Is a x ^ 0 "actually" 1, or is it just a "definition" - a sort of fudge to make the maths work?
I understand how say 10 ^ 2 is 100 and 10 ^ -2 is 0.01, making it intuitive to say 10 ^ 0 is 1, but is it inherently so, or was it agreed that we'd consider it to be 1 for the above reasons?
So the x^n operation is "n copies of x" times 1 (the multiplicative identity). x^0 is 0 copies of x.
Know your field
>Is a x ^ 0 "actually" 1, or is it just a "definition" - a sort of fudge to make the maths work?
It depends on what your definition of ^ is. But even with the basic definition of x^n is x*x*...*x n times (with n > 0),
it's somewhat natural to say x^0 = 1 because of what you said, so we say it's like so
(but be aware, x^-n is just the same, it was defined to be the inverse of x^n to make things work well, in some way).
We can also define ^ with the exponential function, and then x^0 is defined to be 1 without a need to go further.
Yes, before I'd seen a demonstration of 0.01, 0.1 (negative powers), 1, 10, 100, etc, that would have been my "intuitive" reasoning: that raising to the power of 0 is 0. Just like anything times 0 = 0.
It's not necessarily anything big, and this field is not practiced in my country. Therefore, I have no one to ask about it , and my knowledge is self-obtained. Nevertheless, I put some pieces together and would like to publish my stuff (which itself is correct) if possible.
If it is original work and you believe it to be worthwhile to the field post it. It may turn out that that result has already been found, in that case you have independently verified that result. Ergo it is still worthwhile to post.
Bra ket notation. |Ψ> is a vector, is a scalar product
x^0 isn't 0 copies of x, it's the "empty product" (it has in fact nothing to do with x, since it's empty). And the empty product must be 1, as you noticed.
In a maybe simpler way, say you multiply x^0 by x^n. That makes it x^(0+n) = x^n.
So x^0 = 1.
Thank you!