Prove that root n is not a rational number

Prove that root n is not a rational number, if n is not perfect square


Let √n be a rational number.

\sqrt{n} = \frac{p}{q}, where p and q are co-prime integers q≠ 0.

On squaring both side we get

n = \frac{p^2}{q^2}

\Rightarrow p^2 = nq^2….(i)

⇒ n divides p²

⇒ n divides p … (ii)

[If n divides p², then n divides p]

Let p = nm, where m is any integer.

p^2 = n^2m^2

From eq (i)

n^2m^2 = nq^2

\Rightarrow q^2=nm^2

⇒ n divides q²

⇒ n divides q …..(iii)

[If n divides q², then n divides q]

From (ii) and (iii), n is common factor of both p and q which contradicts the assumption that

So, our assumption is wrong.

Hence, √n is an irrational number.

Some other question

Question 1:Prove that 2-√3 is irrational, given that root 3 is irrational

Question 2:Prove that √5 is an irrational number

Question 3: Prove that  √p + √q is irrational

Question 4:Prove that root 2 + root 5 is irrational


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