Real Analysis – Is There a Simple 1-1 Mapping Between Reals and Irrationals?

Question

Is there a simple, constructive, 1-1 mapping between the reals and the irrationals?

I know that the Cantor–Bernstein–Schroeder theorem
implies the existence
of a 1-1 mapping between the reals and the irrationals,
but the proofs of this theorem
are nonconstructive.

I wondered if a simple
(not involving an infinite set of mappings)
constructive
(so the mapping is straightforwardly specified)
mapping existed.

I have considered
things like
mapping the rationals
to the rationals plus a fixed irrational,
but then I could not figure out
how to prevent an infinite
(possible uncountably infinite)
regression.

Answer 1

Map numbers of the form $q + k\sqrt{2}$ for some $q\in \mathbb{Q}$ and $k \in \mathbb{N}$ to $q + (k+1)\sqrt{2}$ and fix all other numbers.