Does it make sense to ask if there are as many real numbers between 1 and 2 as between 0 and 1?

Question

If yes, how does one go about finding the answer?

Can the same question be asked for the hyperreals too?

Answer 1

I think the map $f:(0,1) \to (1,2)$ given by $f(x) = 1+x$ makes a nice bijection between these sets.

Answer 2

hint: $f(x) = x+1$ is bijective.

Answer 3

It depends on how you want to count your reals.

The standard way (using cardinality) is to say that the sets have the same number of elements if there is a bijection between the sets. This seems reasonable but it implies that there are as many real numbers between $0$ and $1$ as between $0$ and $2$, for example.

If this is not desirable, you can define the "number" of real numbers in any given closed interval by the length of that interval. In this case, your answer is still yes and is well defined but does not respect bijections.