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Number Systems : Exercise 1.2 (Mathematics NCERT Class 9th)


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Q.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.

Sol.

(i)     Every irrational number is a real number - True
         Justification : - A real number is either rational or irrational.
(ii)   Every point on the number line is of the form \sqrt m   - False
         Justification : - Numbers  of other types also lie on the number line.
(iii)  Every real number is an irrational number - False
          Justification : - Rational numbers are also real numbers.


Q.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 numbers.
Sol.

No, the square roots of all positive integers are not irrational.
Example : -  4 is a positive integer but \sqrt 4 = 2 is a  natural number.


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

We shall now show how to represent \sqrt 5 on the number line.
We first represent \sqrt 5 on the number line l. We construct a right - angled \Delta OAB, right - angled at A such that OA = 2 and AB = 1  unit (see figure)
ns1
Then,   OB= \sqrt {O{A^2} + A{B^2}} = \sqrt {4 + 1} = \sqrt 5

Now, we cut off a length  OC = OB = \sqrt 5 on the number line.
Then the point C represents the irrational number \sqrt 5 .


Q.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 O{P_1} of unit length. Draw a line segment {P_1}{P_2} perpendicular to O{P_1} of unit  length (see figure). Now draw a line segment {P_2}{P_3} perpendicular to O{P_2}. Then draw a line segment {P_3}{P_4} perpendicular to O{P_3}. Continuing in the manner, you can get the line segment {P_{n - 1}}{P_n} by drawing a line segment of unit length perpendicular to O{P_{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 ,\,.....

ns2Sol.        Classroom activity - Do as directed.