Question

i need a short method to prove √2 as irrational.

Answer 1

let √2 be a rational number.

√2 = p/q

squaring on both sides,

2 = p²/q²
2q² = p²

2 divides p² then 2 also divides p,
so, p = 2a (a is any positive integer)

2q² = p²
2q² = (2a)²
2q² = 4a²
q² = 2a²

2 divides q² then 2 also divides q.
both p and q have 2 as common factor.

but this contradicts the fact that p and q are co-primes.
so,our supposition is false,

therefore, √2 is irrational number.