# Show that Sqrt (2)/3 Is Irrational. - Mathematics

Show that sqrt (2)/3 is irrational.

#### Solution

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
⇒2q^2 = 9p^2
⇒p^2 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
⇒ 2q^2 = 4 (9p^2)

⇒ q^2 = 2 (9p^2)
⇒ q^2 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.

Concept: Revisiting Rational Numbers and Their Decimal Expansions
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