Question

Prove that root 3 is irrational number

Answer 1

This is your answer see it ok

Answer 2

Let us assume that √3 is a rational number.

That is, we can find integersaandb(≠ 0) such that √3 = (a/b)

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.

√3b = a

⇒ 3b2=a2(Squaring on both sides) → (1)


Therefore, a2is divisible by 3

Hence ‘a’ is also divisible by 3.

So, we can write a = 3c for some integer c.

Equation (1) becomes,

3b2=(3c)2

⇒ 3b2= 9c2

∴ b2= 3c2

This means that b2is divisible by 3, and so b is also divisible by 3.

Therefore, a and b have at least 3 asa common factor.

But this contradicts the fact that a and b are coprime.

This contradiction has arisen because of our incorrect assumption that √3 is rational.So, we conclude that √3 is irrational.