Prove that the following are irrational.

√5

Let take √5as rational number equal to , where a, b are positive co-primes. Then,

⇒ √5 b = a

⇒ 5b² = a²[squaring both sides] …I

Therefore, 5 divides a² and according to theorem of rational number, for any prime number p divides a² then it will divide a also.

∴ a = 5c

Put value of a in Eq. I, we get

5b² = (5c)²

⇒ 5b² = 25c²

⇒ [divide by 25 both sides]

Using same theorem we get that b will divide by 5 and we have already get that a is divided by 5. This contradicts our assumption.

Hence, √5 is irrational.