**1. State whether the following statements are true or false. Justify your answers.**

**(i)Every irrational number is a real number.**

**Solution:**

True

According to the definition:

Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0.

i.e., Irrational numbers = π, e, √5, 6+√3, 7.241, 0.0110….

Real numbers – The collection of both rational and irrational numbers are known as real numbers.

i.e., Real numbers = √4, √7, , 0.102…

Hence, every irrational number is real number.

**(ii) Every point on the number line is of the form √m , where m is a natural number.**

**Solution:**

False

Because negative numbers cannot be the square root of any natural number.

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

**Solution:**

False

Because the real numbers include both irrational and rational numbers. Therefore, every real number cannot be 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.**

**Solution:**

No, the square roots of all positive integers are not irrational.

Let us take

√4 = 2 is rational.

Hence, the square roots of positive integer 4 is not irrational.

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

**Solution:**

Step 1: Draw a line OB be of 2 unit on a number line.

Step 2: Draw a perpendicular line AB of length 1 unit at B.

Step 3: Now, join OA

Step 4: we observe that AOB is a right angled triangle. So, applying Pythagoras theorem,

AB^{2}+OB^{2} = OA^{2}

1^{2}+2^{2} = OA^{2} = 5

Hence, OA = √5 .

Thus, OA is a line of length √5 unit.

Step 4: With center O, taking OA as a radius, draw an arc touching the number line. The point at which number line get intersected by arc is at √5 distance from 0.

**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 Fig. 1.9). Now draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in Fig. 1.9 :**

**Solution:** Do it yourself.

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