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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 0k=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 xx as x tends to zero (from the right) is 1. In other words, if we want the xx 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
    SUMn=0 to INFINITY an (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