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

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## 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 â‰ 0. It is 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 terminating or repeating decimal.

For example, âˆš2 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 terminating or repeating decimal.

For examples: 0.10 and 5/3 are rational numbers

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

**Solution:**

**(i)** âˆš7

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

**(ii)** âˆš4

A perfect square root of 2.

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

**(iii)** 2 + âˆš3

Here, 2 is a rational number but âˆš3 is an irrational number

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

**(iv)** âˆš3 + âˆš2

âˆš3 is not a perfect square thus an irrational number.

âˆš2 is not a perfect square, thus an irrational number.

Therefore, sum of âˆš2 and âˆš3 gives an irrational number.

**(v)** âˆš3 + âˆš5

âˆš3 is not a perfect square and hence, it is an irrational number

Similarly, âˆš5 is not a perfect square and also an irrational number.

Since, sum of two irrational number, is an irrational number, therefore âˆš3 + âˆš5 is an irrational number.

**(vi)** (âˆš2 â€“ 2)^{2}

(âˆš2 â€“ 2)^{2} = 2 + 4 â€“ 4 âˆš2

= 6 – 4 âˆš2

Here, 6 is a rational number but 4âˆš2 is an irrational number.

Since, the sum of a rational and an irrational number is an irrational number, therefore, (âˆš2 â€“ 2)2 is an irrational number.

**(vii)** (2 â€“ âˆš2)(2 + âˆš2)

We can write the given expression as;

(2 â€“ âˆš2)(2 + âˆš2) = ((2)^{2} âˆ’ (âˆš2)^{2})

^{2}â€“ b

^{2}]

= 4 â€“ 2 = 2 or 2/1

Since, 2 is a rational number, therefore, (2 â€“ âˆš2)(2 + âˆš2) is a rational number.

**(viii)** (âˆš3 + âˆš2)^{2}

We can write the given expression as;

(âˆš3 + âˆš2)^{2} = (âˆš3)^{2} + (âˆš2)^{2} + 2âˆš3 x âˆš2

= 3 + 2 + 2âˆš6

= 5 + 2âˆš6

[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, therefore, (âˆš3 + âˆš2)^{2} is an irrational number.

**(ix)** âˆš5 â€“ 2

âˆš5 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, âˆš5 â€“ 2 is an irrational number.

**(x)** âˆš23

Since, âˆš23 = 4.795831352331â€¦

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

**(xi)** âˆš225

âˆš225 = 15 or 15/1

âˆš225 is rational number as it can be represented in the form of p/q and q not equal to zero.

**(xii)** 0.3796

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

**(xiii)** 7.478478â€¦â€¦

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

**(xiv)** 1.101001000100001â€¦â€¦

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

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

**Solution**:

**(i)** âˆš4

âˆš4 = 2, which can be written in the form of a/b. Therefore, it is a rational number.

Its decimal representation is 2.0.

**(ii)** 3âˆš18

3âˆš18 = 9âˆš2

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

Therefore, 3âˆš18 is an irrational.

Or 3 Ã— âˆš18 is an irrational number.

**(iii)** âˆš1.44

âˆš1.44 = 1.2

Since, every terminating decimal is a rational number, Therefore, âˆš1.44 is a rational number.

And, its decimal representation is 1.2.

**(iv)** âˆš9/27

âˆš9/27 = 1/âˆš3

Since, we know, quotient of a rational and an irrational number is irrational numbers, therefore, âˆš9/27 is an irrational number.

**(v)** â€“ âˆš64

â€“ âˆš64 = â€“ 8 or â€“ 8/1

Therefore, â€“ âˆš64 is a rational number.

Its decimal representation is â€“8.0.

**(vi)** âˆš100

âˆš100 = 10

Since, 10 can be expressed in the form of a/b, such as 10/1,

Therefore, âˆš100 is a rational number.

And itâ€™s 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 both the sides,

x = âˆš5

âˆš5 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 a rational number.

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

z^{2} = 0.04

Taking square root both the 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 both the sides, we get

u = âˆš17/2

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

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

Taking square root both the sides, we get

v = âˆš3

Since, âˆš3 is not a perfect square root, so v is irrational number.

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

Taking square root both the sides, we get

w = 3âˆš3

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

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

Taking square root both the sides, we get

t = âˆš(4/10)

t = 2/âˆš10

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

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

RD Sharma SolutionsÂ Class 9 Maths Chapter 1 Number System Exercise 1.4 is based on 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.