How would this work when mapping them to the fractional part of the real number 2 ?

Collapsing a countably infinite set to 0 doesn't seem useful or reversable.

The integer part is easy, since we already have the integers. Once you have D=[0,1), then you can define R=ZxD. That is to say, this definition of R seperates out the integral and fractional components of every real number.

fun n => 0

I'm not sure what you're trying to disagree with here.