# Circle plus

Let $h>0$ and $z_1,z_2 \in \mathbb{C}_h$, the Hilger complex plane. Then we define the $\oplus_h$ operation by $$z_1 \oplus_h z_2 = z_1+z_2+z_1 z_2h.$$
Theorem: The structure $(\mathbb{C}_h,\oplus_h)$ is an Abelian group.