Question

Ryiaz claims it is impossible to prove that a rational number multiplied by an irrational number gives an irrational product. How can you prove this?(1 point)
Responses

by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer
by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer

by using a proof by contradiction to show multiplication is a closed operation for rational numbers
by using a proof by contradiction to show multiplication is a closed operation for rational numbers

by choosing random numbers to plug into multiplication problems until he’s convinced
by choosing random numbers to plug into multiplication problems until he’s convinced

He’s correct; it is impossible to prove in any way.

Answer 1

To prove that a rational number multiplied by an irrational number gives an irrational product, we can use a proof by contradiction. Here’s how the proof works:

1. **Definitions**:
- A rational number can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- An irrational number cannot be expressed as a fraction of two integers.

2. **Assumption**:
- Suppose \( r \) is a rational number (let's say \( r = \frac{p}{q} \)) and \( x \) is an irrational number. We will assume that their product \( rx \) is a rational number.

3. **Expression**:
- If \( rx \) is rational, then we can express it as \( \frac{m}{n} \), where \( m \) and \( n \) are integers and \( n \neq 0 \).

4. **Rearranging**:
- From our assumption, we have \( rx = \frac{m}{n} \).
- Therefore, \( x = \frac{rx}{r} \).
- Since both \( r \) and \( rx \) are rational, the quotient \( \frac{rx}{r} \) would also be rational (because the set of rational numbers is closed under multiplication and division by non-zero rationals).

5. **Contradiction**:
- According to our assumption, \( x \) was irrational, but we have derived that \( x \) would be rational, which is a contradiction.

6. **Conclusion**:
- Therefore, our initial assumption that multiplying a rational number \( r \) by an irrational number \( x \) yields a rational product must be false. Thus, the product \( rx \) must be irrational.

This proof shows that multiplying a rational number by an irrational number indeed results in an irrational product. Therefore, the correct response is:
**by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.**