Question

Suppose that p and are rational numbers and ris an irrational number. Which expression represents a rational number?

Answer 1

To determine which expressions involving rational numbers \( p \) and \( q \) (where \( p, q \in \mathbb{Q} \)) and an irrational number \( r \) (where \( r \notin \mathbb{Q} \)) would yield a rational number, we can analyze possible combinations of these types of numbers:

1. \( p + r \): This is a rational number plus an irrational number, which results in an irrational number.
2. \( p - r \): This is also a rational number minus an irrational number, hence this results in an irrational number.
3. \( p \cdot r \): A rational number multiplied by an irrational number results in an irrational number.
4. \( \frac{p}{r} \): A rational number divided by an irrational number results in an irrational number.
5. \( r + r \): This is the sum of two irrational numbers and can be either rational or irrational.
6. \( r - r \): This gives \( 0 \), which is a rational number.
7. \( r \cdot r \): The product of two irrational numbers can be either rational or irrational, depending on the numbers involved.
8. \( p + q \): This is the sum of two rational numbers, which results in a rational number.
9. \( p \cdot q \): The product of two rational numbers also results in a rational number.

Given those combinations, the expressions that represent rational numbers would be:

- \( p + q \) (the sum of two rational numbers)
- \( r - r \) (the difference of an irrational number with itself)

Hence, \( p + q \) and \( r - r \) represent rational numbers among the mentioned combinations.