Question

Prove that √2 is irrational using 'proof by contradiction' technique.

Answer 1

Let us assume √2 to be rational

√2= p/q (where p and q are co-prime)

[Squaring]

2=p²/q²

2q²=p²_____(1)

Here 2 divides p²     So 2 will divide p as well

2x=p

[Squaring]  4x²=p² (substitute the value of 1)

4x²=2q²

2x²=q²

Here 2 divides q²      So it'll divide q as well

Therefore as 2 divides both p and q then they aren't co-primes as they have more than 1 common factor.

Hence our assumption was wrong so what the contradiction appeared and therefore √2 is irrational.