Question

Classify the following numbers as rational or irrational:

(i) $2 - \sqrt5$

(ii) $(3 + \sqrt23) - \sqrt23$

(iii) $\frac{2 \sqrt{7}} { 7 \sqrt7}$

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

(v) 2π

Answer 1

(i) $2 - \sqrt5$ = 2-2.2360679…

=-0.2360679…

As the decimal expansion of this expression is non-terminating non-recurring, therefore,

it is an irrational number.

(ii) $(3 + \sqrt23) - \sqrt23$ $=3=\frac{3}{1}$

As it can be represent in $\frac{p}{q}$

(iii) $\frac{2 \sqrt{7}} { 7 \sqrt7}=\frac{2}{7}$

As it can be represent in $\frac{p}{q}$

(iv) $\frac{1}{\sqrt{2}}=\frac{\sqrt2}{2}$=0.7071067811…

As the decimal expansion of this expression is non-terminating non-recurring,

therefore, it is an irrational number.

(v) 2π = 2(3.1415 …) = 6.2830 …

As the decimal expansion of this expression is non-terminating non-recurring, therefore,

it is an irrational number.