Re: 0^0

The Ghost In The Machine <ewill@xxxxxxxxxxxxxxxxxxxxxxx> writes:

> In sci.math, Androcles wrote:
>>
>> "David Kastrup" <dak@xxxxxxx> wrote:
>> | "Androcles" <Androcles@ MyPlace.org> writes:
>> |
>> | > "David Kastrup" <dak@xxxxxxx> wrote:
>> | > | "Androcles" <Androcles@ MyPlace.org> writes:
>> | > |
>> | > | > "David Kastrup" <dak@xxxxxxx> wrote:
>> | > | > | "Ulrich Sondermann" <usondermann@xxxxxxxxxxxxx> writes:
>> | > | > |
>> | > | > | > The problem with 0^0 = 1 is that it implies 0/0 = 1 ,
>> | > | > |
>> | > | > | Bull***. It does no such thing.
>> | > | >
>> | > | >
>> | > | > x^1 = x
>> | > | > x^(-1) = 1/x
>> | > | > x^(1-1) = x/x
>> | > | > x = 0.
>> | > | > 0^0 = 0/0
>> | > | > Moron!
>> | > |
>> | > | x^2 = x*x
>> | > | x^(-1) = 1/x
>> | > | x^(2-1) = (x*x)/x
>> | > | x = 0.
>> | > | 0^1 = 0/0
>> | > | Moron!
>> | >
>> | > You certainly are, because you are correct. 0^n is undefined.
>> | > If you think otherwise, define 0^-1
>> |
>> | 0^x is undefined for x<0, it is 1 for x=0, and it is 0 for x>0.
>>
>> 0^x is undefined for x<=0, 0 for x> 0, and YOU define
>> 0^0 as 1. Prove your definition is well-defined, running off at
>> the mouth is not a valid proof .
>> Androcles
>
> Pedant point:
>
> 0^z is undefined for Re(z) < 0 and for Re(z) = 0, and is 0
> for Re(z) > 0.
>
> Lim(n -> +oo) (1/n) = 0
> Lim(n -> +oo) (1/log(n)) = 0
>
> Lim(n -> +oo) (1/n)^(1/log(n))
> = Lim(n -> +oo) exp(-log(n) * 1/log(n)) = exp(-1) = 1/e
>
> Or:
>
> Lim(x->0+) 0^x = 0
> Lim(y->0+) y^0 = 1
> Lim( (x,y)->(0+,0+)) x^y = ??

Look up "defined" in a dictionary of your choice. You are putting the
cart before the horse. Defining a value does not involve limits,
since there is no mathematical law that functions have to be
continuous.

Function points have to be defined already before you can even start
taking limits.

And the definitions of powers with integral exponents make abundantly
clear that 0^0 has to be the neutral element for multiplication, since
adding exactly 0 factors of 0 to a product does not change its value.

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
.

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