Consider the set

where the elements of

consist of

where

is rational and between 0 and 1.

Since we define

, then

as well. That verifies the first requirement.

It is countable because there exists a bijection from the rationals of

(which are countable) to the set

. Specifically, the bijection just takes an element of the rationals of

and divides it by pi.

It is irrational since there is no way to express any

as a ratio between two integers.

And it is dense because the rationals are dense, so too will

Note that this could have worked with any irrational number, such as

or

or whatever.

Also note that if a set is dense, it cannot be finite. So by definition, it must be countably infinite or uncountable infinite anyway.