बताइए नीचे दी गई संख्याओं में कौन-कौन परिमेय हैं और कौन-कौन अपरिमेय हैं

$(i)$ $2-\sqrt{5}$

$(ii)$ $(3+\sqrt{23})-\sqrt{23}$

$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$

$(iv)$ $\frac{1}{\sqrt{2}}$

$(v)$ $2 \pi$

$(i)$ $2-\sqrt{5}$

Since it is a difference of a rational and irrational number,

$\therefore $ $2-\sqrt{5}$ is an irrational number.

$(ii)$ $(3+\sqrt{23})-\sqrt{23}$

We have : $(3+\sqrt{23})-\sqrt{23}=3+\sqrt{23}-\sqrt{23}=3,$ which is a rational number.

$\therefore(3+\sqrt{23})-\sqrt{23}$ is a rational number.

$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$

since, $\frac{2 \sqrt{7}}{7 \sqrt{7}}=\frac{2 \times \sqrt{7}}{7 \times \sqrt{7}}=\frac{2}{7},$ which is a rational number.

$\therefore \frac{2 \sqrt{7}}{7 \sqrt{7}}$ is a rational number.

$(iv)$ $\frac{1}{\sqrt{2}}$

$\because$ The quotient of rational and irrational is an irrational number.

$\therefore \frac{1}{\sqrt{2}}$ is an irrational number.

$(v)$ $2 \pi$

$\therefore 2 \pi=2 \times \pi=$ Product of a rational and an irrational (which is an irrational number)

$\therefore 2 \pi$ is an irrational number.