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From Norbert Thiebaud <nthieb...@gmail.com>
Subject Re: Calc behavior: result of 0 ^ 0
Date Wed, 13 Feb 2013 07:28:30 GMT
```On Tue, Feb 12, 2013 at 10:09 PM, Rob Weir <rabastus@gmail.com> wrote:
> On Feb 12, 2013, at 10:39 PM, Pedro Giffuni <pfg@apache.org> wrote:
>
>> (OK, I guess it's better to re-subscribe to the list).
>>
>> In reply to Norbert Thiebaud*:
>>
>> In the Power rule, which *is* commonly used for differentiation, we take a series
>> of polinomials where n !=0. n is not only different than zero, most importantly,
>> it is a constant.

Power Rule : d/dx x^n = n.x^(n-1)  for n != 0  indeed.
so for n=1  (which _is_ different of 0 !)
d/dx X = 1.x^0
for _all_ x. including x=0. (last I check f(x) = x is differentiable in 0.

I know math can be challenging... but you don't get to invent
restriction on the Power Rule just to fit you argument.

>> and we just can't assume every speadsheet
>> user will use a restricted set of capabilities.
>>
>> Now, in a spreadsheet this formula would be used if you have a polinomial and
>> you want to calculate and/or graph it's derivative. Since we don't do symbolic
>> math in the speadsheet you would actually do this by hand and you would resolve
>> the trivial constant^0 cases.

Really... but extending by continuity a function in 0, without
consideration for convergence,  _that_ is something that is done by
iow just because 0^0 is an indeterminate _form_ does not mean that 0^0
can not have a value... it just mean that when searching for a limit
for a function h(x)
if your _transformation_ lead you to 0^0 you cannot conclude from that
_form_ that means that the rule and tools that allow you to jump form
lim -> 0 to a value in 0 do not hold when they lead to that 'form'. I
know math is a tricky thing... but the definition of words and their
scope of application is kind of important in Math.

>>
>> In the case of the set theory book, do note that the author is constructing
>> his own algebra,

The fact that you call 'Nicola Bourbaki' 'the author', is in itself
unlikely to know 'who' Borbaki is.

>> that get outside his set: 0^0 and x/0 are such cases. The text is not
>> a demonstration, it is simply a statement taken out of context.

You ask for a practical spreadsheet example, when one is given you
invent new 'rules' to ignore' it.
You claim that 'real mathematician' consider 0^0=... NaN ? Error ?
And when I gave you the page and line from one of the most rigorous
mathematical body of work of the 20th century (yep Bourbaki... look it
up)
you and hand-wave, pretending the author did not mean it.. or even
better " if this author(sic) *is* using mathematics correctly."

>>
>> I guess looking hard it may be possible to find an elaborated case where
>> someone manages to shoot himself in the foot

Sure, Leonard Euler, who introduced 0^0 = 1 circa 1740, was notorious
for shooting himself in the foot when doing math...

For those interested in the actual Math... in Math words have meaning
and that meaning have often context. let me develop a bit the notion
of 'form' mentioned earlier:
for instance in the expression 'in an indeterminate form', there is
'form' and it matter because in the context of determining extension
by continuity of a function, there are certain case where you can
transform you equation into another 'form' but if these transformation
lead you to an 'indeterminate form', you have to find another
transformation to continue...
hence h = f^g  with f(x)->0 x->inf and g(x)->0 x->inf  then -- once it
is establish that h actually converge in the operating set, and that
is another topic altogether -- lim h(x) x->0 = (lim f)^(lim g).
passing 'to the limit' in each term would yield 0^0 with is a
indeterminable 'form' (not a value, not a number, not claimed to be
the result of a calculation of power(0,0), but a 'form' of the
equation that is indeterminate...) at which point you cannot conclude,
what the limit is. What a mathematician is to do is to 'trans-form'
the original h in such a way that it lead him to a path to an actual
value.

in other words that is a very specific meaning for a very specific
subset of mathematics, that does not conflict with defining power(0,0)
= 1.

wrt to the 'context' of the quote I gave earlier:

"Proposition 9: : Let X and Y be two sets, a and b their respective
cardinals, then the set X{superscript Y} has cardinal a {superscript
b}. "

( I will use x^y here from now on to note x {superscript y} for readability )

"Porposition 11: Let a be a cardinal. then a^0 = 1, a^1 = a, 1^a = 1;
and 0^a = 0 if a != 0

For there exist a unique mapping of 'empty-set' into any given set
(namely, the mapping whose graph is the empty set); the set of
mappings of a set consisting of a single element into an arbitrary set
X is equipotent to X (Chapter II, pragraph 5.3); there exist a unique
mapping of an arbitrary set into a set consisting of a single element;
and finally there is not mapping of a non-empty set into the
empty-set;
* Note in particular that 0^0 = 1
"

Here is the full context of the quote. And if you think you have a
proof that there is a mathematical error there, by all means, rush to
your local university, as surely proving that half-way to the first
volume, on set theory, of a body of work that is acclaimed for it's
rigor and aim at grounding the entire field of mathematics soundly in
the rigor of set theory, there is an 'error', will surely promptly get
you a PhD in math... since that has escaped the attentive scrutiny and
peer review of the entire world of mathematicians for decades...

```
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