Question

Prove that √3 is irrational number​

Answer 1

Answer:

√3 is an irrational number.

Step-by-step explanation:

Let us just assume that √3 is a rational number.

Then, there exist positive integers a and b such that √3 = a/b where, a and b, are co-prime i.e. their HCF is 1.

Now,

√3 = a/b

​⇒ 3 = a²/b²

⇒ 3b² = a²

⇒ 3 divides a²     [∵ 3 divides 3b²]

⇒ 3 divides a     ...(i)

⇒ a = 3c for some integer c

⇒ a² = 9c²

⇒ 3b² = 9c²     [∵a² = 3b²]

⇒ b² = 3c²

⇒ 3 divides b 2     [∵3 divides 3c²]

⇒ 3 divides b     ...(ii)

From (i) and (ii), we observe that a and b have at least 3 as a common factor. But, this disproves the fact that a and b are co-prime. This means that our assumption is not correct.

Hence, √3 is an irrational number.