To prove that a rational number multiplied by an irrational number yields an irrational product, the best approach is:
by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.
Here’s a concise proof:
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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 as a simple fraction and has a non-repeating, non-terminating decimal expansion.
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Assume a Rational Number and an Irrational Number:
- Let \( r = \frac{p}{q} \) be a rational number, and let \( x \) be an irrational number.
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Product of Rational and Irrational Numbers:
- Consider the product \( rx = \frac{p}{q} \cdot x \).
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Assume for Contradiction:
- Suppose \( rx \) is rational. Then, it can be expressed as \( \frac{m}{n} \) for some integers \( m \) and \( n \).
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Solve for \( x \):
- Rearranging gives us \( x = \frac{rx \cdot q}{p} \). Since both \( r \) and \( \frac{p}{q} \) are rational, the product \( rx \) and \( q \) are also rational, hence \( x \) would also be rational.
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Contradiction:
- This contradicts our assumption that \( x \) is irrational. Therefore, the original assumption that \( rx \) is rational must be incorrect.
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Conclusion:
- Hence, the product of a rational number \( r \) and an irrational number \( x \) must be irrational.
This proof demonstrates that the product of a rational number and an irrational number is indeed irrational.