If a contradiction is obtained after assuming that √2 is rational, according to logic which conclusion is correct?
Answer and explanation
Correct answer: √2 is irrational
The proof uses contradiction. To establish that √2 is irrational, assume temporarily that √2 is rational and write it as p/q in lowest terms, with integers p and q and q nonzero. Squaring gives p² = 2q². This implies p is even; writing p = 2k then shows q is also even, contradicting the assumption that p/q was in lowest terms. The contradiction means the initial assumption that √2 is rational must be false. Therefore its negation is true: √2 is irrational. Option B repeats the rejected assumption, while options C and D do not follow from the argument. Hence option A is correct.
Frequently asked questions
What is the correct answer to this question?
√2 is irrational
Why is this the correct answer?
The proof uses contradiction. To establish that √2 is irrational, assume temporarily that √2 is rational and write it as p/q in lowest terms, with integers p and q and q nonzero. Squaring gives p² = 2q². This implies p is even; writing p = 2k then shows q is also even, contradicting the assumption that p/q was in lowest terms. The contradiction means the initial assumption that √2 is rational must be false. Therefore its negation is true: √2 is irrational. Option B repeats the rejected assumption, while options C and D do not follow from the argument. Hence option A 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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