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Number Systems EXERCISE 1.2 Solutions
- Q1(i): State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number.
- Q1(ii): 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.
- Q1(iii): State whether the following statements are true or false. Justify your answers. (iii) Every real number is an irrational number.
- Q2: 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.
- Q3: Show how $\sqrt{5}$ can be represented on the number line.
- Q4: 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 $\sqrt{2}, \sqrt{3}, \sqrt{4}, ...$
