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)
• by choosing random numbers to plug into multiplication problems until he's convinced
• Hes correct; it is impossible to prove in any way.
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

Answer 1

To prove that multiplying a rational number by an irrational number results in an irrational product, we can use a proof by contradiction. Here’s a structured way to do that:

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

2. **Assumption for Contradiction**:
- Let's assume that \( r \) is a rational number and \( x \) is an irrational number, and their product \( y = r \cdot x \) is a rational number.

3. **Expressing the Product**:
- Since \( r \) is rational, we can express \( y \) as:
\[
y = \frac{a}{b} \cdot x
\]
- Rearranging gives:
\[
x = \frac{y \cdot b}{a}
\]
- Here, \( y \) is the product we assumed to be rational, and \( a \) and \( b \) are integers.

4. **Analysis**:
- The right-hand side of the equation \( x = \frac{y \cdot b}{a} \) is the ratio of two integers (since \( y \) is assumed to be rational, and \( a \) and \( b \) are integers), which means \( x \) is rational.

5. **Contradiction**:
- This leads to a contradiction because we started with \( x \) being irrational, but derived that it must also be rational.

6. **Conclusion**:
- Therefore, our initial assumption that \( r \cdot x \) (the product of a rational and an irrational number) is rational must be incorrect. This means \( r \cdot x \) must be irrational.

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