Cosets of Positive Reals in Multiplicative Group of Complex Numbers

Theorem
Let $S$ be the positive real axis of the complex plane:


 * $S = \set {z \in \C: z = x + 0 i, x \in \R_{>0} }$

Consider the algebraic structure $\struct {S, \times}$ as a subgroup of the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$.

The cosets of $\struct {S, \times}$ are the sets of the form:


 * $S = \set {z \in \C: \exists \theta \in \hointr 0 {2 \pi}: z = r e^{i \theta}, r \in \R}$

That is, the sets of all complex numbers with a constant argument.