Simplify the following expressions :

$(i)$ $(5+\sqrt{7})(2+\sqrt{5})$

$(ii)$ $(5+\sqrt{5})(5-\sqrt{5})$

$(iii)$ $(\sqrt{3}+\sqrt{7})^{2}$

$(iv)$ $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$

Find the decimal expansions of $\frac{10}{3},\, \frac{7}{8}$ and $\frac{1}{7}$.

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

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

Is zero a rational number ? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$ ?

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 ?