No, it is uncountable. The set of real numbers is uncountable and the set of rational numbers is countable, since the set of real numbers is simply the union of both, it follows that the set of irrational numbers must also be uncountable. (The union of two countable sets is countable.)

It is uncountable, because it contains infinite amount of numbers

They are not. They are countably infinite. That is, there is a one-to-one mapping between the set of rational numbers and the set of counting numbers.

Proof By Contradiction:

Claim: R\Q = Set of irrationals is countable.

Then R = Q union (R\Q)

Since Q is countable, and R\Q is countable (by claim), R is countable because the union of countable sets is countable.

But this is a contradiction since R is uncountable (Cantor's Diagonal Argument).

Thus, R\Q is uncountable.

Uncountable