# Cardinality of rational number and irrational number

Since there is a rational number between any two irrational numbers, so can we say that rational number and irrational number have the same cardinality?

No. $\mathbb Q$ is countable. Suppose that the irrational numbers have the same cardinality, then $\mathbb R \setminus \mathbb Q$ is also countable. This gives

$\mathbb R = \mathbb Q \cup (\mathbb R \setminus \mathbb Q)$ is countable, a contradiction.

• I know your argument. But what I mean is intuitively if there is a rational number between any two irrational numbers, then there are as many rational numbers as irrational numbers. – Bella Mar 20 '17 at 10:26

Of course NOT.

The argument is not correct. For example, I can say there always exists a rational number between any two distinct real numbers, but apparently $\mathbb{Q}$ and $\mathbb{R}$ don't have the same cardinality.