Irrationally is the adverb: in a way that is not based on clear thought. It tells you how someone feels, acts, or decides. “She was irrationally angry” means the anger was too big for the facts. People also say “irrationally afraid” or “irrationally sure.” The word does not claim the person is foolish all the time; it claims that this one feeling or act is not sitting on reason. It is a little formal. Friends might say “I don’t know why, I just…” instead. You place it next to a verb (act irrationally) or an adjective (irrationally hopeful).