Consider this: between any two irrational numbers there exist a rational number.
I'm assuming you're referring to the set of irrational numbers; in that case you can write the irrationals as a union of two disjoint sets, say . Now recall that a set is connected iff the only sets that are both open and closed (in ) are the empty set and itself.
From this it should clear whether the irrationals (as well as the rationals, actually) are connected or disconnected.