Prove that \(\sqrt{2}\) is irrational.
Proof:
Assume, \(\sqrt{2}\) is .
By Definition, \(\sqrt{2}=\) be , where \(p\) and \(q\) are and \(q\neq 0\)
Let, \(p=2k\), where \(k\) is any integer.
On simplification, \(q^2=2k^2\)
This means that \(q^2\) is divisible by and hence, \(q\) is divisible by .
This implies that \(p\) and \(q\) have as a common factor.
And this is a contradiction to the fact that \(p\) and \(q\) are .
Hence, \(\sqrt{2}\) cannot be expressed as .
Therefore \(\sqrt{2}\) is .