Menu

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 \(\sqrt 2\), \(\sqrt 3\), \(\sqrt 4\), …

1. Write the following in decimal form and say what kind of decimal expansion each has:

3. Express the following in the form \(p\over q\), where p and q are integers and q ≠ 0.

7. Write three numbers whose decimal expansions are non-terminating non-recurring.

9. Classify the following numbers as rational or irrational:

1. Classify the following numbers as rational or irrational:

(ii) \((3+\sqrt{23})-\sqrt{23}\)

(iii) \({2\sqrt 7}\over{7\sqrt 7}\)

2. Simplify each of the following expressions:

(i) \((3+\sqrt 3)(2+\sqrt 2)\)

(ii) \((3+\sqrt 3)(3-\sqrt 3)\)

(iv) \((\sqrt 5-\sqrt 2)(\sqrt 5+\sqrt 2)\)

4. Represent \(\sqrt {9.3}\) on the number line.

5. Rationalise the denominators of the following:

(ii) \(1\over{\sqrt 7-\sqrt 6}\)