Question

prove that √3 is irrational

Answer 1

since it cannot be expressed in fraction form it is irrational

Answer 2

Answer:

Let √3 be a rational number

√3 = a/b (a and b are integers and co-primes and b ≠ 0)

On squaring both the sides, 3 = a²/b²

⟹ 3b² = a²

⟹ a² is divisible by 3

⟹ a is divisible by 3

We can write a = 3c for some integer c.

⟹ a² = 9c²

⟹ 3b² = 9c²

⟹ b² = 3c²

⟹ b² is divisible by 3

⟹ b is divisible by 3

From (i) and (ii), we get 3 as a factor of ‘a’ and ‘b’ which is contradicting the fact that a and b are co-primes.

Hence our assumption that√3 is an rational number is false.

∴ √3 is an irrational number.