0^0 = 1?

[Chris Dreher]

(04-28-2017 04:57 AM)ttw Wrote:  How zeros (and ones) are treated depends on use. The most common definition for 0^0 by mathematicians is 1. Basically this comes from combinatorial arguments. The only case where another definition would make sense is in looking at 0^0 with both the base and exponent as reals. Then there is a path-dependent answer.

To get the binomial coefficients to come out right, one should define 0^0 as 1. Some computer languages define 0^0 as 1 if both zeros are integers and zero if the exponent is zero and the base is real. An real exponent is undefined as there are many branch points to the exponential.

I'd go for 1 in general; it's rare that another number would be more useful. Similarly the sum of an empty list should be 0 and the product of an empty list should be 1.

Actually, most mathematician regard 0^0 as an Indeterminate Form. They do not regard it as 1 (or 0) unless there are additional constraints.