Question

Show that `sqrt(2)/3` is irrational.

Answer 1

Let `sqrt(2)/3` is a rational number.

∴`sqrt(2)/3 = p/q` where p and q are some integers and HCF(p, q) = 1   ...(1)

⇒ `sqrt(2)q = 3p`

⇒ `(sqrt(2)q)^2 = (3p)^2`

⇒ 2q² = 9p²

⇒ p² is divisible by 2

⇒ p is divisible by 2   ...(2)

Let p = 2m, where m is some integer.

∴ `sqrt(2)q = 3p`

⇒ `sqrt(2)q = 3(2m)`

⇒ `(sqrt(2)q)^2 = [3(2m)]^2`

⇒ 2q² = 4(9p²)

⇒ q² = 2(9p²)

⇒ q² is divisible by 2

⇒ q is divisible by 2   ...(3)

From (2) and (3), 2 is a common factor of both p and q, which contradicts (1).

Hence, our assumption is wrong.

Thus, `sqrt(2)/3` is irrational.