Question

let X be rational and y be irrational.Is xy necessarily irrational? justify your answer by an example​

Answer 1

Answer: No, xy is necessarily an irrational only when x ≠ 0.

Let x be a non-zero rational and y be an irrational. Then, we have to show that xy be an irrational. If possible, let xy be a rational number. Since, quotient of two non-zero rational number is a rational number.

So, xy/x is a rational number

=> y is a rational number.

But, this contradicts the fact that y is an irrational number. Thus, our supposition is wrong. Hence, xy is an irrational number. But, when x = 0, then xy = 0, a rational number.