#### 2. 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.

#### Class 9

#### Number Systems

#### Exercise 1.2

1. State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number.

(ii) Every point on the number line is of the form \(\sqrt{m}\), where m is a natural number.

(iii) Every real number is an irrational number.

3. Show how \(\sqrt5\) can be represented on the number line.

**4. 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_{1}P_{2 }perpendicular to OP_{1 }of unit length (see Fig. 1.9). Now draw a line segment P_{2}P_{3 }perpendicular to OP_{2}. Then draw a line segment P_{3}P_{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 \(\sqrt 2\), \(\sqrt 3\), \(\sqrt 4\), …