Prove that $\displaystyle 4-2 \sqrt{5}$ is an irrational number given that $\displaystyle \sqrt{5}$ is irrational.

Sol. Let \(\displaystyle 4-2 \sqrt{5}\) be a rational number. \(\displaystyle \therefore 4-2 \sqrt{5}=\frac{\mathrm{a}}{\mathrm{b}}\) where a and b are integers and \(\displaystyle \mathrm{b} \neq 0\). \(\displaystyle \sqrt{5}=\frac{4 \mathrm{~b}-\mathrm{a}}{2 \mathrm{~b}}\) RHS is rational but LHS is an irrational which is a contradiction to our supposition. Hence \(\displaystyle 4-2 \sqrt{5}\) is an irrational number.