Insert two irrational number between 2 and 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 ?

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

Classify the following numbers as rational or irrational:

$\begin{matrix} \text{(i)} & \sqrt{23} \\[1.5em] \text{(ii)} & \sqrt{225} \\[1.5em] \text{(iii)} & 0.3796 \\[1.5em] \text{(iv)} & 7.478478… \\[1.5em] \text{(v)} & 1.101001000100001… \\[1.5em] \text{(vi)} & 345.0\overline{456} \\[1.5em] \end{matrix}$

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