Chapter 1 ex 1.3 1question

Suppose root 5 is a rational number Therefore root 5= p/ , p,q are integers, & q is not equals to 0 On reducing-- root 5= a/b , a & b are coprime no. => b root 5= a Squaring both sides 5b^2= a^2 ------------(i) => b^2 = a^2 /5 Since 5 divides a^2 Therefore 5 will also divide a (by theorem 1.3) ----(ii) => a/5= c => a= 5c => a^2 = 25c^2 => 5b^2 = 25c^2 (by eq.^n (i)) => b^2= 5c^2 => b^2 /5= c^2 Therefore 5 divides b^2 Therefore 5 will also divide b (by theorem 1.3) -----(iii) From eq.^n (ii) & (iii) 5 divides a & b both Therefore 5 is common factor of a & b But this contradict the fact that a & b are coprime Therefore this is due to our wrong assumption that root 5 is rational. Therefore we conclude that root 5 is rational.