Question

Which of the following statements are always true?

• A. Product of two irrational numbers is an irrational number.
• B. The sum of two irrational numbers is irrational.
• C. The sum or difference of a rational and irrational number is rational.
• D. The product and quotient of a non-zero rational and irrational number is irrational.

Answer 1

The correct option is D

The product and quotient of a non-zero rational and irrational number is irrational.

- Option A does not always hold true as $\sqrt{2}\times \sqrt{2}=2$, which is rational.

- Sum of two irrational number can also be rational. Here is one example among many possibilities: $(5-\sqrt{2})+(4+\sqrt{2})=9$. Thus, option B does not always hold true.

- The sum or difference of a rational and an irrational number is always irrational. Thus, option C is incorrect.

For different rational and irrational values, statements in options $A,B,C$ fail to hold. But option $D$ is true always.