Which assumption is rejected by the contradiction obtained after assuming (\sqrt{3}) rational?
Answer and explanation
Correct answer: (\sqrt{3}) is rational
A proof by contradiction temporarily assumes the opposite of the statement that is to be proved. Here the target is to show that \(\sqrt{3}\) is irrational, so the proof begins by assuming that \(\sqrt{3}\) is rational. It is then written as a fraction \(p/q\) in lowest terms, with \(q\neq0\). The algebraic argument eventually forces both \(p\) and \(q\) to be divisible by 3.
That conclusion conflicts with the choice of a lowest-terms fraction, so the temporary assumption cannot be true. The contradiction does not reject the fact that \(\sqrt{3}>0\), that it is real, or that the denominator is non-zero; those facts remain valid. It rejects only the assumption of rationality. Therefore option A is the correct choice.
Frequently asked questions
What is the correct answer to this question?
(\sqrt{3}) is rational
Why is this the correct answer?
A proof by contradiction temporarily assumes the opposite of the statement that is to be proved. Here the target is to show that \(\sqrt{3}\) is irrational, so the proof begins by assuming that \(\sqrt{3}\) is rational. It is then written as a fraction \(p/q\) in lowest terms, with \(q\neq0\). The algebraic argument eventually forces both \(p\) and \(q\) to be divisible by 3.
That conclusion conflicts with the choice of a lowest-terms fraction, so the temporary assumption cannot be true. The contradiction does not reject the fact that \(\sqrt{3}>0\), that it is real, or that the denominator is non-zero; those facts remain valid. It rejects only the assumption of rationality. Therefore option A is the correct choice.
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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