Question

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

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

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

Answer 1

To prove that the product of a rational number and an irrational number is irrational, you can use a proof by contradiction. Here's how you can do it:

**Proof by Contradiction:**

1. **Assume the contrary:** Suppose that \( r \) is a rational number and \( x \) is an irrational number, and let's assume that their product \( rx \) is rational.

2. **Definition of rational number:** Since \( r \) is rational, we can express it as \( r = \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).

3. **Assuming the product is rational:** According to our assumption, let \( rx = q \), where \( q \) is a rational number.

4. **Rearranging the equation:** Hence, we can write:
\[
x = \frac{q}{r} = \frac{q}{\frac{a}{b}} = \frac{qb}{a}
\]
Since \( q \), \( a \), and \( b \) are all integers and \( b \) and \( a \) are not zero, \( \frac{qb}{a} \) is a ratio of integers, which is a rational number.

5. **Contradiction arises:** This implies that \( x \) must also be rational, which contradicts our assumption that \( x \) is irrational.

6. **Conclusion:** Therefore, our initial assumption that the product \( rx \) is rational must be false. Hence, the product of a rational number and an irrational number must be irrational.

This reasoning shows that it is indeed possible to prove that the product of a rational number and an irrational number is irrational. Thus, the correct response would be:

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