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Consider this:

0^0 = 1
0^1 = 0
0^2 = 0
0^n = 0 (for int n > 0)

So how come 0^0 = 1, how can you get something out of nothing? Isn't it a mathematical heresy?

Also what does it mean to raise something to the power of 0? Like 2^2 = 2 * 2 = 4, I understand that, but what 2^0=? in that case?

OutFall
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3 Answers3

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Simple, you are absolutely right. $0^0$ is not defined, it doesn't really make sense.

But on the other hand, it is often handy to take $0^0=1$. For example, if you write a polynomial as: $$ p(x) = \sum_{0 \le k \le n} a_k x^k $$ you want to have $p(0) = a_0$, i.e., you are implicitly taking $0^0 = 1$. In other situations it is nice to take $0^0 = 0$. Just be careful.

There have been several threads here on this subject, look around.

vonbrand
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If you take the limit of $x^x$ as $x$ goes to zero, that comes out to 1. That's one rationale for letting $0^0$ equal 1. For nonzero $x$, $x^0 = x^1/x = 1$.

StumpyLeg
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$0^0$ is not necessarily equal to $1$, it is indeterminate, at least in the context of Algebra and Calculus. Take for example this question, as an example of why $0^0$ stays undefined when it comes up in Algebra and Calculus. If you take a limit that appears to approach $0^0$, and we were to define $0^0=1$, we might end up with something like $1=\frac12$.

Community
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Ephraim
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  • In set theory, or related fields $0^0$ is absolutely equal to $1$. Because it's defined as the cardinality of $\varnothing^\varnothing={\varnothing}$, which is precisely $1$. – Asaf Karagila Jun 04 '14 at 21:25
  • @Asaf - fair enough. I edited my answer to specify that it is undefined specifically for the purposes of algebra and calculus. – Ephraim Jun 04 '14 at 21:43