Find six rational numbers between $3$ and $4$.

Show that $0.2353535 \ldots=0.2 \overline{35}$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$.

Classify the following numbers as rational or irrational :

$(i)$ $2-\sqrt{5}$

$(ii)$ $(3+\sqrt{23})-\sqrt{23}$

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

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

$(v)$ $2 \pi$

Show that $3.142678$ is a rational number. In other words, express $3.142678$ in the form $\frac {p }{q }$, where $p$ and $q$ are integers and $q \ne 0$.

Visualize the representation of $5.3 \overline{7}$. on the number line upto $5$ decimal places, that is, up to $5.37777$.

Recall, $\pi$ is defined as the ratio of the circumference (say $c$ ) of a circle to its diameter
(say $d$ ). That is, $\pi=\frac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction ?