Why irrational number multiply by Rational won give you a rational number?
You can’t just willy nilly declare things to be rational and irrational numbers as you see fit. They are words that have specific meanings attached to them. Let A be an arbitrary irrational number, that means there do not exist integers M and N such that A = M/N (This is NOT the case for ANY integers) Let P,Q be any arbitrary integers and suppose that A*(P/Q) is a rational number (meaning it has the form M/N with M and N integers) Then we have A*(P/Q) = M/N Now cross multiply or whatever it’s called to get A = (Q*M)/(N*P) but since M,N,Q,P are all integers, so are Q*M and N*P so A is a rational number which contradicts the fact we said it was irrational. This means that if A is any irrational number there is no rational number we can multiply it by and still get a rational number out. The fact that everything involved {A,M,N,Q,P} was arbitrary means the proof holds for any sets of irrational and irrational numbers. I’m not really sure what you want in terms of examples. It holds for al
