelling at first sight, but it is conditional on there needing to exist a single undisputed power function for all occasions, and I think this is simply false. When working with the binomial theorem, the meaning of exponentiation must certainly be taken to involve 00 being 1; but in other contexts this need not be the case, it can be undefined or zero or 42 or whatever the hell it likes, and we resolve the dispute with Knuth by pointing out that his exponentiation and ours aren't necessarily the same thing. Where it matters, make it clear.
Anybody who wants the binomial theorem (x + y)^n = sum_(k = 0)^n (n\choose k) x^k y^(n - k) to hold for at least one nonnegative integer n must believe that 0^0 = 1
is flawed of course: they might just be disappointed…
1) IANAMathmo, so know nothing 2) "1" seems the obvious answer as a^0=1 for a!=0 3) but 0^a=0 for a>0, so I could see an intuitive case for "0" 4) If you're doing this in code, you deserve to lose (TM), so I can see a case for "undefined" 5) clearly therefore "depends why you're asking" and "eh?" are the right answers 6) see 1.
Damn, I know I remember someone else talking about this, but can't remember where. I would say:
* There is no obvious choice, because in z=x^y you get a discontinuity whatever you do. (I mention this because high-school-mathmos often act as if one choice is inevitable, which it isn't) * You may choose 0, 1 or undefined, whichever fits best. * This will often be clear from context, but to be rigorous you should state it, or define your function in such a way it doesn't come up * When you choose, 1 may be better because that's a more common choice (regardless of whether it's sensible in more mathemetical functions, which it may be, but I haven't thought about fully enough to say so)
I tickyboxed 0 for exclusive-or as in C, and undefined for exponentiation. You didn't provide a "pair of glasses" option, or I'd have ticked that, too.
I'd have written 00 but the poll creator form didn't like it. It didn't like the answer of 0 either though posting the raw poll markup with it in still worked. I suspect crappy Perl programmers.
There are various other places where 0 isn't allowed around LiveJournal. (I think the poll creator used not to like scales starting with 0, though that might have been fixed now; the username 0 doesn't work well, I think; etc.)
I just voted for 1 mainly for economic reasons, :). I'm sure that the amount of time wasted considering this question (thousands if not millions of dollars, my sources Messers I. and P. News inform me) has far outweighed the effort in just working around defining it to have one value.
The precident for this kind of agrument is that the null graph is now almost universally not considered a graph, mainly because almost no graph theorems applied to it, so it was considered a waste of time and ink, :).
We can't allow ourselves to be paralysed by choice, or we'll never get our flying cars or rocket suits! Therefore: one.
That's a good point. Particularly given what happened to Arianne Five when it did just that. Equally, though, I guess, that all the rocket-suit manufacturers are assuming the same value for 0^0 than each having their own, in the style of the famous lb/in^2,N mixup at JPL.
I sometimes wonder if the space industry exists primarily to keep comp.risks full of interesting articles.
I'm going to remain blissfully unaware of what the answer actually is, but I'll guess at 0 or possibly 'everything breaks' like if you try to divide by zero. My maths skills are not strong, though. If I tried to do that in a program, though, I reckon it would explode.
Or perhaps it's all an elaborate ruse, and 0^0 is actually a crazy new smiley that represents a bird looking upwards? :)
While IANAmathmo, surely the only answer here can be "undefined". As far as I can remember, 0^x=0, and x^0=1, so both answers make some sort of sense. Or is there anything here that I have missed that is known only to students of arcane high-level maths?
"Undefined" isn't quite the right word, because mathematicians can and do give definitions for 0⁰, for example simont noted above that Donald Knuth (http://www-cs-faculty.stanford.edu/~uno/) defines 0⁰ = 1 in his book Concrete Mathematics (http://cs.ioc.ee/yik/lib/1/Graham1.html) so that the binomial theorem (http://en.wikipedia.org/wiki/Binomial_theorem) can be extended to negative numbers.
So it's not undefined, it's that you have a choice of definitions. That's because 0⁰ is an indeterminate form: it's an expression for which you can get different results if you take the limit as you let the subexpressions tend to their proper values. Wikipedia (http://en.wikipedia.org/wiki/Indeterminate_form) explains this pretty clearly.
(However, 0⁰ = 1 has useful applications — the product of no numbers is 1, there's only one way to choose no objects from an empty set — and as far as I know other values don't.)
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(no subject)
Date: 2006-06-05 12:47 pm (UTC)(no subject)
Date: 2006-06-05 12:57 pm (UTC)(no subject)
Date: 2006-06-05 01:01 pm (UTC)(no subject)
Date: 2006-06-05 04:38 pm (UTC)(no subject)
Date: 2006-06-05 04:42 pm (UTC)(no subject)
Date: 2006-06-05 04:57 pm (UTC)(no subject)
Date: 2006-06-05 12:59 pm (UTC)2) "1" seems the obvious answer as a^0=1 for a!=0
3) but 0^a=0 for a>0, so I could see an intuitive case for "0"
4) If you're doing this in code, you deserve to lose (TM), so I can see a case for "undefined"
5) clearly therefore "depends why you're asking" and "eh?" are the right answers
6) see 1.
(no subject)
Date: 2006-06-05 01:00 pm (UTC)* There is no obvious choice, because in z=x^y you get a discontinuity whatever you do. (I mention this because high-school-mathmos often act as if one choice is inevitable, which it isn't)
* You may choose 0, 1 or undefined, whichever fits best.
* This will often be clear from context, but to be rigorous you should state it, or define your function in such a way it doesn't come up
* When you choose, 1 may be better because that's a more common choice (regardless of whether it's sensible in more mathemetical functions, which it may be, but I haven't thought about fully enough to say so)
(no subject)
Date: 2006-06-05 01:08 pm (UTC)(no subject)
Date: 2006-06-05 02:32 pm (UTC)(no subject)
Date: 2006-06-05 02:36 pm (UTC)(no subject)
Date: 2006-06-06 12:39 pm (UTC)So do I.
There are various other places where 0 isn't allowed around LiveJournal. (I think the poll creator used not to like scales starting with 0, though that might have been fixed now; the username 0 doesn't work well, I think; etc.)
(no subject)
Date: 2006-06-12 09:43 am (UTC)(no subject)
Date: 2006-06-12 09:45 am (UTC)(no subject)
Date: 2006-06-05 01:12 pm (UTC)(no subject)
Date: 2006-06-05 01:13 pm (UTC)The precident for this kind of agrument is that the null graph is now almost universally not considered a graph, mainly because almost no graph theorems applied to it, so it was considered a waste of time and ink, :).
We can't allow ourselves to be paralysed by choice, or we'll never get our flying cars or rocket suits! Therefore: one.
(no subject)
Date: 2006-06-05 08:43 pm (UTC)(no subject)
Date: 2006-06-06 09:42 am (UTC)I sometimes wonder if the space industry exists primarily to keep comp.risks full of interesting articles.
(no subject)
Date: 2006-06-05 01:16 pm (UTC)=>
0^0 = e^(0 ln 0) = e^0 = 1
(no subject)
Date: 2006-06-05 01:38 pm (UTC)(no subject)
Date: 2006-06-05 03:43 pm (UTC)(no subject)
Date: 2006-06-05 02:20 pm (UTC)Or perhaps it's all an elaborate ruse, and 0^0 is actually a crazy new smiley that represents a bird looking upwards? :)
(no subject)
Date: 2006-06-05 02:38 pm (UTC)(no subject)
Date: 2006-06-05 03:39 pm (UTC)(no subject)
Date: 2006-06-05 03:22 pm (UTC)(no subject)
Date: 2006-06-05 04:16 pm (UTC)(no subject)
Date: 2006-06-05 06:09 pm (UTC)(no subject)
Date: 2006-06-05 10:42 pm (UTC)So it's not undefined, it's that you have a choice of definitions. That's because 0⁰ is an indeterminate form: it's an expression for which you can get different results if you take the limit as you let the subexpressions tend to their proper values. Wikipedia (http://en.wikipedia.org/wiki/Indeterminate_form) explains this pretty clearly.
(However, 0⁰ = 1 has useful applications — the product of no numbers is 1, there's only one way to choose no objects from an empty set — and as far as I know other values don't.)
(no subject)
Date: 2006-06-05 07:32 pm (UTC)