To prove that a rational number multiplied by an irrational number results in an irrational product, we can use a direct proof.
Let’s proceed with the proof:
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Definition of a Rational Number: A rational number can be expressed as \( r = \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
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Definition of an Irrational Number: An irrational number cannot be expressed as a fraction of two integers.
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Assume: Let’s denote the irrational number as \( x \).
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Product: Now consider the product \( r \cdot x = \frac{a}{b} \cdot x \).
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Multiplication with Rational Number: Suppose \( r \cdot x \) is rational. This means that \( r \cdot x \) can also be expressed as some rational number \( \frac{c}{d} \), where \( c \) and \( d \) are integers, and \( d \neq 0 \).
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Isolate \( x \): Rearranging this gives us: \[ x = \frac{b}{a} \cdot \frac{c}{d} \] which is a product of rational numbers \( \frac{b c}{a d} \). Since \( b, c, a, d \) are all integers (and \( a \neq 0 \)), the result \( \frac{bc}{ad} \) is also rational.
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Contradiction: This leads to a contradiction because we assumed \( x \) to be irrational, but we derived that it can be expressed as a rational number. Therefore, our assumption that \( r \cdot x \) is rational must be false.
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Conclusion: Hence, \( r \cdot x \) must be irrational.
Therefore, we have proven that the product of a rational number and an irrational number is irrational.
So the correct answer would be: by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.
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