Uncountability of irrationals

$\mathbb{Q}$ countable. List rationals as $p/q$ with a diagonal enumeration.

$\mathbb{R}$ uncountable. Cantor's diagonal argument on decimal expansions.

Irrationals. $\mathbb{R}\setminus\mathbb{Q}$ is uncountable: if it were countable, $\mathbb{R}$ would be countable union of two countable sets, contradiction.

Conclusion. Almost all real numbers are irrational (in the measure-theoretic sense, the irrationals have full measure).