​ Verify by method of contradiction p ∶ √11 is irrational. ​

Question

Verify by method of contradiction p ∶ \(\sqrt{11}\) is irrational.

Answer 1

Let the given statement be false i.e., ~ p ∶ \(\sqrt{11}\)

Is rational. The \(\sqrt{11}\) = \(\frac{p}{q}\) where p and q are coprime and q ≠ 0.

⇒ 11 = \(\frac{p^2}{q^2}\)

⇒ p² = 11q²

⇒ 11 divides p ...(1)

∴ r ∈ z such that−

p = 11r

⇒ p² = 121r

⇒ 11q² = 121r²

⇒ q² = 11r²

⇒ 11 divides q … (2)

From (1) & (2), we arrive at a contradiction, since p and q are coprime.

⇒ \(\sqrt{11}\) is irrational.