0^0 = 1?

Mudd Math Fun Fact gives arguments to show that the answer to Zero to the Zero Power "should" be 1.

• The alternating sum of binomial coefficients from the n-th row of Pascal's triangle is what you obtain by expanding (1-1)ⁿ using the binomial theorem, i.e., 0ⁿ. But the alternating sum of the entries of every row except the top row is 0, since 0^k=0 for all k greater than 1. But the top row of Pascal's triangle contains a single 1, so its alternating sum is 1, which supports the notion that (1-1)⁰=0⁰ if it were defined, should be 1.
• The limit of x^x as x tends to zero (from the right) is 1. In other words, if we want the x^x function to be right continuous at 0, we should define it to be 1.
• The expression mⁿ is the product of m with itself n times. Thus m⁰, the "empty product", should be 1 (no matter what m is).
• Another way to view the expression mⁿ is as the number of ways to map an n-element set to an m-element set. For instance, there are 9 ways to map a 2-element set to a 3-element set. There are NO ways to map a 2-element set to the empty set (hence 0²=0). However, there is exactly one way to map the empty set to itself: use the identity map! Hence 0⁰=1.
• Here's an aesthetic reason. A power series is often compactly expressed as
  SUM_{n=0 to INFINITY} a_n (x-c)ⁿ.
  We desire this expression to evaluate to a₀ when x=c, but the n=0 term in the above expression is problematic at x=c. This can be fixed by separating the a₀ term (not as nice) or by defining 0⁰=1.

Link to Google 0^0 search via Gseeker & Digg

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