Same amount of irrational numbers in equal size intervals?

Question

Let's say you have the intervals $(q_1, q_2),\ q_i,\in \mathbb{Q}$ and $(q_1+a,q_2+a),\ a\in\mathbb{Q}$.

Is the cardinality of $(q_1,q_2)\cap \mathbb{I}$ equal to the cardinality of $(q_1+a,q_2+a)\cap \mathbb{I}$?

Answer 1

For your specific example, there is a simple 1-1 correspondence between irrationals in these intervals.

Specifically, the map $x \mapsto x+a$ does the trick. That's because if $a$ is rational, then $x$ is irrational if and only if $x+a$ is irrational.

So, yes, if by "amount" you mean cardinality.

Answer 2

The set of irrational numbers between $q_1$ and $q_2$ has Lebesgue measure $q_2-q_1$ and cardinality $\mathfrak c$.