**Question 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 √m , where m is a natural number.**

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

**Solution: **

(i)Every irrational number is a real number.

True

Irrational numbers are the number that cannot be written in the form of p/q , p and q are the integers and q ≠ 0.

Some examples of irrational numbers are π, √3, e, √2, 011011011…..Real numbers include both rational numbers and irrational numbers.

Thus, 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.

False

We can represent both negative and positive numbers on a number line.

Positive numbers can be written as √16=4 that is a natural number, But √3=1.73205080757 is not a natural number.

But negative numbers cannot be expressed as the square root of any natural number, as if we take square root of an negative number it will become complex number that will not be a natural number (√5=5i is a complex number).

(iii)Every real number is an irrational number.

False

Every irrational number is a real number but every real number is not an irrational number as real numbers include both rational and irrational number.

**Question 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 in not irrational. For example √9 = 3, √25 = 5, hence square roots of all positive integers is not irrational.

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

**Solution: **

To represent √5 on the number line follow the following steps:**Step 1:** Draw a number line.

**Step 2: **Let line AB of 2 units on the number line.

**Step 3:** Now draw a perpendicular of unit 1 on point B and mark the other end as point C.

**Step 4:** Join AC, you will have triangle ABC which is a right-angle triangle. as shown below:

**Step 5:** Apply Pythagoras Theorem on triangle ABC,

⇒ AB

^{2}+ BC^{2}= AC^{2}

⇒ AC^{2}= 2^{2}+ 1^{2}

⇒ AC^{2}= 5

⇒ AC = √5

Therefore, line AC is of √5 unit length.

**Step 6: **Now take line AC as the radius and an arc and intersect it to number line, the point where the arc will intersect the number line is of lengths √5 from point 0 to the intersection as point A is the centre of the radius.

**Question 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 Fig. 1.9). Now draw a line segment P_{2} P_{3} perpendicular to OP_{2}. Then draw a line segment P_{3} P_{4} perpendicular to OP_{3}. 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 √2, √3, √4, …

_{1}of unit length. Draw a line segment P

_{1}P

_{2}perpendicular to OP

_{1}of unit length (see Fig. 1.9). Now draw a line segment P

_{2}P

_{3}perpendicular to OP

_{2}. Then draw a line segment P

_{3}P

_{4}perpendicular to OP

_{3}. 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 √2, √3, √4, …

**Solution: **

**Step 1:** First lets mark a point O on the larger sheet of paper, this point will be the centre of the square root spiral.

**Step 2:** Draw point P_{1} from point O of 1 unit. OP_{1}=1 unit.

**Step 3:** Similarly as in the above problem from P_{1 }draw a perpendicular of 1 unit, P_{1}P_{2 }= 1 unit.

**Step 4: **Now join OP_{2} = √2

**Step 5: **From point P_{2} draw a perpendicular of 1 unit. P_{2}P_{3} = 1 unit.

**Step 6: **Now join OP_{3} = √3

**Step 7:** Now repeat the step to make √4, √5, √6, …….