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

Multiply $6 \sqrt{5}$ by $2 \sqrt{5}$.

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

Classify the following numbers as rational or irrational:
$(i)$ $2-\sqrt{5}$
$(ii)$ $(3+\sqrt{23})-\sqrt{23}$
$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$
$(iv)$ $\frac{1}{\sqrt{2}}$
$(v)$ $2 \pi$

Write the following in decimal form and state what kind of decimal expansion each has:
$(i)$ $\frac{36}{100}$
$(ii)$ $\frac{1}{11}$
$(iii)$ $4 \frac{1}{8}$
$(iv)$ $\frac{3}{13}$
$(v)$ $\frac{2}{11}$
$(vi)$ $\frac{329}{400}$

Rationalise the denominators of the following:
$(i)$ $\frac{1}{\sqrt{7}}$
$(ii)$ $\frac{1}{\sqrt{7}-\sqrt{6}}$
$(iii)$ $\frac{1}{\sqrt{5}+\sqrt{2}}$
$(iv)$ $\frac{1}{\sqrt{7}-2}$