On Wednesday, September 18, 2013 4:53:13 PM UTC+2, Dan Christensen wrote: > For the natural numbers, exponents greater than 1 are naturally defined for x in N as follows: > > > > x^2=xx > > x^3=xxx > > x^4=xxxx > > x^5=xxxxx > > > > and so on. > > > > More formally, exponentiation can be defined as a binary function on the set of natural number N such that: > > > > (1) Ax in N: x^2=x*x > > > > (2) Ax,y in N: x^(y+1) = x^y * x > > > > It can then be shown that: > > > > (1) Ax in N:(x=/=0 => x^1=x) > > > > (2) Ax in N:(x=/=0 => x^0=1) > > > > (3) Ax,y,z in N:(x^(y+z) = x^y * x^z) => 0^1=0 /\ (0^0=0 \/ 0^0=1) > > > > (Formal proof to follow.) > > > > Thus, if the Product of Powers Rule is to hold on N, 0^0 will be ambiguous -- being either 0 or 1. Unless one of these alternatives can be formally proven or shown to give rise to a contradiction, the prudent course is to leave 0^0 undefined. > > > > Dan > > Download my DC Proof 2.0 software at http://www.dcproof.com
In this subjective of mathematics, we don't talk about proof. It is about definition. And in most of mathematics 0^0 = 1. However, I am not troubled if you define it as something else. It's like 0! In some area of mathematics, it's defined as 1 and in other areas it's defined different. I don't get the creeps over this.