Example 1. Which of the following is an irrational number?
(i) 3.14
(ii) 3.14
(iii) 3.14141414...
(iv) 3.14114111411114...

Solution The correct answer is (iv) 3.14114111411114... as it is a nonterminating, nonrepeating number


Example 2. Which of the following is an irrational number?
(i) $\sqrt{\frac{9}{16}}$
(ii) $\sqrt{49}$
(iii) $\frac{\sqrt{63}}{\sqrt{7}}$
(iv) $\sqrt{7}$

Solution The correct answer is (iv) $\sqrt{7}$ as it is a nonterminating, nonrepeating number


Example 3. Which of the following is a rational number?
(i) 0.10100100010000...
(ii) $\sqrt{5}$
(iii) $\frac{\sqrt{63}}{\sqrt{9}}$
(iv) $\sqrt{361}$

Solution The correct answer is (iv) $\sqrt{361}$ as it is a perfect square. $19^2 = 361$


Example 4. Find an irrational number between $\frac{1}{5}$ and $\frac{2}{5}$

Solution We have
$\frac{1}{5} = 0.2$ and
$\frac{2}{5} = 0.4$ and

To find an irrational number between $0.2$ and $0.4$, we find a number which is non-terminating and non-repeating. There are infinite numbers which satisfy this criteria.
One such number is 0.31311311131111...

Example 5. Rationalise the denominator $\frac{1}{2 + \sqrt{3}}$
Solution To rationalise the denominator, we multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of $(2 + \sqrt{3})$ is $(2 - \sqrt{3})$
Hence we have
$\begin{align*} & \frac{1}{2 + \sqrt{3}} \\ & = \frac{1}{2 + \sqrt{3}} * \frac{2 - \sqrt{3}}{2 - \sqrt{3}} \\ & = \frac{2 - \sqrt{3}}{2^2 - \sqrt{3}^2} \\ & = \frac{2 - \sqrt{3}}{4 - 3} \\ & = 2 - \sqrt{3} \end{align*} $