Regarding the theory of real addition with a nonzero constant

Question

Let us consider the theory of the structure $(\mathbb{R}, +, r)$, where $r$ is a nonzero constant. Does it matter which $r$ it is? In other words, is any theory of that form the same as any other theory of that form? This is different from a previous question, because that one was a universal algebra question, while this is a model theory question.

Answer 1

For any $c\in\mathbb{R}\setminus\{0\}$, the map $$x\mapsto cx$$ is an automorphism of $(\mathbb{R}, +)$; this means $(\mathbb{R}; +,r)\cong(\mathbb{R}; +, 1)$ for any $r\not=0$, so the answer to your question is "no" in the strongest way possible.