0^0?
mjh...@dahlia.waterloo.edu
Have you people all done undergraduate courses in mathematics (specifically
calculus of course)? Look:
lim (x ^ 0) = 1 <> lim (0 ^ x) = 0
x->0 x->0
Both have x approaching from above for consistency. Now does this look well
defined to you? I have done many problems asking to solve: lim f(x)^g(x)
where lim f(x)=0 and lim g(x)=0 but have gotten both answers 0 and 1 (for
different questions of course).
0^0 is not a well defined expression. It's about as well defined as 0/0. I
hope your calculus professors don't catch your postings to the contrary :)
Paul Hsieh
The opinions expressed above belong to those who believe them.
Ronald BODKIN
>
>Have you people all done undergraduate courses in mathematics (specifically
>calculus of course)? Look:
>
> lim (x ^ 0) = 1 <> lim (0 ^ x) = 0
> x->0 x->0
to be defined? That is the whole point of this discussion, that we
can define consistently 0^0 even though f(x,y)=x^y is a discontinuous
function at the origin. The answer is definitely 1.
Ron
Andrew P. Mullhaupt
> Ah, but do you understand that a function need not be continuous
> to be defined? That is the whole point of this discussion, that we
> can define consistently 0^0 even though f(x,y)=x^y is a discontinuous
> function at the origin. The answer is definitely 1.
> Ron
Yeah, but it looks like a bonehead move to tie your own hands with
the assumption that you'll always look at cases where 0^0=1. To
some people it looks stupid not to pick the 'right' answer and
call all the cases (which abound) where you want another value
pathological. We've had postings on both sides of this issue from
real authorities and their legitimate surrogates, but we're not
approaching resolution. It would seem time to 'agree to disagree'
for isn't it possible that this is one point on which reasonable
men can disagree?
Remember that mathematics is a plastic language. When it serves us,
(in GF(2) for example), we define 1+1=0 instead of 2. Definition
is not a matter of fact, but of choice. (This is not to excuse the
ragingly bad taste of some choices...) It might, for example be
useful to consider number theory from the point of view where 2 is
'defined' by a limit of reciprocals of zeros of the zeta function-
the benefit being that the Riemann hypothesis is universally true.
Now there are fully consistent arithmetics under this point of view,
we just don't know about the special case of the "usual" complex
numbers.
One presumes that Knuth is playing this kind of game when he puts
0^0=1, in order to 'save' the binomial theorem, but since I'm not
going to trash the binomial theorem for the sake of some endpoint,
I wish he hadn't bothered. The benefit of this rash action is a
little strained in view of the quality of the above 'proof' of the
Riemann hypothesis.
Hardy had the right idea - he encouraged the use of log x as a kind
of surrogate for x^0, but only as a guide to the intuition. This is
often more useful than trying to 'legitimize' the value at the endpoint
by legislation.
This thing takes on the 'I say tomato and you say tomato...' aspect
after about one round. Indeed, let's call the whole thing off.
Later,
Andrew Mullhaupt
Joe Keane
> x->0
`Come on' yourself! You've fallen for the 0^0 = 0 `proof' without examining
it very closely. The `limit' you quote is just plain wrong. Last time i
checked, zero to even a small negative power is infinite. When you keep this
in mind the 0^0 = 0 claim falls apart. We know that 0^x is very small when x
is negative, and very large when x is positive, so it's not hard to guess that
it's somewhere in between when x is zero. Besides, the counter-argument is
rock solid. x^0 = 1 always, whether x is positive, negative, or even complex.
Why not when it's zero? It'd be _really_ stupid, and a pain in the butt to
boot, to leave that one point out. Anyway, enough from me. Please keep in
mind what i've said before following up.
Peter Montgomery
Maple treats 0^0 as undefined internally. Here is an
example where that convention leads to a software bug.
My Maple source code is:
with(student):
# First powsubs call is successful
powsubs_1 := powsubs(x^3 = cube, x^2 + x^4);
# Second fails with 0^0 error, presumably doing x^2 = x^2 * (x^3)^0.
powsubs_2 := powsubs(x^3 = 0, x^2 + x^4);
quit;
The "with" statement identifies which software package to use.
The first powsubs call uses the relation "x^3 = cube"
to simplify "x^2 + x^4". It properly returns "x^2 + cube*x".
The second is similar, and should return "x^2".
But it instead gets an error. My output (on a SUN 3) is
|\^/|
._|\| |/|_. Department of Mathematics, U.C.L.A.
\ MAPLE / Version 4.2 --- Dec 1987
<____ ____> For on-line help, type help();
|
# First powsubs call is successful
2
powsubs_1 := x + cube x
# Second fails with 0^0 error, presumably doing x^2 = x^2 * (x^3)^0.
Error, (in student[powsubs]) 0^0 is undefined
words used=22093, alloc=16384, time=.367
Presumably powsubs uses the identity
3n + k n k
x = (x^3) x
to do the replacements. So it replaces x^4 by (x^3)^1 * x,
and x^2 by (x^3)^0 * x^2. Alas, if 0^0 is undefined, then this
algorithm does not work when the replacement field is identically zero.
--
--------
Peter Montgomery
pmon...@MATH.UCLA.EDU
Department of Mathematics, UCLA, Los Angeles, CA 90024
Ronald BODKIN
>Yeah, but it looks like a bonehead move to tie your own hands with
>the assumption that you'll always look at cases where 0^0=1.
that in any abstract algebraic system where you might define ^:AxA -> A
that ^:(0,0) |-> 1. However, in |N, |R, |Q, |Z, |C, the class of ordinals
and cardinals, I would say that it IS correct to insist that 0^0 is
consistently and correctly defined as 1 -- just like I would insist that
1+1 = 2 in all the above systems with the usual intended interpretation.
As for GF(2) and other abstract/different systems, they are a different
question. The big point is that (at least the first 5) correspond
strongly to some very real properties and so exponention does -- no one
is upset by 1+1=2 in these systems, I might add.
Ron
Aaron Watters
>and cardinals, I would say that it IS correct to insist that 0^0 is
>consistently and correctly defined as 1 -- just like I would insist that
>1+1 = 2 in all the above systems with the usual intended interpretation.
``Just like?'' Remarkable. The average 3 year old can tell you
why 1+1=2 (a better example might be 2+2=4
since 1+1 could be called a definition of 2), probably in several ways.
Your average PhD Mechanical Engineer might have trouble telling you
why 0^0=1. It seems to me these things are different in kind.
Or perhaps 0^0 is the definition of 1? -aaron.
david Casperson
In what sense does 1+1 not equal 2?
--
--david Casperson
Dave Seaman
>People keep insisting that 1+1 doesn't equal 2 in GF(2).
>In what sense does 1+1 not equal 2?
Actually, 1+1 does equal 2. It's just that in GF(2), 2 = 0.
--
Dave Seaman
a...@seaman.cc.purdue.edu