Classify the following numbers as rational or irrational :
$(i)$ $2-\sqrt{5}$
$(ii)$ $(3+\sqrt{23})-\sqrt{23}$
$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$
$(iv)$ $\frac{1}{\sqrt{2}}$
$(v)$ $2 \pi$
Classify the following numbers as rational or irrational :
$(i)$ $2-\sqrt{5}$
$(ii)$ $(3+\sqrt{23})-\sqrt{23}$
$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$
$(iv)$ $\frac{1}{\sqrt{2}}$
$(v)$ $2 \pi$
$(i)$ $2-\sqrt{5}$
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.
Similar Questions
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$(ii)$ $(3+\sqrt{3})(3-\sqrt{3})$
$(iii)$ $(\sqrt{5}+\sqrt{2})^{2}$
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