The simplest definition of natural numbers is as the free algebraic structure generated by a nullary and unary operation (conventionally named "zero" and "succ"

The simplest definition of natural numbers is as the free algebraic structure generated by a nullary and unary operation (conventionally named "zero" and "succ"), which makes it natural to associate zero as the value of the nullary operation, so that one can be it's successor and addition by one works as expected. Not having zero and associating one as the result of the nullary operation works as well, but then the natural definition of addition will result in x + y - 1 rather than x + y and it seems likely extension to rational numbers would not give an intuitive result.