>>11416134Let be the group of integers under addition, and let where . It is clear that the identity elements of these groups is and respectively. There is a natural homomorphism defined by . We can easily verify the homomorphism property, which for integer exponents follows trivially from association: . Thus our map is a homomorphism as desired. Now, we show why then exponentiation by integers must yield 1 when taking the 0 exponent. We first claim that this homomorphism must map an identity element 0 to the identity element 1.
Proof: . We see that is both the left and right identity for itself here, but that is only possible when it is an identity element by definition, so we conclude . This gives us directly that . An almost identical proof extends this to rational exponents. Exponentials are important from a fundamental theory perspective partially because of the above: they translate additive structure into multiplicative structure.
>>11416185The recursive definition gives way for integer and rational exponentiation to make sense. The analytic definition follows from basic analysis on top of the recursive definition for integers and rationals. Without going into more detail, it’s clear to see that the analytic definition is exactly the rational one when .
