$\mathbb{R^+}$ is the disjoint union of two nonempty sets, each closed under addition. I saw Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition. and have a question related to the answer (I'm not sure if this is the right place to post it);
Why don't we just take
$\mathcal{A}=\{A\subseteq \mathbb{R^+}\ :\text{A is closed under addition and all elements of A are irrational}\}$,
$\mathcal{B}=\{B\subseteq \mathbb{R^+}\ :\text{B is closed under addition and all elements of B are rational}\}$
and order both of them with the partial order $\subseteq$ so that they will satisfy the chain condition? Then, they have maximal elements $\bigcup{\mathcal{A}}$ and $\bigcup{\mathcal{B}}$ respectively. So it remains to verify that $\bigcup{\mathcal{A}}$ is the set of all irrationals and $\bigcup{\mathcal{B}}$ is the set of all rationals. Is this approach correct?

While the sum of rational numbers is always rational, the sum of irrational numbers is not always irrational.
If you want to split $\Bbb R^+$ completely you will have to mix the two, as Lord_Farin commented $\pi$ and $4-\pi$ are both irrational, but their sum is rational. 
