Question

prove that√5 is irrational​

Answer 1

Let take √5 as rational number

If a and b are two co prime number and b is not equal to 0.

We can write √5 = a/b

Multiply by b both side we get

b√5 = a

To remove root, Squaring on both sides, we get

5b² = a² ……………(1)

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

That means 5 will divide a. So we can write

a = 5c

and plug the value of a in equation (1) we get

5b² = (5c)²

5b² = 25c²

Divide by 25 we get

b²/5 = c²

again using same theorem we get that b will divide by 5

and we have already get that a is divide by 5

but a and b are co prime number. so it is contradicting .

Hence √5 is a non rational number