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zero_to_the_zero_equals_one
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andru235
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since most science reduces to math; and since most science views sentience as a strictly biological entity; science tends to doubt in multi-life
yet; given that zero raised to the power of itself equals one, it would seem that even *math* supports infinite existence (i know, i'm taking a flying leap; yet, am i?!?)
the effect of 'nothing' on 'nothing' literally is 'something'!!!
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050606
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science geek
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it's undefined, actually
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050606
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andru235
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really? hmm. there are several such anomalies on calculators, i guess.
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050607
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anne-girl
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it's because x^0 = x^(1-1)
= x^1 * x^(-1)
= x/x
so when x = 0
it's 0/0
but for any other value of x it's 1
explains just in case anyone cares
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050607
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andru235
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[from Dr. Math]
The following is a list of reasons why 0^0 should be 1.
Rotando & Korn show that if f and g are real functions that vanish at the origin and are analytic at 0 (infinitely differentiable is not sufficient), then f(x)^g(x) approaches 1 as x approaches 0 from the right.
From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):
Some textbooks leave the quantity 0^0 undefined, because the functions 0^x and x^0 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.
Published by Addison-Wesley, 2nd printing Dec, 1988.
As a rule of thumb, one can say that 0^0 = 1 , but 0.0^(0.0) is undefined, meaning that when approaching from a different direction there is no clearly predetermined value to assign to 0.0^(0.0) ; but Kahan has argued that 0.0^(0.0) should be 1, because if f(x), g(x) -- 0 as x approaches some limit, and f(x) and g(x) are analytic functions, then f(x)^g(x) -- 1 .
The discussion of 0^0 is very old. Euler argues for 0^0 = 1 since a^0 = 1 for a not equal to 0 . The controversy raged throughout the nineteenth century, but was mainly conducted in the pages of the lesser journals: Grunert's Archiv and Schlomilch's Zeitshrift. Consensus has recently been built around setting the value of 0^0 = 1 .
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050607
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andru235
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[and from www.math.hmc.edu]
We'll give several arguments to show that the answer "should" be 1.
The alternating sum of binomial coefficients from the n-th row of Pascal's triangle is what you obtain by expanding (1-1)n using the binomial theorem, i.e., 0n. But the alternating sum of the entries of every row except the top row is 0, since 0k=0 for all k greater than 1. But the top row of Pascal's triangle contains a single 1, so its alternating sum is 1, which supports the notion that (1-1)0=00 if it were defined, should be 1.
The limit of xx as x tends to zero (from the right) is 1. In other words, if we want the xx function to be right continuous at 0, we should define it to be 1.
===
The expression mn is the product of m with itself n times. Thus m0, the "empty product", should be 1 (no matter what m is).
===
Another way to view the expression mn 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 02=0). However, there is exactly one way to map the empty set to itself: use the identity map! Hence 00=1.
===
Here's an aesthetic reason. A power series is often compactly expressed as
SUMn=0 to INFINITY an (x-c)n.
We desire this expression to evaluate to a0 when x=c, but the n=0 term in the above expression is problematic at x=c. This can be fixed by separating the a0 term (not as nice) or by defining 00=1.
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050607
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anne-girl
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ouch... i seem to have been defeated, by volume if nothing else :)
looks up the binomial theorem, and tries to make sense of it all
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050607
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anne-girl
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even google calculator says 0^0 = 1... it must be true
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050607
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on the road
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What Andru basically said in that stuff from Dr Math was that the LIMIT of the function 0^0 is 1.
That means that it equals one in the tiny neighbourhood around the origin (ie 0) but itself it is undefined.
These limits(aka approximations) are often used for many other undefined quantities too.
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050607
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anne-girl
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basically what the page says, it seems, is that there are a bunch of mathematical theorems which would have exceptions at 0 if 0^0 were not 1
and because this would complicate things, make them more ugly and such, it's the convention to say that 0^0 = 1, despite the fact that if you want to look at it a certain way, it's undefined
it's kind of like saying 0! = 1... it doesn't make any sense, given the definition of factorial, but it makes functions which use factorials (like, er, taylor series from what i've seen) not requre exceptions for the case when n = 0
thinks that's it
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050607
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andru235
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it is a quandry which opens the door to the possibility that there may be mathematical situations that have multiple, mutually exclusive truths
indeed i don't really care; my doubts about the existence of 'nothing' remain intact;
what's curious to me is the attempts we all make, myself included, to conquer 'abstract' holes by filling them with 'concrete', usually an unsuccessful endeavor
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050607
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anne-girl
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i really don't know anything at all
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060614
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jane
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oh, i see now.
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080501
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hsg
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it is most useful to define 0^0 as the value of 1/ infinity. whereas its quantitative value is zero but in issues of quality even an infinitely small detail has importance.
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080502
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tourist
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Does This Mean I CAN Have My Cake
AND Eat It Too?
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080502
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