I know the real numbers and the rational numbers have different cardinalities and I understand the proofs that the rational numbers are countable and the real numbers are uncountable.
I also know that the rational numbers are dense in the real numbers. However I am having trouble understanding how/why the density of the rationals does not contradict the difference in cardinality between the rationals and the reals.
I can't seem to wrap my mind around this so any motivation would be extremely helpful!
Salaam!
The set of rational numbers $\mathbb{Q}$ is countable because it has the same cardinality as the natural numbers $\mathbb{N}$.
The set of rational numbers $\mathbb{Q}$ is dense in the set of real numbers $\mathbb{R}$.
However, the cardinality of the rationals $\mathbb{Q}$ is not equal to the cardinality of the reals $\mathbb{R}$.
I can see why you find it difficult to make sense of it, but I would remind you that the set of irrational numbers is dense in the set of real numbers $\mathbb{R}$, too! Thus, casually speaking, it kind of makes the "size" of the reals $\mathbb{R}$ much larger than that of the rationals $\mathbb{Q}$, and so the cardinality of the reals $\mathbb{R}$ is much bigger than that of the rationals $\mathbb{Q}$.
I hope you find this make sense!