Question

Prove that√3 i s irrational number ​

Answer 1

Answer:

refer to the attachment

Answer 2

Let us assume to the contrary that √3 is rational.

So, we can find co - prime integers a and b such that a/b =√3 .

On squaring both sides :

(a/b)2 = 3

=> a2 = 3b2

3 divides a square....

3 also divides a

let c be any positive integer, such that a =3c

By Substituting the value of a , we get

3b2 =9c2

=> b2 = 3c2

3 divides b square.....

3 also divides b

Therefore a and b has atleast 3 as a common factor.

But this contradicts the fact that a and b are co prime.

This contradiction has arisen because of our wrong assumption that √3 is irrational.

So, √3 is irrational.