Example of a bijection from the set of real numbers to a subset of irrationals I need an example of a bijection from the set of real numbers to a subset of the irrationals.
I tried something like
$f(x)=x+\sqrt{2}$,
but where should I map $-\sqrt{2}$?

Let $A_i = \pi^i {\mathbb Q}$ and $ A =\cup_{i\ge0} A_i$.  Then the map $h:{\mathbb R}\backslash{\mathbb Q}\rightarrow {\mathbb R}$
$$
h(x) = \left\{
\begin{array}{ll}
\pi x & x\in A \\
x & x \in B={\mathbb R}\backslash A\\
\end{array}
\right.
$$
is a bijection.
