**NCERT Solutions for Class 9 Maths Chapter-1 Number System Exercise 1.2**

**Q1. 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 √m, Where m is a natural number.**

**(iii) Every real number is an irrational number.**

Sol. (i) True, real numbers are a collection of rational and irrational numbers.

(ii) False, the square root of a negative number can be a natural number.

(iii) False, For example, 2 is a real number but not an irrational number.

**Q2. 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, √25 = 5 is a rational number.

**Q3. Show how √5 can be represented on the number line.**

Sol. Step 1: Let line AB be of 2 units on a number line.

Step 2: At B, draw a perpendicular line BC of length 1 unit.

Step 3: Join CA

Step 4: Now, ABC is a right-angled triangle. Applying Pythagoras theorem,

⇒ AB^{2}+BC² = CA²

⇒ 2²+1²= CA²

⇒ CA² = 5

⇒ CA = √5.

Thus, CA is a line of length √5 units.

Step 5: Taking CA as a radius and A as a center draw an arc touching the number line. The point at which the number line gets intersected by are is at √5 distance from 0 because it is a radius of the circle whose center was A.

Thus, √5 is represented on the number line as shown in the figure.

**Q4. 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 PIP2 perpendicular to OP, of unit length (see Fig. 1.9). Now draw a line segment P₂P, perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP. Continuing in Fig. 1.9:**

**Constructing this manner, you can get the line segment P. Pn by square root spiral drawing a line segment of unit length perpendicular to OP. In this manner, you will have created the points Ps P.Pn. and joined them to create a beautiful spiral depicting √2, 3, 4….**

Sol. Do Yourself.

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