In irrationality of (\sqrt{3}), which assumption is rejected by the contradiction?
Answer and explanation
Correct answer: (\sqrt{3}) is rational
The key idea is that an irrationality proof begins by temporarily assuming the opposite of what we want to prove. Here, we assume that \(\sqrt{3}\) is rational. A rational number can be written as \(p/q\), where \(p\) and \(q\) are integers, \(q\ne0\), and the fraction is in lowest terms. The contradiction finally shows that this assumption cannot be true.
Thus, the rejected assumption is “\(\sqrt{3}\) is rational,” which is option C. The proof does not reject the fact that \(\sqrt{3}\) is real; it is certainly a real number. Also, \(\sqrt{3}>0\) and \(q\ne0\) are valid facts or conditions used in the argument, not the assumption being disproved. Therefore the supplied answer is correct.
Frequently asked questions
What is the correct answer to this question?
(\sqrt{3}) is rational
Why is this the correct answer?
The key idea is that an irrationality proof begins by temporarily assuming the opposite of what we want to prove. Here, we assume that \(\sqrt{3}\) is rational. A rational number can be written as \(p/q\), where \(p\) and \(q\) are integers, \(q\ne0\), and the fraction is in lowest terms. The contradiction finally shows that this assumption cannot be true.
Thus, the rejected assumption is “\(\sqrt{3}\) is rational,” which is option C. The proof does not reject the fact that \(\sqrt{3}\) is real; it is certainly a real number. Also, \(\sqrt{3}>0\) and \(q\ne0\) are valid facts or conditions used in the argument, not the assumption being disproved. Therefore the supplied answer is correct.
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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