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

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+k2 for some qQ and kN to q+(k+1)2 and fix all other numbers.

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

Leave a Comment