Irrationality exists in all of us and effects many aspects of our lives. Since everything we do without considering all its
ramifications, and the odds of its working as we want it to is essentially irrational, many things we might consider good are irrational by definition. One cannot prove that what he
hopes for will come about, in fact in most cases one has more than one reason to believe it won't, but one still hopes, therefore all
Hope is irrational.
One cannot prove the existence of god, and certainly not that the way one worships one's god is the right way, and still millions of people have irrational faith in their gods and religions. A rational being will state that everything that cannot be proved cannot exist, and yet there you are faith in god is still around. More puzzling than faith in divinities, is faith in the soul itself. We all instinctively believe that we have souls, and yet we cannot prove it. Physicalism is much more rational than any other metaphysical viewpoint, and yet many people belive they have souls.
Even more essential than the above, yet related to them, is self-confidence, how can one be self assured when he is totally aware of his faults, short-comings, inabilities and fears? being self-confident is probably the most irrational quality one can have!
Yet more surprising in its irrationality is human ability to forgive and to renew trust. How can one do these things and rely on someone that has failed in the past and therefore might do this again?
The reason is that we need our irrationality. Its what keeps us sane. A completely rational human being will be hospitalized for severe depression (assuming the men in white suits get to him before he throws himself out the window).