# I just found out that $0^0$ equals $1$, why is this? [duplicate]

This question already has an answer here:

I have done a lot of math so far, but I never stumbled on something this simple and yet mind boggling. Can someone tell me why $0^0$ equals $1$? I always knew that everything raised to a power of $0$ is $1$, but I never thought of the zero case itself.

I guess it's not the biggest problem in math, but I would like to have a good argument about why a $0$ can be turned into a $1$?

-

## marked as duplicate by Nate Eldredge, Simon Markett, martini, Sami Ben Romdhane, amWhyJun 8 '13 at 20:18

This question was marked as an exact duplicate of an existing question.

It's usually by definition. It prevents a lot you from making an exception or separate remark for $0^0$ when stating proofs. There is a good set-theoretic way to interpret $0^0$ which is a good indicator for why $0^0 = 1$. We can think of $a^b$ as the number of functions from a set with $a$ number of elements to a set with $b$ number of elements. The set with no elements is the empty set and there is only one function from the empty set to the empty set: the empty function. – Cameron Williams Jun 8 '13 at 20:15
I've been told it's related to the empty product. – hasnohat Jun 8 '13 at 20:22
You could think it this way- actually you cannot multiply 0 with itself 0 times. So if the no. is not even multiplied with itself, then the thing which remains in the end in your answer is just 1. – Rohinb97 Jun 8 '13 at 21:04

This depends on the situation. $0^0$ is often considered an indeterminate form. I believe a combinatorial interpretation leads to $0^0=1$. Namely that the number of total outcomes from $m$ events with $k$ outcomes each is $k^m$. Thus zero events having zero outcomes each still produce $1$ total outcome and so $0^0=1$.