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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.
(ii) Every real number is an irrational number:

Sol.
(i) True, as real numbers consist of rational and irrational numbers.
(ii) False, as  ${3 \over 2}$  on the number line cannot be a square root of a natural number.
Also, a negative number cannot be a square root of natural number, $\sqrt m$ represents a positive value.
(iii) False, as 2 is a real number but not an irrational number.

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.

Sol. No. For example, $\sqrt 9$ = 3, $\sqrt {16}$ = 4, etc., 3, 4 are rational numbers.

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

Sol. For $\sqrt 5$, we have 5 = 5 × 1. On a number line, take 0 at position 0 and OA = 5 and OB = 1. With AB as diameter draw a semicircle. Draw OP ⊥ AB meeting semicircle at P. With 0 as centre and OP as radius an arc is drawn meeting the number line at Q.

Then, OQ = OP = √5  and Q represents  √5  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 OP1 of unit length. Draw a line segment P1P2  perpendicular to OP1, of unit  length (see Figure).

Now, draw a line segment P1P2 perpendicular to OP2 of unit length. Then, draw a line segment P3P4, perpendicular to OP3 of unit length. Continuing in this manner, you can get the line segment Pn-1Pn by drawing a line segment of unit length perpendicular to OPn -1. In this manner, you will have created the points P2, P3……,Pn,….,and joined them to create a beautiful spiral
depicting $\sqrt 2 ,\sqrt 3 ,\sqrt 4 ,........$ ,