Question

Let x be a rational number and y be irrational. Is xy necessary irrational? Justify with example.

Answer 1

The product of a rational and an irrational number is always irrational.

Answer 2

No,

(xy) is necessarily an irrational only when x ≠ 0.

Example :
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,
$( \frac{xy}{x} ) \: \: is \: \: a \: \: rational \: \: number$

=> 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.