Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Show how $\sqrt{5}$ can be represented on the number line.

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

State whether the following statements are true or false. Justify your answers.
(iii) Every real number is an irrational number.

State whether the following statements are true or false. Justify your answers.
(ii) Every point on the number line is of the form $\sqrt{m}$, where $m$ is a natural number.

State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.