Counting Cosets of $\langle\tfrac12\rangle$, in $\Bbb{R}$ and in $\Bbb{R}^{\times}$

Question

Describe the cosets of the subgroups described:

1. The subgroup $\langle\frac{1}{2}\rangle$ of $\mathbb{R}^{\times}$, where $\mathbb{R}^{\times}$ is the group of non-zero real numbers with multiplication.
2. The subgroup $\langle\frac{1}{2}\rangle$ of $\mathbb{R}$, where $\mathbb{R}$ is the group of real numbers with addition

Answer 1

1. A coset of this subgroup is a set of all real numbers who's pairwise ratio is a power of $2$ (really a power of $\frac{1}{2}$, but that's the same thing). For instance, the coset containing $\pi$ also contains $16\pi$ and $\frac{\pi}{128}$, because $16$ and $\frac{1}{128}$ are both powers of $2$.

2. Same thing, but this time the defining feature is that pairwise differences is a multiple of $\frac{1}{2}$. That means that the coset containing for instance $\log 53$ also contains $\frac{2197}{2}+\log 53$.

Answer 2

If I understand correctly, you mean the subgroups $$H_1=\langle\tfrac{1}{2}\rangle\subset\mathbb{R}^{\times}\qquad\text{ and }\ H_2= \langle\tfrac{1}{2}\rangle\subset\mathbb{R}^+.$$ So as sets, these subgroups look like $$H_1=\left\{2^k:\ k\in\mathbb{Z}\right\}\qquad\text{ and }\qquad H_2=\left\{\tfrac{k}{2}:\ k\in\mathbb{Z}\right\}.$$ Can you now determine what the cosets of these subgroups look like? Can you describe a set of representatives for the cosets?