Class 8 Maths Solution | Lesson 3 Rational and Irrational Numbers | Curriculum Development Centre (CDC)

Lesson 3 Rational and Irrational Numbers

Exercise 2.1

1. Write true or false to the following statements.

(a) True - Both rational and irrational numbers are part of the set of real numbers.

(b) True - The sets of rational and irrational numbers do not overlap.

(c) False - Whole numbers (natural numbers including zero) are a subset of integers (which also include negative numbers).

(d) True - Integers are a subset of rational numbers, and rational numbers are a subset of real numbers.

(e) False – √4 = 2, which is a rational number.

(f) False - Whole numbers (ℕ_0) are a subset of integers (ℤ).

(g) True - All rational numbers are indeed real numbers.

2. Identify the rational and irrational numbers from the following.

(a) 3/4 – Rational

(b) √2 – Irrational

(c) √5 – Irrational

(d) 2/5 – Rational

(e) 10/20 – Rational

(f) 3.57 – Rational

(g) 3.5982 – Rational

(h) 15 – Rational

(i) 0.735 – Rational

(j) √169 – Rational (because √169 = 13)

(k) √3 – Irrational

(l) √26 – Irrational

(m) 2.5 – Rational

(n) 35/9 – Rational

(o) ³√9 – Irrational

3. Reduce the following decimal numbers to fractions.

(a) 0.5̅

Let x = 0.5̅

x = 0.555... (i)

Multiplying equation (i) by 10,

10x = 5.555... (ii)

Subtracting (ii) from (i),

10x - x = 5.555 - 0.555

or, 9x = 5

or, x = 5/9

Therefore, 0.5 = 5/9

(b) 0.7̅

Let x = 0.7̅

x = 0.777... (i)

Multiplying equation (i) by 10,

10x = 7.777... (ii)

Subtracting (ii) from (i),

10x - x = 7.777 - 0.777

or, 9x = 7

or, x = 7/9

Therefore, 0.7 = 7/9

(c)0.2̅4̅

Let x = 0.2̅4̅

x= 0.242424... (i)

Multiply equation (i) by 100,

100x = 24.242424...(ii)

Subtracting (ii) from (i),

100x - x = 24.242424 - 0.242424

or, 99x = 24

or, x = 24/99

(d) 0.1̅3̅2̅

Let x = 0.1̅3̅2̅

x = 0.132132... (i)

Multiplying equation (i) by 1000,

1000x=132.1322... (ii)

Subtracting (ii) from (i),

1000x - x = 132.132 −0.132

or, 999x = 132

or, x = 132 /999

(e)0.2̅7̅

Let x = 0.2̅7̅

x = 0.2727... (i)

Multiply (i) by 100:

Multiplying equation (i) by 100,

100x = 27.2727... (ii)

Subtracting (ii) from (i),

or, 99x = 27

or x = 27/99

Therefore, 0.27 = 3/11

(f) 1.5̅7̅

Let x = 1.5̅7̅

x = 1.5757......(i)

Multiplying equation (i) by 100,

100x = 157.57... (ii)

Subtracting (ii) from (i),

or, 100x−x = 157.57 −1.57

or, 99x = 156

or, x = 156/99

(g) 0.3̅6̅5̅

Let x = 0.3̅6̅5̅

x = 0.365 (i)

Multiplying equation (i) by 1000,

1000x = 365.365... (ii)

Subtracting (ii) from (i),

or, 1000x−x = 365.365−0.365

or, 999x = 365

or, x = 365/999

Therefore, 0.365 = 4/11

(h) 4.7̅8̅

Let x = 4.7̅8̅

x = 4.7878... (i)

Multiplying equation (i) by 10,

Subtracting (ii) from (i),

100x−x = 478.787878 −4.787878

or, 99x = 474

or, x = 474/99

(i) 0.4̅4̅5̅

Let x = 0.4̅4̅5̅

x = 0.445.... (i)

Multiplying equation (i) by 1000,

1000x = 445.445445445 (ii)

Subtracting (ii) from (i),

or, 999x = 445

or, x = 445/999

(j) 1.5̅25̅

Let x = 1.5̅25̅

This means x = 1.525252... (i)

Multiply equation (i) by 1000:

1000x = 1525.252525... (ii)

Subtract equation (i) from equation (ii):

1000x - x = 1525.252525 - 1.525252

or, 999x = 1524

or, x = 1524 / 999