Question

prove

root 5 is an irrational number ​

Answer 1

Hence,√5 is irrational.

HOpe ThIS miGhT heLp you

Answer 2

Suppose, 5 represent a rational number. Then, 5 can expressed in the form

p/q, where p & q are integer and have no common factor, q ≠ 0.

                         √5 = p/q

On squaring both sides, we get

     5  = p²/q²

=> p² = 5q²                                                                                                               …(i)

  5 divides p² ⇒5 divides p                                                                (thereom 1.1) …(ii)

On putting the value of p² in Eqn (i) , we get

                                  25m² = 5q²

                                    5m²= q²

5 divides q² ⇒5 divides q                                                               (thereom 1.1)...(iii)

Thus, from Eq. (ii), 5 divides p and from Eq. (iii), 5 also divides q. It means 5 is a common factor of p and q. This contradicts the supposition so there is no common factor of p and q. Hence, 5 is an irrational number.

Hence proved.