As well it should be! Although some analysts seem timid about defining
0^0, logicians and set theorists are not. Defining 0^0 as 1 is not some
pecularity, but is actually fairly natural. I believe Paul Halmos's
book, Naive Set Theory, was the first place I saw it. So if my
explanation below is opaque, you can consult his book.
Exponentiation is defined first on the non-negative integers: n^m is
the size of the set of all functions from a set of size m to a set of
size n . So for example, there are 2^3 functions from the set {0,1,2}
to the set {0,1}.
Now n^0 should be the number of functions (i.e., sets of ordered
pairs...) from a set of size 0 to a set of size n. There is exactly one
set that has an ordered pair for each element in the (empty) domain,
namely the empty set. Thus n^0 is 1 for any non-negative integer n. In
particular, 0^0 = 1.
I think the reluctance to accept this definition stems largely from the
fact that the function f(x,y) = x^y has no limit at the origin, so in
particular is not continuous at the origin. Thus when evaluating limits
with the indeterminate form 0^0, the result might well be different from
1. If we taught our students that 0^0 = 1, we would probably have even
more difficulty than now in preventing them from "plugging in" to
evaluate limits of g(x)^(h(x)) when both g and h go to zero.
We've probably all seen numerous articles and/or texts that define power
series of a real or complex variable with sigma notation, the sum from n
= 0 to infinity of a_n * z^n . When z = 0, the value of the power
series is a_0. We are implicitly accepting that z^0 = 1 for all z, not
just all non-zero values of z. This is a notational convention rather
than a definition, but the definition exists and is useful even outside
of the realm of logicians.
Personally, I'm quite satisfied that the TI-92's evaluates 0^0 as 1.
Bruce Yoshiwara
Los Angeles Pierce College
>>
>Lawrence G. Gilligan
>Professor of Mathematics
>
>University of Cincinnati
>OMI College of Applied Science
>2220 Victory Parkway
>Cincinnati, OH 45206
>
>
>(513) 556-4868
>
>FAX: (513) 556-4878
>
>http://www.uc.edu/~gilligan
>
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byoshiwara@hotmail.com
http://www.lapc.cc.ca.us/usr/yoshibw
"...and I have assuredly found an admirable proof of this, but the
margin is too narrow to contain it." --Pierre de Fermat
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