How can $\mathbb{Q}$ be dense in $\mathbb{R}$ and vice versa yet $|\mathbb{Q}| < |\mathbb{R}|$

Question

I am confused about the density and size of $\mathbb{Q}$ and $\mathbb{R}$; between any two rationals there is an irrational number and between any two irrationals there is a rational number. Does from this not follow that rationals and irrationals alternate? That cannot be the case however, as $|\mathbb{Q}| < |\mathbb{R}|$ as provable through a diagonal proof on the digits of the irrationals and if they were alternating they could be paired up and thus would be equal in quantity. So, how can there be a rational between any two irrationals and vice versa yet irrationals and rationals not be alternating?

Answer 1

This is a copy-paste of dxiv's comment and should serve as an answer too.
Between any two rationals there is an irrational number and between any two irrationals there is a rational number. This may be easier to rationalize if you think of it as "between any two rationals there is an (uncountable) infinity of irrational numbers, and between any two irrationals there is a (countable) infinity of rational numbers".