Which statement is used most in the proof of (\sqrt{2})?
Answer and explanation
Correct answer: If (n^2) is even then (n) is even
The standard proof assumes, for contradiction, that \(\sqrt{2}\) is rational and writes it as \(\frac{p}{q}\), where p and q are integers with no common factor. Squaring gives \(p^2=2q^2\). The right side is even, so \(p^2\) is even. A key elementary result is that if the square of an integer is even, then the integer itself is even. Therefore p is even.
Writing \(p=2k\) and substituting back shows that \(q^2\), and hence q, is also even. This means p and q have a common factor 2, contradicting the assumption that the fraction was in lowest terms. Thus the statement in option A is the central parity fact used in the proof. The other statements are false or unrelated to this argument.
Frequently asked questions
What is the correct answer to this question?
If (n^2) is even then (n) is even
Why is this the correct answer?
The standard proof assumes, for contradiction, that \(\sqrt{2}\) is rational and writes it as \(\frac{p}{q}\), where p and q are integers with no common factor. Squaring gives \(p^2=2q^2\). The right side is even, so \(p^2\) is even. A key elementary result is that if the square of an integer is even, then the integer itself is even. Therefore p is even.
Writing \(p=2k\) and substituting back shows that \(q^2\), and hence q, is also even. This means p and q have a common factor 2, contradicting the assumption that the fraction was in lowest terms. Thus the statement in option A is the central parity fact used in the proof. The other statements are false or unrelated to this argument.
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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