Question

Prove that root 2is irrational

Answer 1

   

First assume for reaching the contradiction that √2 is irrational. So that √2 can be written as p/q, where p, q are co-prime integers and q ≠ 0.

⇒ p/q = √2

⇒ (p/q)² = (√2)²

⇒ p²/q² = 2

⇒ p² = 2q²

Seems that p² is an even number. As p is an integer, if p² is even, then so will be p. Let p = 2m.

⇒ p² = 2q²

⇒ (2m)² = 2q²

⇒ 4m² = 2q²

⇒ 2m² = q²

Also seems that q² is even. As q is also an integer, if q² is even, then so will be q.

But this contradicts our earlier assumption that p, q are co-prime integers, because now it seems that both p and q are even numbers.

Thus we proved that √2 is irrational.

Thank you......

$\textsc{\#adithyasajeevan}$

Answer 2

please see attachment

i hope u satisfied