(Q9).Prove that √5 is irrational .

Let √5 be a rational number.

So it can be expressed in the form
p
q
 where p,q are co-prime integers and q ≠ 0
√5 =
p
q
5 =
p2
q2

5q2 = p2

So 5 divides p2,p is a multiple of 5

Let p = 5m

p2 = 25m2

5q2 = 25m2

q2 = 5m2

q2 is multiple of 5

q is multiple of 5

Thus p,q have common factor .

This contradicts our assumption that they are co-primes

Therefore,
p
q
 is not a number

Hence, √5 is an number

So √5 is an number