Question

Which of the following statements is not correct?

• A. The sum of a rational number and an irrational number is an irrational number.
• B. The sum of two irrational numbers need not be an irrational number.
• C. The product of a non-zero rational number and an irrational number is an irrational number.
• D. The product of two irrational numbers is always an irrational number.

Answer 1

The Correct Option is D

Let's analyze each statement:

Statement (1): The sum of a rational number and an irrational number is an irrational number.

This is correct. Proof: If rational + irrational = rational, then irrational = rational - rational = rational, which is a contradiction.

Statement (2): The sum of two irrational numbers need not be an irrational number.

This is correct. Example: \( (1 + \sqrt{2}) + (1 - \sqrt{2}) = 2 \), which is rational.

Statement (3): The product of a non-zero rational number and an irrational number is an irrational number.

This is correct. Proof: If rational × irrational = rational, then irrational = rational ÷ rational = rational, which is a contradiction.

Statement (4): The product of two irrational numbers is always an irrational number.

This is not correct. Counterexample: \( \sqrt{2} × \sqrt{2} = 2 \), which is rational.

Conclusion

The statement that is not correct is (4) The product of two irrational numbers is always an irrational number.