What is the correct difference between the roles of (d\neq0) and (\gcd(c,d)=1) in the proof of (\sqrt{2})?

(d\neq0) keeps the fraction defined and (\gcd(c,d)=1) is the basis of contradiction

A fraction \(c/d\) represents a number only when its denominator is non-zero, so \(d\neq0\) is required to make the expression defined. This condition does not say that the fraction is reduced, and it does not imply any equality between \(c\) and \(d\). The condition \(\gcd(c,d)=1\) has a different purpose: it says that numerator and denominator have no common factor and that the fraction is in lowest terms.

Assuming \(\sqrt{2}=c/d\), squaring gives \(c^2=2d^2\). The parity argument shows that \(c\) is even and then that \(d\) is even. Thus both have the common factor 2, contradicting \(\gcd(c,d)=1\). The contradiction rests on the lowest-terms condition, while the non-zero condition only keeps the fraction meaningful. Hence option B is correct.

This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Proof of irrationality of square root 2 and square root 3.