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**

Map numbers of the form q+k√2 for some q∈Q and k∈N to q+(k+1)√2 and fix all other numbers.

**Attribution***Source : Link , Question Author : marty cohen , Answer Author : WimC*