Prove that root n is not a rational number, if n is not perfect square
Let √n be a rational number.
∴ , where p and q are co-prime integers q≠ 0.
On squaring both side we get
⇒ n divides p²
⇒ n divides p … (ii)
[If n divides p², then n divides p]
Let p = nm, where m is any integer.
From eq (i)
⇒ 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