Use of zero to the power of zero

Question

To my understanding, the value of $0^0$ is usually undefined, however sometimes people define it to be equal to either $1$ or $0$, when they need certain formulas to work for practical purposes.

My question is, can defining $0^0=0$ or $0^0=1$ be used while proving theorems and why?

Slight clarification: Imagine that some well-known mathematician publishes a proof to one of the millennium problems. However, their proof requires that $0^0=1$. Will such proof be generally accepted, given that there are no other flaws in the proof?

Answer 1

You can define it to be whatever you want. All you need to do is to say what you defined it to be, and use it consistently and in a way consistent with your choice.

By the way, usually $0^0$ is defined as $1$, since this makes consistent use most convenient (in many persons' minds).

To illustrate what I mean, you can say $0^0 = 13$. But if you do this, then you cannot compute $0^0 \cdot 0^0 = 0^{0+0} = 0^0$, because $0^0 \cdot 0^0 = 0^{0+0}$ just is not true under this choice.

If instead you choose $0^0 = 1$ then it is true that $0^0 \cdot 0^0 = 0^{0+0}$, which is intuitive and convenient and makes a more reasonable choice than $0^0 = 13$.

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On the clarified question: yes, it is basically a choice of notation; it is not really a requirement as such. Whether one accepts the proof or not would also not depend on one's own preferred choice. It is not some 'believe' or anything like that, it's just a convention.

Yet note that $0^0 = 1$ must not be confused with the assertions like:

For $(a_n)_{n \in \mathbb{N}}$, $(b_n)_{n \in \mathbb{N}}$ sequences of positive reals. If $\lim_{n \to \infty} a_n = 0$ and $\lim_{n \to \infty} b_n = 0$, then $\lim_{n \to \infty} a_n^{b_n} = 1$.

This is just wrong in general, and assuming it to be true would invalidate the proof.

Answer 2

Thats tricky but easy to prove that 0th power of 0 equals to one.The image attached herewith the answer contains it's calculation. calculation of zeroth power of zero