proove that root 2 is a …

Let √2 be a rational number  Therefore, √2= p/q  [ p and q are in their least terms i.e., HCF of (p,q)=1 and q ≠ 0 On squaring both sides, we get                     p²= 2q²                                                                                    ...(1) Clearly, 2 is a factor of 2q² ⇒ 2 is a factor of p²                                                                    [since, 2q²=p²] ⇒ 2 is a factor of p  Let p =2 m for all m ( where  m is a positive integer) Squaring both sides, we get              p²= 4 m²                                                                                          ...(2) From (1) and (2), we get             2q² = 4m²      ⇒      q²= 2m² Clearly, 2 is a factor of 2m² ⇒       2 is a factor of q²                                                      [since, q² = 2m²] ⇒       2 is a factor of q  Thus, we see that both p and q have common factor 2 which is a contradiction that H.C.F. of (p,q)= 1      Therefore, Our supposition is wrong Hence √2 is not a rational number i.e., irrational number. Read more on Brainly.in - https://brainly.in/question/2367037#readmore