Numbers to the Power of Zero I have been a witness to many a discussion about numbers to the power of zero, but I have never really been sold on any claims or explanations. This is a three part question, the parts are as follows...


*

*Why does $n^{0}=1$ when $n\neq 0$?  How does that get defined?


*What is $0^{0}$?  Is it undefined?  If so, why does it not equal $1$?


*What is the equation that defines exponents?  I can easily write a small program to do it (see below), but what about in equation format?

I just want a little discussion about numbers to the power of zero, for some clarification.

Code for Exponents: (pseudo-code/Ruby)
def int find_exp (int x, int n){
    int total = 1;
    n.times{total*=x}
    return total;
}


I've always been told that the definition of a power is: $a^{b+c}=a^b\times{a^c}$.
My reasoning works as follows

*

*Set $b=1$ and $c=0$ (so $b+c= 1+0 = 1$)

*This would mean $a^1=a^1*a^0$

*There are only two options where one can get the same result by multiplying by another number:


*

*That 'other number' is 1 which would mean that $a^0=1$

*That 'other number' is 0 and the first number is also 0 which mean that $a^1=0$ and $a^0=0$


*

*We now have 2 valid options left, which means we must expand the definition:

*In the second case every power would be 0 because $a^x=a^{1+1+1+...}$ with $x$ amount of 1's. This makes the whole idea of a power rather 'useless' so we can skip this option

*For the 1st option only the $a^1=a$ makes sense because it's the only way to invalidate the 2nd option and not break the original definition.

So the full definition of a power now becomes $a^{b+c}=a^b\times{a^c}$ and $a^1=a$.
This would also lead to $a^0=a^{1-1}=\frac{a^1}{a^1}=1$
