Which conclusion is correct in the proof that √2 is irrational?
Answer and explanation
Correct answer: √2 is irrational
The standard proof uses contradiction. Assume that √2 is rational and write √2=p/q, where p and q are coprime integers and q≠0. Squaring gives p²=2q². Thus p² is even, which means p is even; let p=2k. Substitution gives 4k²=2q², so q²=2k², and q is also even. This contradicts the assumption that p and q have no common factor. Therefore the assumption is false and √2 is irrational. Option C states this conclusion. Options A and B conflict with the contradiction proof, and √2 is clearly not zero because its square is 2.
Frequently asked questions
What is the correct answer to this question?
√2 is irrational
Why is this the correct answer?
The standard proof uses contradiction. Assume that √2 is rational and write √2=p/q, where p and q are coprime integers and q≠0. Squaring gives p²=2q². Thus p² is even, which means p is even; let p=2k. Substitution gives 4k²=2q², so q²=2k², and q is also even. This contradicts the assumption that p and q have no common factor. Therefore the assumption is false and √2 is irrational. Option C states this conclusion. Options A and B conflict with the contradiction proof, and √2 is clearly not zero because its square is 2.
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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