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 $q \neq 0$.

Find five rational numbers between $1$ and $2$.

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

Rationalise the denominator of $\frac{5}{\sqrt{3}-\sqrt{5}}$.

Classroom activity (Constructing the 'square root spiral') : Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point $O$ and draw a line segment $OP_1$ of unit length. Draw a line segment $P_1P_2$ perpendicular to $OP_1$ of unit length (see Fig.). Now draw a line segment $P_2P_3$ perpendicular to $OP_2$. Then draw a line segment $P_3P_4$ perpendicular to $OP_3$. Continuing in this manner,you can get the line segment $P_{n-1}P_n$ by drawing a line segment of unit length perpendicular to $OP_{n-1}$. In this manner,you will have created the points $P_2, P_3, ..., P_n, ...$,and joined them to create a beautiful spiral depicting $\sqrt{2}, \sqrt{3}, \sqrt{4}, ...$.

Rationalise the denominator of $\frac{1}{7+3 \sqrt{2}}$