Let us assume that √5 is a rational number.Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0⇒√5=p/qOn squaring both the sides we get,⇒5=p²/q²⇒5q²=p² —————–(i)p²/5= q²So 5 divides pp is a multiple of 5⇒p=5m⇒p²=25m² ————-(ii)From equations (i) and (ii), we get,5q²=25m²⇒q²=5m²⇒q² is a multiple of 5⇒q is a multiple of 5Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number√5 is an irrational number

Let us assume that √5 is a rational number.we know that the rational numbers are in the form of p/q form where p,q are integers.so, √5 = p/q p = √5qwe know that \’p\’ is a rational number. so √5 q must be rational since it equals to pbut it doesnt occurs with √5 since its not an integertherefore, p =/= √5qthis contradicts the fact that √5 is an irrational numberhence our assumption is wrong and √5 is an irrational number.\xa0