Prove that 1/(√ 5) is an irrational number.

Let \(\displaystyle \frac{1}{\sqrt{5}}\) be a rational number. \(\displaystyle \therefore \frac{1}{\sqrt{5}}=\frac{\mathrm{p}}{\mathrm{q}}\), where \(\displaystyle \mathrm{q} \neq 0\) and let p & q be the co-primes. \(\displaystyle 5 \mathrm{p}^{2}=\mathrm{q}^{2} \Rightarrow \mathrm{q}^{2}\) is divisible by 5. ⇒ q is divisible by 5. let \(\displaystyle \mathrm{q}=5 \mathrm{a}\), where 'a' is some integer. \(\displaystyle 25 \mathrm{a}^{2}=5 \mathrm{p}^{2} \Rightarrow \mathrm{p}^{2}=5 \mathrm{a}^{2} \Rightarrow \mathrm{p}^{2}\) is divisible by 5. ⇒ p is divisible by $\displaystyle 5$ (i) and (ii) leads to contradiction as p and q are coprimes. \(\displaystyle \therefore \frac{1}{\sqrt{5}}\) is an irrational number