# Number Systems : Exercise 1.2 (Mathematics NCERT Class 9th) 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) 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 $\sqrt 2 ,\sqrt 3 ,\sqrt 4 ,\,....$. Sol.        Classroom activity - Do as directed.

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