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Zero to zero power

Sorry for asking this simple question, but googling this question yields conflicting answers.
Some say it's indeterminate, other's say it's $1$.

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marked as duplicate by Clive Newstead, amWhy, Jonas Meyer, rschwieb, user1551 Jan 6 '13 at 20:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

The answer is "indeterminate and $1$". – Michael Hardy Jan 6 '13 at 22:51

1 Answer 1

The short answer: 'yes'. The slightly longer answer: $0^0$ has different meanings in different contexts, depending on whether the power represents the analytical power (defined either through continuation of the function $x^y$ from rational values of $y$ to real values) or the combinatorial power (where $A^B$ is defined as the number of maps from the set $A$ to the set $B$). The 'analytical' $0^0$ is considered undefined because the double-limit $\displaystyle\lim_{x,y\to 0+}x^y$ will yield different values (in fact, it can yield all possible values) depending on what path is taken towards the origin. By contrast, the combinatorial power $0^0$ is taken to be $1$; there is considered to be one map from the empty set to itself, the identity map. This convention substantially simplifies formulas of a combinatorial nature; for instance, the binomial identity $(1+x)^n = x^0+nx^1+{n \choose 2}x^2+\ldots$.

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