2011-03-14, 03:39 PM
The 0^0 argument is like the why .999... = 1 argument (sort of). In that people tend to argue craploads for multiple sides and no one like to agree with the other side. 0^x = 0 for any nonzero, thus people say that 0 ^ 0 should be 0. But x ^ 0 = 1 for any non zero x, thus people say that 0^0 should be 1. There's a few more arguments about why it should be 1 though:
-lim x^x = 1. Thus if we want the x^x function continuous from the right side we'll need to define 0^0 as 1.
x->0+
(Had to look this one up since i forgot the details of it)
-Another way to view the expression m^n is as the number of ways to map an n-element set to an m-element set. For instance, there are 9 ways to map a 2-element set to a 3-element set. There are NO ways to map a 2-element set to the empty set (hence 0^2=0). However, there is exactly one way to map the empty set to itself: use the identity map! Hence 0^0=1.
-Look at the taylor series expansion of x^x:
f(x) = 1+ xlnx + (1/2) x^2 (lnx)^2 + (1/3!) x^3 (lnx)^3 +....
take the limit as x approaches zero from the left to get 1. Sadly you can't just plug in x =0.
Overall though, 0 ^ 0 is an indeterminate form. 0 ^ 0 has no actual answer. If you try it in your google search bar it'll give you 1, which isn't actually true... not sure why they did that.
-lim x^x = 1. Thus if we want the x^x function continuous from the right side we'll need to define 0^0 as 1.
x->0+
(Had to look this one up since i forgot the details of it)
-Another way to view the expression m^n is as the number of ways to map an n-element set to an m-element set. For instance, there are 9 ways to map a 2-element set to a 3-element set. There are NO ways to map a 2-element set to the empty set (hence 0^2=0). However, there is exactly one way to map the empty set to itself: use the identity map! Hence 0^0=1.
-Look at the taylor series expansion of x^x:
f(x) = 1+ xlnx + (1/2) x^2 (lnx)^2 + (1/3!) x^3 (lnx)^3 +....
take the limit as x approaches zero from the left to get 1. Sadly you can't just plug in x =0.
Overall though, 0 ^ 0 is an indeterminate form. 0 ^ 0 has no actual answer. If you try it in your google search bar it'll give you 1, which isn't actually true... not sure why they did that.