CBSE Class 10 Answered
let us assume that root 3 is rational.
so we can find integers a and b (b not equal to zero) such that root 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.
therefore, b* root3 = a
squaring both sides we get, 3 (b)^2 = (a)^2------------------------1
therefore (a)^2 is divisible by 3
By prime factorization we get that a is divisible by 3.
so we can write a=3c for some integer c
Substituting the value of a in above 1 we get,
this implies (b)^2= 3(c)^2
therefore (b)^2 is divisible by 3 so b is divisible by 3.
so a and b have atleast 3 as a common factor.
But this contradicts the fact that a and b are coprime which is a contradiction that root3 is rational.
Hence, root 3 is irrational.