What is the main mistake in writing (m=3n) directly from (m^2=3n^2) in the proof of (\sqrt{3})?
Answer and explanation
Correct answer: Taking a wrong linear conclusion from a squared equation
The equation is obtained while proving that sqrt{3} cannot be rational. From the equality of squares, it is not valid to remove the squares and claim that the two expressions are equal in the same form. In particular, the statement m=3n is much stronger than what the equation gives and is generally false.
The correct number-theory conclusion is that 3 divides m^2, because m^2=3n^2. Since 3 is prime, this implies that 3 divides m. Writing m=3k and substituting then produces the required contradiction with coprimality. Thus option B identifies the error: it takes an incorrect linear conclusion from a squared equation. The denominator is not being assumed zero, and the contradiction method itself is not the mistake.
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What is the correct answer to this question?
Taking a wrong linear conclusion from a squared equation
Why is this the correct answer?
The equation is obtained while proving that sqrt{3} cannot be rational. From the equality of squares, it is not valid to remove the squares and claim that the two expressions are equal in the same form. In particular, the statement m=3n is much stronger than what the equation gives and is generally false.
The correct number-theory conclusion is that 3 divides m^2, because m^2=3n^2. Since 3 is prime, this implies that 3 divides m. Writing m=3k and substituting then produces the required contradiction with coprimality. Thus option B identifies the error: it takes an incorrect linear conclusion from a squared equation. The denominator is not being assumed zero, and the contradiction method itself is not the mistake.
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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