The best approach to prove that a rational number multiplied by an irrational number yields an irrational product is to create a proof that shows directly how multiplying a rational and an irrational number leads to an irrational answer.
Here’s a concise logical proof:
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Definition of Rational and Irrational Numbers: A rational number can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b \neq 0\). An irrational number cannot be expressed as a fraction of two integers.
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Assumption: Let \( r = \frac{a}{b} \) be a rational number (where \( a, b \in \mathbb{Z} \), \( b \neq 0 \)), and let \( x \) be an irrational number.
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Product: We are interested in the product \( rx \), which is equal to \( \frac{a}{b} x = \frac{ax}{b} \).
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Proof by Contradiction: Assume that \( rx \) is rational. This means that \( \frac{ax}{b} \) can be expressed as \( \frac{m}{n} \) where \( m, n \in \mathbb{Z} \) and \( n \neq 0 \).
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Rearrange the Equation: From the assumption that \( \frac{ax}{b} = \frac{m}{n} \), we can rearrange this to find \( ax = \frac{bm}{n} \).
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Implying \( x \) is Rational: This equation indicates that \( x = \frac{bm}{an} \) (assuming \( a \) is not zero), which is a ratio of two integers and thus a rational number.
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Contradiction: This contradicts our initial assertion that \( x \) is irrational.
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Conclusion: Therefore, our assumption that \( rx \) is rational must be incorrect. Thus, the product of a non-zero rational number and an irrational number is irrational.
In conclusion, this demonstrates that it is indeed possible to prove that a rational number multiplied by an irrational number yields an irrational product.