Prove that $\frac{1}{\sqrt{5}}$ is an irrational number.

Question

Accepted Answer

Step 1: Let $\frac{1}{\sqrt{5}} = \frac{p}{q}$ be a rational number, where $q 
eq 0$ and $p$, $q$ are co-primes.
Step 2: $\Rightarrow 5p^2 = q^2 \Rightarrow q^2$ is divisible by $5 \Rightarrow q$ is divisible by $5$ ... (i)
Step 3: Let $q = 5a$, where $a$ is some integer. Then $25a^2 = 5p^2 \Rightarrow p^2 = 5a^2 \Rightarrow p^2$ is divisible by $5 \Rightarrow p$ is divisible by $5$ ... (ii)
Step 4: (i) and (ii) lead to a contradiction as $p$ and $q$ are co-primes. $	herefore \frac{1}{\sqrt{5}}$ is an irrational number.

Question

Solution