Believing on faith is something we all do in our lives. It really means reliance on a source of information without knowing it personally for ones self. Most people put faith in their doctors even to the point of putting their life in their hands without knowing first hand that the information being told to them is the truth. Its not irrational, its because there is a reasonable logic that the doctor would be a reliable source for that information. The same thing with religious faith. For example a jew may be convinced that its hard to fake the whole bible on the mountain in front of thousands of people experience, especially since there is an unbroken chain of tradition from then, so he's convinced from reason its true. Now he'll believe details in that tradition even if each detail he can't prove on its own.

Its impossible to prove God does not exist because its definition is beyond the realm of science. Science can only establish what can exist within the rules of science. It has nothing to say about something beyond the realm of science. One can choose to believe there is nothing beyond but that's just an opinion and not provable.