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

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

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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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