Why is it necessary to take (\frac{p}{q}) in lowest form in the proof of (\sqrt{3})?

So contradiction with coprime assumption can be shown

In the standard contradiction proof, assume that \(\sqrt{3}=p/q\), where \(p\) and \(q\) are integers and the fraction is in lowest form. Lowest form means that \(p\) and \(q\) have no common factor. Squaring gives \(p^2=3q^2\). This shows that 3 divides \(p^2\), and therefore 3 divides \(p\). Writing \(p=3k\) then shows that 3 also divides \(q\).

That conclusion contradicts the original statement that \(p/q\) was in lowest form. The contradiction proves that \(\sqrt{3}\) cannot be rational. Thus option B is correct. Without the coprime assumption, finding a common factor would not create a contradiction, so taking lowest form is essential. The supplied explanation is accurate.

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