Question

Root3-2is iraational number

Answer 1

So hence proved friend

Answer 2

Assume the contrary that √3-2 is rational

√3-2=p/q, where p and q are co-prime, q≠0

√3=p/q+2

√3=p+2q/q

p+2q/q is rational hence p and q are integers ∴√3 is rational

√3=p/q ,p and q are co-prime, q≠0

√3q=p

squaring,(√3q)²=p²

3q²=p²

q²=p²/3

3 divides p²⇒3 divides p (theorem 1)

put p=3m

q²=(3m)²/3

q²=9m²/3

m²=q²/3

3 divides q²⇒3 divides q(theorem 1)

∴3 is a common factor for p and q.

This contradicts the fact that p and q are co-prime.∴our assumption is wrong.

∴√3 is irrational.

i.e.,√3=p+2q/q is irrational.

∴√3-2 is irrational .

Hence the proof.