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.