What is the highest-level description of the similarity between the proofs of √2 and √3?
Answer and explanation
Correct answer: In both, the rational assumption gives a common prime factor contradicting a lowest-term fraction
Both arguments use the same proof architecture: assume that the square root is rational, express it as a fraction in lowest terms, square the equation, and use prime divisibility to force a common factor in the numerator and denominator. For √2, the forced prime is 2; for √3, it is 3. In either case, the new common factor contradicts gcd(numerator, denominator) = 1. Thus option A gives the highest-level similarity while still identifying the essential mechanism. The denominator is not assumed to be zero, decimal termination is not the basis of the proof, and equality of numerator and denominator is never required.
Frequently asked questions
What is the correct answer to this question?
In both, the rational assumption gives a common prime factor contradicting a lowest-term fraction
Why is this the correct answer?
Both arguments use the same proof architecture: assume that the square root is rational, express it as a fraction in lowest terms, square the equation, and use prime divisibility to force a common factor in the numerator and denominator. For √2, the forced prime is 2; for √3, it is 3. In either case, the new common factor contradicts gcd(numerator, denominator) = 1. Thus option A gives the highest-level similarity while still identifying the essential mechanism. The denominator is not assumed to be zero, decimal termination is not the basis of the proof, and equality of numerator and denominator is never required.
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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