Give an example of :
(i) Two rationals whose sum is rational.
(ii) Two irrationals whose sum is rational.
(iii) Two irrationals whose product is rational.

Answer

Let's solve the question step by step.

### Step 1: Example of Two Rationals Whose Sum is Rational
- **Choose two rational numbers**: Let's take \( a = 5 \) and \( b = \frac{3}{2} \).
- **Calculate their sum**:
\[ a + b = 5 + \frac{3}{2} \]
To add these, we need a common denominator. The least common multiple (LCM) of 1 and 2 is 2.
\[ 5 = \frac{10}{2} \quad \text{(converting 5 to have a denominator of 2)} \]
Now we can add:
\[ a + b = \frac{10}{2} + \frac{3}{2} = \frac{10 + 3}{2} = \frac{13}{2} \]
- **Conclusion**: \( \frac{13}{2} \) is a rational number.

### Step 2: Example of Two Irrationals Whose Sum is Rational
- **Choose two irrational numbers**: Let \( a = 3 + \sqrt{2} \) and \( b = 3 - \sqrt{2} \).
- **Calculate their sum**:
\[ a + b = (3 + \sqrt{2}) + (3 - \sqrt{2}) \]
The \( \sqrt{2} \) terms cancel out:
\[ a + b = 3 + 3 + \sqrt{2} - \sqrt{2} = 6 \]
- **Conclusion**: 6 is a rational number.

### Step 3: Example of Two Irrationals Whose Product is Rational
- **Choose two irrational numbers**: Let \( a = 5 + \sqrt{7} \) and \( b = 5 - \sqrt{7} \).
- **Calculate their product**:
\[ a \times b = (5 + \sqrt{7})(5 - \sqrt{7}) \]
Using the difference of squares formula:
\[ a \times b = 5^2 - (\sqrt{7})^2 = 25 - 7 = 18 \]
- **Conclusion**: 18 is a rational number.

### Summary of Examples:
1. Two rationals whose sum is rational: \( 5 \) and \( \frac{3}{2} \) (Sum: \( \frac{13}{2} \)).
2. Two irrationals whose sum is rational: \( 3 + \sqrt{2} \) and \( 3 - \sqrt{2} \) (Sum: \( 6 \)).
3. Two irrationals whose product is rational: \( 5 + \sqrt{7} \) and \( 5 - \sqrt{7} \) (Product: \( 18 \)).