What is the most important exam caution in the proofs of √2 and √3?
Answer and explanation
Correct answer: While assuming rationality, write the fraction in lowest coprime form
The correct answer is A. In a contradiction proof, assume √2 or √3 equals p/q, where p and q are integers, q is nonzero, and the fraction is already in lowest terms, meaning gcd(p,q) = 1. For √2 the argument eventually shows both numerator and denominator are even; for √3 it shows both are divisible by 3. The contradiction is meaningful only because a lowest-form fraction cannot have a common prime factor. If the fraction is not reduced at the start, the later divisibility result may simply describe a non-reduced representation and no contradiction follows. A denominator cannot be zero, decimals are not a substitute for proof, and numerator and denominator need not be equal.
Frequently asked questions
What is the correct answer to this question?
While assuming rationality, write the fraction in lowest coprime form
Why is this the correct answer?
The correct answer is A. In a contradiction proof, assume √2 or √3 equals p/q, where p and q are integers, q is nonzero, and the fraction is already in lowest terms, meaning gcd(p,q) = 1. For √2 the argument eventually shows both numerator and denominator are even; for √3 it shows both are divisible by 3. The contradiction is meaningful only because a lowest-form fraction cannot have a common prime factor. If the fraction is not reduced at the start, the later divisibility result may simply describe a non-reduced representation and no contradiction follows. A denominator cannot be zero, decimals are not a substitute for proof, and numerator and denominator need not be equal.
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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