Prove that 3 is an irrational number. - Vidyadip

Question

Prove that $\sqrt{3}$ is an irrational number.

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Answer

Let $\sqrt{ } 3$ be a rational number.
$\therefore \sqrt{ } 3=\frac{p}{q}$, where $q \neq 0$ and let $p \& q$ be co-prime.
$3 q^2=p^2 \Rightarrow p^2$ is divisible by $3 \Rightarrow p$ is divisible by $3----$ (i)
$\Rightarrow p=3 a$, where ' $a$ ' is some integer
$9 a^2=3 q^2 \Rightarrow q^2=3 a^2 \Rightarrow q^2$ is divisible by $3 \Rightarrow q$ is divisible by $3---$ (ii)
(i) and (ii) leads to contradiction as ' $p$ ' and ' $q$ ' are co-prime.

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