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

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

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

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

Answer 1

To prove that the product of a rational number and an irrational number is always irrational, we can use a proof by contradiction. Here’s how it can be structured:

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

2. **Let’s assume** the contrary: that multiplying a rational number \( r \) by an irrational number \( x \) results in a rational number \( y \). So, we have:
\[
y = r \cdot x
\]
where \( y \) is rational.

3. **Rearranging**: Since \( r \) is rational (i.e., \( r = \frac{p}{q} \)), we can rewrite the equation:
\[
x = \frac{y}{r}
\]

4. **Analyzing \( \frac{y}{r} \)**: Since both \( y \) (assumed to be rational) and \( r \) (which is also rational) are rational numbers, their quotient \( \frac{y}{r} \) will also be rational (as long as \( r \neq 0 \)).

5. **Contradiction**: This leads us to conclude that \( x \) is rational (because it equals \( \frac{y}{r} \)), which contradicts the assumption that \( x \) is irrational.

6. **Conclusion**: Therefore, our assumption must be incorrect. Thus, it follows that the product of a rational number and an irrational number must be irrational.

The correct response to the initial question is:
**by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.**