If assuming (√2) rational gives a contradiction, which conclusion is correct?
Answer and explanation
Correct answer: (√2) is irrational
A proof by contradiction begins by assuming the opposite of the statement to be proved. Here the assumption is that √2 is rational. If valid reasoning from that assumption produces an impossibility, the assumption must be false. Therefore √2 is not rational; it is irrational, so option A is correct. The contradiction does not imply that √2 is zero, an integer, or a natural number. In fact, every integer and every natural number is rational, so options B and D conflict with the conclusion. Option C is also numerically false because the square of zero is 0, not 2. The logical structure is assumption, contradiction, rejection of assumption, and conclusion.
Frequently asked questions
What is the correct answer to this question?
(√2) is irrational
Why is this the correct answer?
A proof by contradiction begins by assuming the opposite of the statement to be proved. Here the assumption is that √2 is rational. If valid reasoning from that assumption produces an impossibility, the assumption must be false. Therefore √2 is not rational; it is irrational, so option A is correct. The contradiction does not imply that √2 is zero, an integer, or a natural number. In fact, every integer and every natural number is rational, so options B and D conflict with the conclusion. Option C is also numerically false because the square of zero is 0, not 2. The logical structure is assumption, contradiction, rejection of assumption, and conclusion.
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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