>>2 >>8
Nooooo. 0/0, 0^0, 1^infinity, etc. are all /indeterminate./ Indeterminate quanitities do not exist, that's true, but they exist in an entirely different ways from undefined quantities. There is no value that satisfies an undefined quantity, whereas there are infinitely many values that satisfy an indeterminate form.
0^0 = 0/0 = Any real number at all.
On the other hand,
n/0 (n != 0) = Nothing.
They both don't exist, but they are very, very different, and not just in theory.
That's incorrect. The lim(x^x,x,0) = 1 only from the RHS. If the absolute limit is to be defined for when x->0 then the limit from the LHS must equal RHS in which case it does not. Trust me on this, I have a degree in math.
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Anonymous2005-11-09 13:09
lim(x^x,x,-0) is undefined, hence no limit at all.
0^0 is often defined to be 1, because zero over zero is one (i.e. there's only one way to order nothing)
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Anonymous2005-11-09 17:42
but 0^0^0 = 0
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Anonymous2005-11-12 11:10
OH SHIT YOU GUYS BETTER STOP DIVIDING BY ZERO!!!!
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Anonymous2005-11-13 0:35
I personally think of the exponential in that case as a kind of special function which is just defined to be 1 at zero.
Not very formal, but I never was.
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Anonymous2005-11-13 13:34
>>19
0/0 is indeterminate; read up on limits for more detail.
Consensus says 0^0 = 1, but there seems to be no proof as yet.
btw in mathematics you can just define things to be something if it's convenient. you don't need to prove it
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Anonymous2005-11-14 14:42
>>25
But through expansion of the meaning of a^n in such a way that it still holds true for all values a != 0, you can achieve a solution that will also work for a=0.
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Anonymous2005-11-26 23:53
Well, multiplication is iterated addition, right?
Therefore...
2+0 = 2x1 = 2
2+2 = 2x2 = 4
2+2+2 = 2x3 = 6
So exponents are iterated multiplication, which is iterated addition.
Some textbooks leave the quantity 0^0 undefined, because the functions x^0 and 0^x have different limiting values when x decreases to 0. But this is a mistake. We must define x^0=1 for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = -y. The theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant.