If the contradiction proof of \(\sqrt{2}\) succeeds, what is the final logical conclusion?
Answer and explanation
Correct answer: \(\sqrt{2}\) is irrational
In a contradiction proof, we assume that \(\sqrt{2}=p/q\) is rational, where \(p\) and \(q\) are coprime integers. The argument shows that both \(p\) and \(q\) must be even, contradicting their being coprime. Hence the original assumption is false, so \(\sqrt{2}\) is irrational. Exam tip: A contradiction rejects the initial assumption, not the statement being proved.
Frequently asked questions
What is the correct answer to this question?
\(\sqrt{2}\) is irrational
Why is this the correct answer?
In a contradiction proof, we assume that \(\sqrt{2}=p/q\) is rational, where \(p\) and \(q\) are coprime integers. The argument shows that both \(p\) and \(q\) must be even, contradicting their being coprime. Hence the original assumption is false, so \(\sqrt{2}\) is irrational. Exam tip: A contradiction rejects the initial assumption, not the statement being proved.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Proof of irrationality of square root 2 and square root 3.
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