Question

if x and y are rational and irrational numbers respectively , then is xy necessarily an irrational number? justify your answer???​

Answer 1

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.

Answer 2

Yes!

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

prove : Let x (which is rational) be

2 and y (which is irrational)

be root 2

Now,

2× root 2 =》

{ root 2 = 1.14.....}

So,

=》 2.14.....

which is irrational. ......

[Note: if x=0 than xy is a irrational. ]
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