Question
Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example.
✓
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.
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.
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