Let $D$ be a squarefree integer. Prove that $\{ a+b\sqrt{D} \ |\ a,b \in \mathbb{Z}\} \cup \{ (m + \frac{1}{2}) + (n + \frac{1}{2})\sqrt{D} \ |\ m,n \in \mathbb{Z} \} = \{ c + d(1 + \sqrt{D})/2 \ |\ c,d \in \mathbb{Z}\}$.
Let $\omega = (1+\sqrt{D})/2$.
$(\subseteq)$ Evidently, $a+b\sqrt{D} = a-b + b\omega$, where $a,b \in \mathbb{Z}$, and $(m + \frac{1}{2}) + (n+\frac{1}{2})\sqrt{D} = m-n + (n+1)\omega$.
$(\supseteq)$ If $d = 2k$, then evidently $c+d\omega = c+k + k\sqrt{D}$. If $d = 2k+1$, then evidently $c+d\omega = c+k+\frac{1}{2} + (k + \frac{1}{2})\sqrt{D}$.