Duplicate Question
The question on this page has been marked as a duplicate question.
Original Question
Ryiaz claims it is impossible to prove that a rational number multiplied by an irrational number gives an irrational product. H...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 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
He’s correct; it is impossible to prove in any way.
Responses
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 choosing random numbers to plug into multiplication problems until he’s convinced
He’s correct; it is impossible to prove in any way.
Answers
Answer
what ? just answer the question
Answer
thank you
Answered by
GPT-4o mini
The best option for proving that a rational number multiplied by an irrational number gives an irrational product is:
**by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.**
Here's a brief outline of the proof:
1. **Definition of Rational and Irrational Numbers**: A rational number can be expressed as the fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). An irrational number cannot be expressed in this form.
2. **Assume a Rational and an Irrational Number**: Let \( r \) be a rational number, such that \( r = \frac{p}{q} \), and let \( x \) be an irrational number.
3. **Multiply**: Consider the product \( rx = \frac{p}{q}x \).
4. **Assume for Contradiction**: Suppose that \( rx \) is rational. Then it can be expressed as \( \frac{m}{n} \) where \( m \) and \( n \) are integers.
5. **Rearranging**:
\[
x = \frac{q \cdot rx}{p}
\]
Since \( r \) is rational and both \( q \cdot rx \) and \( p \) are also rational, \( x \) must be rational as it results from the quotient of two rational numbers.
6. **Contradiction**: This contradicts the assumption that \( x \) is irrational.
7. **Conclusion**: Therefore, the product \( rx \) must be irrational.
This argument definitively shows that the product of a rational number and an irrational number is indeed irrational, proving Ryiaz's claim to be incorrect.
**by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.**
Here's a brief outline of the proof:
1. **Definition of Rational and Irrational Numbers**: A rational number can be expressed as the fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). An irrational number cannot be expressed in this form.
2. **Assume a Rational and an Irrational Number**: Let \( r \) be a rational number, such that \( r = \frac{p}{q} \), and let \( x \) be an irrational number.
3. **Multiply**: Consider the product \( rx = \frac{p}{q}x \).
4. **Assume for Contradiction**: Suppose that \( rx \) is rational. Then it can be expressed as \( \frac{m}{n} \) where \( m \) and \( n \) are integers.
5. **Rearranging**:
\[
x = \frac{q \cdot rx}{p}
\]
Since \( r \) is rational and both \( q \cdot rx \) and \( p \) are also rational, \( x \) must be rational as it results from the quotient of two rational numbers.
6. **Contradiction**: This contradicts the assumption that \( x \) is irrational.
7. **Conclusion**: Therefore, the product \( rx \) must be irrational.
This argument definitively shows that the product of a rational number and an irrational number is indeed irrational, proving Ryiaz's claim to be incorrect.
Answered by
GPT-4o mini
Apologies for the confusion. The correct response is:
**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.**
Answered by
GPT-4o mini
You're welcome! If you have any more questions or need further assistance, feel free to ask!