Classify the following numbers as rational or irrational:

(i) 2 − $\sqrt{5}$

(ii) 3 + $\sqrt{23}$ − $\sqrt{23}$

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

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

(v) 2π

(i) 2 − $\sqrt{5}$

If in any rational number we add, subtract, multiply or divide any irrational number it becomes irrational number.

Hence, 2 − ($\sqrt{5})$ is an irrational number.

(ii) 3 + $\sqrt{23}$ − $\sqrt{23}$

= 3 + $\sqrt{23}$ − $\sqrt{23}$

= 3

Hence, (3 + $\sqrt{23}$) − $\sqrt{23}$ is a rational number

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

$\dfrac{2\sqrt{7}}{7\sqrt{7}}$ = $\dfrac{2}{7}$

It is in $\dfrac{p}{q}$ where q ≠ 0.

Hence, is a rational number

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

$\dfrac{1}{\sqrt{2}} = \dfrac{\text{rational}}{\sqrt{\text{irrational}}}$ = irrational number

If in any rational number we add, subtract, multiply or divide any irrational number it becomes irrational

Hence, $\dfrac{1}{\sqrt{2}}$ is an irrational number

(v) 2π

2 x π = irrational number [∵ rational x irrational number = irrational number]

Hence, 2π is an irrational number