**RD Sharma Solutions for Class 9 Mathematics Chapter 1 Exercise 1.4 Number System** are provided here. Students can use this study material as a reference tool for studying as well as practising questions on irrational numbers. The students can enhance their exam preparations by practising RD Sharma Solutions for Class 9 Chapter 1 textbook questions and score well in the annual exam.

## RD Sharma Solutions for Class 9 Maths Chapter 1 Number System Exercise 1.4

### Access Answers to RD Sharma Solutions for Class 9 Maths Chapter 1 Number System Exercise 1.4

## Exercise 1.4

**Question 1: Define an irrational number.**

**Solution:**

A number which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0. It is a non-terminating or non-repeating decimal.

**Question 2: Explain how irrational numbers differ from rational numbers.**

**Solution:**

An irrational number is a real number which can be written as a decimal but not as a fraction i.e. it cannot be expressed as a ratio of integers.

It cannot be expressed as a terminating or repeating decimal.

For example, √2 is an irrational number

A rational number is a real number which can be written as a fraction and as a decimal, i.e. it can be expressed as a ratio of integers.

It can be expressed as a terminating or repeating decimal.

For example, 0.10 and 5/3 are rational numbers

**Question 3: Examine whether the following numbers are rational or irrational:**

**Solution:**

**(i)** √7

Not a perfect square root, so it is an irrational number.

**(ii)** √4

A perfect square root of 2.

We can express 2 in the form of 2/1, so it is a rational number.

**(iii)** 2 + √3

Here, 2 is a rational number, but √3 is an irrational number

Therefore, the sum of a rational and irrational number is an irrational number.

**(iv)** √3 + √2

√3 is not a perfect square, thus an irrational number.

√2 is not a perfect square, thus an irrational number.

Therefore, the sum of √2 and √3 gives an irrational number.

**(v)** √3 + √5

√3 is not a perfect square, and hence, it is an irrational number

Similarly, √5 is not a perfect square, and it is an irrational number.

Since the sum of two irrational numbers is an irrational number, √3 + √5 is an irrational number.

**(vi)** (√2 – 2)^{2}

(√2 – 2)^{2} = 2 + 4 – 4 √2

= 6 – 4 √2

Here, 6 is a rational number, but 4√2 is an irrational number.

Since the sum of a rational and an irrational number is an irrational number, (√2 – 2)2 is an irrational number.

**(vii)** (2 – √2)(2 + √2)

We can write the given expression as;

(2 – √2)(2 + √2) = ((2)^{2} − (√2)^{2})

^{2}– b

^{2}]

= 4 – 2 = 2 or 2/1

Since 2 is a rational number, (2 – √2)(2 + √2) is a rational number.

**(viii)** (√3 + √2)^{2}

We can write the given expression as;

(√3 + √2)^{2} = (√3)^{2} + (√2)^{2} + 2√3 x √2

= 3 + 2 + 2√6

= 5 + 2√6

[using identity, (a+b)^{2}= a

^{2}+ 2ab + b

^{2}]

Since the sum of a rational number and an irrational number is an irrational number, (√3 + √2)^{2} is an irrational number.

**(ix)** √5 – 2

√5 is an irrational number, whereas 2 is a rational number.

The difference of an irrational number and a rational number is an irrational number.

Therefore, √5 – 2 is an irrational number.

**(x)** √23

Since, √23 = 4.795831352331…

As the decimal expansion of this number is non-terminating and non-recurring, it is an irrational number.

**(xi)** √225

√225 = 15 or 15/1

√225 is a rational number as it can be represented in the form of p/q, and q is not equal to zero.

**(xii)** 0.3796

As the decimal expansion of the given number is terminating, it is a rational number.

**(xiii)** 7.478478……

As the decimal expansion of this number is a non-terminating recurring decimal, it is a rational number.

**(xiv)** 1.101001000100001……

As the decimal expansion of the given number is non-terminating and non-recurring, it is an irrational number

**Question 4: Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:**

**Solution**:

**(i)** √4

√4 = 2, which can be written in the form of a/b. Therefore, it is a rational number.

Its decimal representation is 2.0.

**(ii)** 3√18

3√18 = 9√2

Since the product of a rational and an irrational number is an irrational number, 3√18 is an irrational number.

Or 3 × √18 is an irrational number.

**(iii)** √1.44

√1.44 = 1.2

Since every terminating decimal is a rational number, √1.44 is a rational number.

And its decimal representation is 1.2.

**(iv)** √9/27

√9/27 = 1/√3

Since the quotient of a rational and an irrational number is irrational numbers, √9/27 is an irrational number.

**(v)** – √64

– √64 = – 8 or – 8/1

Therefore, – √64 is a rational number.

Its decimal representation is –8.0.

**(vi)** √100

√100 = 10

Since 10 can be expressed in the form of a/b, such as 10/1, √100 is a rational number.

And its decimal representation is 10.0.

**Question 5: In the following equation, find which variables x, y, z etc. represent rational or irrational numbers:**

**Solution**:

**(i)** x^{2} = 5

Taking square root on both sides,

x = √5

√5 is not a perfect square root, so it is an irrational number.

**(ii)** y^{2} = 9

y^{2} = 9

or y = 3

3 can be expressed in the form of a/b, such as 3/1, so it is a rational number.

**(iii)** z^{2} = 0.04

z^{2} = 0.04

Taking square root on both sides, we get

z = 0.2

0.2 can be expressed in the form of a/b, such as 2/10, so it is a rational number.

**(iv)** u^{2} = 17/4

Taking square root on both sides, we get

u = √17/2

Since the quotient of an irrational and a rational number is irrational, u is an Irrational number.

**(v)** v^{2} = 3

Taking square root on both sides, we get

v = √3

Since √3 is not a perfect square root, v is an irrational number.

**(vi)** w^{2} = 27

Taking square root on both sides, we get

w = 3√3

Since the product of a rational and irrational is an irrational number, w is an irrational number.

**(vii)** t^{2} = 0.4

Taking square root on both sides, we get

t = √(4/10)

t = 2/√10

Since the quotient of a rational and an irrational number is an irrational number t is an irrational number.

## RD Sharma Solutions for Class 9 Maths Chapter 1 Number System Exercise 1.4

**RD Sharma Solutions Class 9 Maths Chapter 1 Number System Exercise 1.4** is based on the following topics and subtopics:

- Irrational numbers.
- A number is an irrational number if it has a non-terminating and non-repeating decimal representation.
- Some useful results on irrational numbers.

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