Let x be set of irrational numbers
elements belongs to it root3,root5 etc
by this way it was unlimited

You have probably shown:
1) The set Q of rational numbers is countable.
2) The set R of real numbers is uncountable.
3) The union of two countable sets is countable.
Now if both the set of rational numbers and the set of irrational numbers were countable would you be able to get a contradiction using fact 2 and 3? You should be able to use this contradiction to show that the set of irrational numbers must be uncountable.

Read more: