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:
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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 in this form.
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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.
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Multiply: Consider the product \( rx = \frac{p}{q}x \).
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Assume for Contradiction: Suppose that \( rx \) is rational. Then it can be expressed as \( \frac{m}{n} \) where \( m \) and \( n \) are integers.
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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.
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Contradiction: This contradicts the assumption that \( x \) is irrational.
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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.