Question

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

Answer 1

Given: x is rational number and y is irrational number.

Yes, xy is necessarily an irrational number.

Example: Let x = 2, which is rational.

Let y = `sqrt(2)`, which is irrational.

Then, x × y = `2 xx sqrt(2) = 2sqrt(2)`, which is again irrational.

Also, consider the case when x = 0.

Then xy = 0, which is rational.

∴ Product of a rational and an irrational number is always irrational, only if the rational number is not zero.