Classify the following numbers as rational or irrational: <br/> <br/>(i)$2-\sqrt{5}$<br/> <br/>(ii) $(3+\sqrt{23})-\sqrt{23}$ <br/> <br/> (iii) $\frac{2 \sqrt{7}}{7 \sqrt{7}}$ <br/> <br/> (iv) $\frac{1}{\sqrt{2}}$ <br/> <br/> (v) $2 \pi$

Solution:

(i) $2-\sqrt{5}=2-2.2360679 \ldots$

$=-0.2360679 \ldots$

As the decimal expansion of this expression is non-terminating non-recurring, therefore, it is an irrational number.

(ii) $(3+\sqrt{23})-\sqrt{23}=3=\frac{3}{1}$

As it can be represented in $\frac{p}{q}$ form, therefore, it is a rational number.

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

As it can be represented in $\frac{p}{q}$ form, therefore, it is a rational number.

(iv) $\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}=0.7071067811 \ldots$

As the decimal expansion of this expression is non-terminating non-recurring, therefore, it is an irrational number.

(v) $2 \pi=2(3.1415 \ldots)$

$=6.2830 \ldots$

As the decimal expansion of this expression is non-terminating non-recurring, therefore, it is an irrational number.