The order type of the irrational numbers $(\mathbb R\backslash \mathbb Q,\leq)$ is denoted $\theta$. We have the following algebraic identities:
$\theta^\ast = \theta$, as evidenced by the mapping $x\mapsto -x$ which preserves irrationality
$\theta+\theta=\theta$, which can be shown by proving the intermediate result that the ordering of positive irrationals $(\mathbb R^+\backslash\mathbb Q,\leq)$ has order type $\theta$
$\theta+1+\theta=\theta$, which can be shown by proving that $((x,\infty)\backslash\mathbb Q,\leq)$ has order type $\theta$ when $x$ is irrational
$(1+\theta)\cdot\zeta=\theta$, which follows from the previous identity and the fact that $\theta\cdot\zeta=\theta$