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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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.)