Derive that \(\sqrt{5}\) is an irrational number.
Proof:
Let \(\sqrt{5}=\frac{p}{q}\) be a number, where \(p\) and \(q\) are coprime and \(q\neq 0\).
Squaring on both sides, we get: \(5q^2=\) ......(1)
Therefore, \(p^2\) is divisible by \(5\). Hence,\(p\) can be divided by \(5\).
Substitute \(p = 5k\) in equation (1) and simplifying, we get: \(=\) \(5k^{2}\)
This means that \(q^2\) is divisible by \(5\) and hence, \(q\) is divisible by \(5\).
This implies that \(p\) and \(q\) have \(5\) as a .
And this is a contradiction to the fact that \(p\) and \(q\) are .
Hence, \(\sqrt{5}\) cannot be expressed as .
Therefore \(\sqrt{5}\) is .