Let √2 be a rational number p/q, where p and q are integers. p and q are relatively prime to each other, coprime. no common factors among them.
√2 = p / q
p² = 2 q²
p * p = 2 * q * q
p and q are co-prime. no common factors. Then 2 on RHS must be a factor of p.
then p = 2 k
2k * 2k = 2 * q * q
2 k * k = q * q
2 on LHS must be a factor of q on RHS. So q = 2 * m.
But we have started √2 = p/q with p and q such that they are co-prime. They have no common factors. But if √2 is rational, they have a common factor 2.
So the assumption is wrong.
√2 cannot be a rational number.