Which of the following is`//`are correct?

Options

A. Product of two irrational numbers is always an irrational number.
B. The sum of two irrational numbers is always an irrational number.
C. Sum of rational and irrational number is always an irrational number.
D. None of these

Answer

To determine which of the given statements about irrational and rational numbers are correct, we will analyze each option step by step.

### Step 1: Analyze the first statement
**Statement 1:** The product of two irrational numbers is always irrational.

**Example:** Let's take two irrational numbers, √2 and √3.
- The product is:
\[ \sqrt{2} \times \sqrt{3} = \sqrt{6} \]
which is irrational.

However, if we take:
- √2 and -√2, their product is:
\[ \sqrt{2} \times (-\sqrt{2}) = -2 \]
which is rational.

**Conclusion:** The first statement is **incorrect** because the product of two irrational numbers can be rational.

### Step 2: Analyze the second statement
**Statement 2:** The sum of two irrational numbers is always irrational.

**Example:** Let's take two irrational numbers, √2 and -√2.
- The sum is:
\[ \sqrt{2} + (-\sqrt{2}) = 0 \]
which is rational.

**Conclusion:** The second statement is **incorrect** because the sum of two irrational numbers can be rational.

### Step 3: Analyze the third statement
**Statement 3:** The sum of a rational number and an irrational number is always irrational.

**Example:** Let’s take a rational number, 1/2, and an irrational number, √2.
- The sum is:
\[ \frac{1}{2} + \sqrt{2} \]
This sum cannot be expressed as a fraction (p/q) where p and q are integers, hence it remains irrational.

**Conclusion:** The third statement is **correct** because the sum of a rational number and an irrational number is always irrational.

### Final Conclusion
- **Statement 1:** Incorrect
- **Statement 2:** Incorrect
- **Statement 3:** Correct