On the Density of Q in R and the Uncountability of R, Countability of Q

I just finished my first course in Analysis as an undergraduate student and I've got something that I can't quite reconcile in my mind. Maybe someone can help me understand how this is possible?

We learned that regarding the set of real numbers (R) and the set of rational numbers (Q), Q is dense in R. That is, for every two real numbers, there exists a rational number between them.

We also learned R is uncountable, and Q is countable. That is, there are more real numbers than there are natural numbers.

It seems to me that the Density of Q in R implies that Q has the same cardinality as R (or has an even larger cardinality). My thinking is that if we have r1 and r2, there is a q1 between them. If we have r1, r2, r3, r4, then there must at least exist q1, q2, and q3 between them. (In fact, there could also be q4, q5, ..., qn and not violate the density of Q in R).

This has been driving me crazy lately, I can't figure out how all of these statements can be true.