# On the cardinality of rationals vs irrationals

I understand that there are infinitely many more irrational numbers than there are rationals - there are many ways in which one can intuitively understand this. However, consider the following:

LEMMA: For every pair of distinct irrational numbers, there exists a rational number between them.

PROOF: Let $$a,b$$ be distinct irrational numbers such that $$b>a$$ and let $$\delta = b-a$$. Then for some large enough $$N$$, we have $$N\delta >1$$ so there exists an integer in the interval $$[Na,Nb]$$ and as such there exists a rational number in the interval $$[a,b]$$.

By this logic, we can find a rational number for every pair of distinct irrational numbers - why does this not imply there are as many rationals as irrationals?

In order to get an appropriate bijection, we need to assign to a pair of irrationals $$(\alpha,\beta)$$ a rational $$q$$ which is unique, in the sense that a different pair $$(\alpha',\beta')$$ will always get a different rational $$q'$$. Intuitively, there are a lot of tasks to be done (= pairs of irrationals to separate) and few workers (= rationals), but each worker can do a lot of tasks (= separate continuum many pairs).