**CBSE Class 9 Maths Lab Manual – An Irrational Number**

**Objective**

To represent an irrational number on the number line. (To represent √2 on number line).

**Prerequisite Knowledge**

Concept of Pythagoras theorem:

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides containing right angle.

In a right angled triangle, if the base and perpendicular are of 1 unit each, the hypotenuse will be

√(1^{2} +1^{2}) = √2.

Now, by using this concept, we will represent √2 on the number line.

**Materials Required**

A sheet of white paper, pencil, compass, eraser and ruler etc.

**Procedure**

- Draw a straight line X’OX on the white sheet of paper.
- Divide that line into equal parts from point O by paper folding activity taking each part as 1 unit. Mark the points as 1,2,3,…. etc.
- Draw the perpendicular at the point marked as ‘1’ by paper folding.
- Unfold the paper, and draw the line at the crease so formed. Mark a point A on this crease at 1 unit from line X’OX.

- Join O and A, we get OA = √2 units (By Pythagoras theorem).
- With O as centre, OA as radius, draw an arc intersecting the line X’OX at M.

**Observation**

We observe that OA = OM = √2 units.

**Result**

An irrational number √2 is represented on the number line.

**Learning Outcome**

Students can represent any irrational number on number line by using above method.

e.g., (√3)^{2} = (√2)^{2} +(1)^{2}

At M, by paper folding draw perpendicular BM on the number line of 1 unit. Join OB. With O as centre and OB as radius draw an arc intersecting the line at N.

Thus OB = ON = √3 on the number line.

**Activity Time**

Represent √5, √7 on the number line.

**Viva Voce**

**Question 1.**

What are real numbers ?

**Answer:**

The collection of all rational numbers and irrational numbers.

**Question 2.**

What do you mean by rational and irrational numbers ?

**Answer:**

Decimal expansion of rational numbers are either terminating or recurring. Irrational numbers are non-terminating and non-recurring.

**Question 3.**

Is π a rational number ?

**Answer:**

No, π is an irrational number.

**Question 4.**

Can the sum of two irrational numbers be zero ?

**Answer:**

Yes, e.g., (2 + √2)+ (-√2 – 2) = 0

**Question 5.**

Can the square root of any natural number be negative ?

**Answer:**

No

**Question 6.**

Is the square root of -5 is real ?

**Answer:**

No.

**Question 7.**

Write √45 in mixed surd ?

**Answer:**

3√5.

**Question 8.**

What do you mean by surd ?

**Answer:**

If the positive nth root of a number is an irrational number it is called a surd or radical.

**Question 9.**

Who showed that corresponding to every real number, there is a point on the real number line and corresponding to every point on the number line, there exists a real number ?

**Answer:**

Two German Mathematicians named as Centor and Dedekind.

**Multiple Choice Questions**

**Question 1:**

Irrational numbers are:

(i) terminating decimals.

(ii) non-terminating and non-recurring decimals.

(iii) non-recurring decimals.

(iii) none of these.

**Question 2:**

Who discovered Pythagoras’ theorem ?

(i) Pythagoras

(ii) Issac Newton

(iii) Euclid

(iv) none of these

**Question 3:**

In which triangle, the Pythagoras’ theorem is applicable ?

(i) right triangle

(ii) obtuse triangle

(iii) acute triangle

(iv) none of these .

**Question 4:**

Without actual division, check whether \(\frac { 47 }{ 14 }\) is terminating or not:

(i) terminating

(ii) non-terminating

(iii) irrational

(iv) none of these

**Question 5:**

Write the Pythagorean triplets for √2:

(i) (√2, 1, 1)

(ii) (1, √2, 3)

(iii) (1, 1, 3)

(iv) none of these

**Question 6:**

Write the Pythagorean triplets for √5:

(i) (√2, 1, √3)

(ii) (√5, √2, 1)

(iii) (√5, √4, 1)

(iv) none of these

**Question 7:**

Give one example each of rational number and irrational number:

(i) √8, √10

(ii) √2, √6

(iii) √4, √2

(iv) none of these

**Question 8:**

The product of’ a rational number and an irrational number is always:

(i) a rational number

(ii) an irrational number

(iii) an integer

(iv) none of these

**Question 9:**

Is \(\frac { 3 }{ 20 }\) an irrational number?

(i) no

(ii) yes

(iii) not real number

(iv) none of these

**Question 10:**

Two irrational numbers whose sum is a rational are:

(i) √2 and √3

(ii) √2 and (-√2)

(iii) √3and (√3 – √5)

(iv) none of these

**Answers**

- (ii)
- (i)
- (i)
- (ii)
- (i)
- (iii)
- (iii)
- (ii)
- (i)
- (ii)

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