Why is (b\neq0) necessary in the proof of (\sqrt{3})?

Because the denominator of a fraction cannot be zero

In a fraction written as \(a/b\), the denominator tells how many equal parts are being considered. Division by zero is not defined, so a denominator equal to zero would not represent a valid fraction. Therefore, when a rational number is written in the form \(a/b\), it is essential to state that \(b\neq0\). This condition is about the meaning of the fraction, not about whether \(b\) is positive, negative, even, or equal to 3.

In the proof, we suppose that \(\sqrt{3}\) is rational and write it as \(a/b\). The symbol \(a/b\) is meaningful only when \(b\neq0\). Thus option A is correct because the denominator of a fraction cannot be zero. This condition is separate from the later condition that \(a\) and \(b\) have no common factor.

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