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 $x \neq 0.$
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
$\Rightarrow y$ 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
$\Rightarrow y$ 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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