# NUMBER SYSTEMS – Exercise 1.2 – (MATHEMATICS) – 9th Class

**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 , where m is a natural number.

(ii) Every real number is an irrational number:

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**Sol.**

(i) True, as real numbers consist of rational and irrational numbers.

(ii) False, as 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, 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.**

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**Sol.**No. For example, = 3, = 4, etc., 3, 4 are rational numbers.

**3. Show how can be represented on the number line.**

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**Sol.**For , 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 OP_{1} of unit length. Draw a line segment P_{1}P_{2} perpendicular to OP_{1}, of unit length (see Figure).

Now, draw a line segment P_{1}P_{2} perpendicular to OP_{2} of unit length. Then, draw a line segment P_{3}P_{4}, perpendicular to OP_{3} of unit length. 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 ,

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**Sol.**