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√2 = p/q

squaring on both sides

2 = p²/q²

2q² = p² --------- (1)

q² = p²/2

2 divides p² ⇒ 2 divides p ------- (2)

let p = 2m ⇒p² = 4m²

sub p² = 4m² in ----(1)

2q² = 4m²

q² = 2m²

m² = q²/2

2 divides q² ⇒ 2 divides q ------------ (3)

from (2) and (3) we can say that p and q had an other common factor other than 1. it is not true because p and q are co primes and doesnt have any common factor.

it is contradiction to our assumption.

√2 is an irrational number.