Write a pair of irrational numbers whose sum is irrational .

Answer

To solve the question of writing a pair of irrational numbers whose sum is also irrational, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Irrational Numbers**:
- Irrational numbers are numbers that cannot be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \). Common examples include numbers like \( \sqrt{2} \), \( \sqrt{3} \), and \( \pi \).

2. **Choose Two Irrational Numbers**:
- Let's select the irrational numbers \( \sqrt{3} + 5 \) and \( \sqrt{5} - 3 \).

3. **Calculate the Sum of the Two Numbers**:
- Now, we will add these two irrational numbers together:
\[ (\sqrt{3} + 5) + (\sqrt{5} - 3) \]

4. **Simplify the Expression**:
- Combine the terms:
\[ = \sqrt{3} + \sqrt{5} + 5 - 3 \]
\[ = \sqrt{3} + \sqrt{5} + 2 \]

5. **Determine if the Sum is Irrational**:
- The sum \( \sqrt{3} + \sqrt{5} + 2 \) includes the irrational components \( \sqrt{3} \) and \( \sqrt{5} \). Since the sum of two irrational numbers (in this case, \( \sqrt{3} \) and \( \sqrt{5} \)) is generally irrational, we conclude that the entire expression remains irrational.

6. **Final Answer**:
- Thus, the pair of irrational numbers we have chosen is \( \sqrt{3} + 5 \) and \( \sqrt{5} - 3 \), and their sum \( \sqrt{3} + \sqrt{5} + 2 \) is also irrational.

### Final Answer:
The pair of irrational numbers is \( \sqrt{3} + 5 \) and \( \sqrt{5} - 3 \), whose sum is \( \sqrt{3} + \sqrt{5} + 2 \), which is irrational.