analysis - Ways to arrive at the existence of irrational numbers

I know about a way Greeks arrived at the existence of irrational numbers by showing that sometimes two line segments can be incommensurable.

And about a simple way by which it can be shown that some numbers are irrational, for example, as is usually shown that $\sqrt2$ is irrational.

Also, it can be shown that some numbers are transcendental and because all rationals are algebraic that shows that there are some non-rational, that is, irrational numbers.

And also there is a way that shows that all rational numbers have periodic expansion in every base and since there are non-periodic expansions that also shows the existence of irrationals.

And there is countability/uncountability way.

Are there some other ways?