For Discrete Exponents
In most settings not involving continuity in the exponent, interpreting 00 as 1 simplifies formulas and eliminates the need for special cases in theorems. (See the next paragraph for some settings that do involve continuity.) For example:
- Regarding b0 as an empty product assigns it the value 1, even when b = 0.
- The combinatorial interpretation of 00 is the number of empty tuples of elements from the empty set. There is exactly one empty tuple.
- Equivalently, the set-theoretic interpretation of 00 is the number of functions from the empty set to the empty set. There is exactly one such function, the empty function.
- The notation for polynomials and power series rely on defining 00 = 1. Identities like and and the binomial theorem are not valid for x = 0 unless 00 = 1.
- In differential calculus, the power rule is not valid for n = 1 at x = 0 unless 00 = 1.
Read more about this topic: Complex Numbers Exponential, Zero To The Power of Zero
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