**NCERT Class 9 Maths Lab Manual – Represent Some Irrational Numbers on the Number Line**

**OBJECTIVE**

To represent some irrational numbers on the number line.

**Materials Required**

- Two cuboidal wooden strips
- Thread
- Nails
- Hammer
- Two photocopies of a scale
- A screw with nut
- Glue
- Cutter

**Prerequisite Knowledge**

- Concept of number line.
- Concept of irrational numbers.
- Pythagoras theorem.
- Representation of rational number on number line.

**Theory**

- The concept of number line refer to Activity 1.
- For concept of irrational numbers refer to Activity 1.
- For Pythagoras theorem refer to Activity 1.
- Representation of Irrational Number on Number Line.

We know that a real number is either rational or irrational. So, we can say that every real number is represented by a unique point on the number line.

Also, every point on the number line represents a unique real number. So, we can locate some of the irrational number of the form √n, where n is a positive integer on the number line by using following steps.

**Step I –** Write the given number (without root) as the sum of the squares of two natural numbers (say a and b, where a > b).

**Step II –** Take the distance equal to these two natural numbers on the number line (a on number line and b vertically) starting from 0 (say OA and AB) in such a way that one is perpendicular to other (say AB ⊥ OA).

**Step III –** Use Pythagoras theorem to find the distance OB.

**Step IV –** Take O as centre and OB as radius, draw an arc, which cut the number line at C (say).

Thus, the point C will represent the location of √n on the number line, (see Fig. 2.1)

**Procedure**

- Make a straight slit on the top of one of the the wooden strips. Now, fix another wooden strip on the slit perpendicular to the former strip with a screw at the bottom, so that it can move freely along the slit, (see Fig. 2.2)
- Paste one photocopy of the scale on the horizontal (movable) strip and another photocopy of the scale on the perpendicular strip, (see Fig. 2.2)
- Fix nails at a distance of 1 unit each, starting from O, on both the strips, (see Fig. 2.2)
- Now, tie a thread at the nail at 0 on the horizontal strip, (see Fig. 2.2)

**Demonstration**

- With the help of screw, fix the perpendicular wooden strip at 1, which is 1 unit on horizontal scale.
- Tie the other end of the thread to unit 1 on the perpendicular strip.
- Remove the thread from unit 1 on the perpendicular strip and place it on the horizontal strip to represent √2 on the horizontal strip.

ΔOAB is a right angled triangle.

So, from Pythagoras theorem, OB² = OA² + AB²

Here, OA = 1 unit, AB = 1 unit,

OB² = (1)² + (1)² => OB² =2 => OB = √2 Similarly, to represent √3, fix the perpendicular wooden strip at √2 and repeat the above process.

Thus, we conclude to represent √a, a > 1 fix the perpendicular scale at \(\sqrt { a-1 }\) and proceed as above to get √a.

**Note:** To find √a such as √13 by fixing the perpendicular strip at 3 on the horizontal strip and tying the other end of thread at 2 on the vertical strip.

**Observation**

On actual measurement, we get a -1 = ……….. , √a =…………

**Result**

Any irrational number can be represented on the number line by using this method.

Application

This activity may help to student in representing some irrational numbers like √2, √3, √5, √6, √7,…, etc., on the number line.

**Viva Voce**

**Question 1:**

Is every irrational number, a real number?

**Answer: **

Yes, because real numbers consist of both rational and irrational numbers.

**Question 2:**

Can we apply Pythagoras theorem in any triangle?

**Answer:**

No, Pythagoras theorem is applicable only in right angled triangle.

**Question 3:**

Isrca rational or an irrational number? What is value of TC up to three decimal places?

**Answer:**

π is an irrational number. The value of π is 3.142.

**Question 4:**

“Sum of two irrational numbers is an irrational number”. Is this statement true?

**Answer:**

No, its not true, sum of two irrational numbers may be irrational or rational.

**Question 5:**

Is the product of two irrational numbers, an irrational number?

**Answer:**

No, it is not necessary that, the product of two irrational numbers is irrational number.

**Question 6:**

Does the square roots of all positive integers, irrational? Give reason.

**Answer:**

No, square roots of all positive integers are not irrational, e.g. S = √9 = 3², which is a rational number.

**Question 7:**

How would you find a base of a right angled triangle, if hypotenuse and perpendicular are given?

**Answer:**

Base = \(\sqrt { (hypotenuse)2-(perpendicular)2 }\)

**Question 8:**

Is it possible to say that √∞ is defined on number line?

**Answer:**

No

**Question 9: **

Is it possible that the sum of two irrational numbers can be represented on number line?

**Answer:**

Yes

**Suggested Activity**

Represent other irrational numbers such as √3, √5, √7,…, etc., on the number line.

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