Find a rational number between $\sqrt{2}$ and $\sqrt{3}$.

The following real numbers have decimal expansions as given below. In each case, state whether they are rational or not. If they are rational and expressed in the form $\dfrac{p}{q}$, where p, q are integers, q ≠ 0 and p, q are co-prime, then what can you say about the prime factors of q ?

$\begin{matrix} \text{(i)} & 37.09158 \\[1.5em] \text{(ii)} & 423.\overline{04567}\\[1.5em] \text{(iii)} & 8.9010010001… \\[1.5em] \text{(iv)} & 2.3476817681… \\[1.5em] \end{matrix}$

Insert an irrational number between the following :

(i) $\dfrac{1}{3}$ and $\dfrac{1}{2}$

(ii) $-\dfrac{2}{5}$ and $\dfrac{1}{2}$

(iii) 0 and 0.1

Write two irrational numbers between $\dfrac{4}{9}$ and $\dfrac{7}{11}$.