Show that $0.3333... = 0.\overline{3}$ can be expressed in the form $\frac{p}{q}$,where $p$ and $q$ are integers and $q \ne 0$.

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 $\sqrt{2}, \sqrt{3}, \sqrt{4}, ...$.

Locate $\sqrt{3}$ on the number line.

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?

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

Locate $\sqrt{2}$ on the number line.