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

### Question 1: Define an irrational number.

Solution:

A real number that cannot be expressed in the form of fractions i.e. p/q, where p and q are integers and q â‰  0. It is a non-terminating or non-repeating decimal. i.e. for example:

1.1000120010211…..

### Question 2: Explain, how irrational numbers differ from rational numbers?

Solution:

An irrational number is a real number that cannot be expressed in the form of fractions i.e. p/q, where p and q are integers and q â‰  0 i.e it cannot be expressed as a ratio of integers. It is a non-terminating or non-repeating decimal.

For example, âˆš2 is an irrational number

A rational number is a real number that can be expressed as a fraction and as a decimal i.e. it can be expressed as a ratio of integers. It is a terminating or repeating decimal.

For examples: 0.101 and 5/4 are rational numbers

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

(i) âˆš7

(ii) âˆš4

(iii) 2 + âˆš3

(iv) âˆš3 + âˆš2

(v) âˆš3 + âˆš5

(vi) (âˆš2 â€“ 2)2

(vii) (2 â€“ âˆš2)(2 + âˆš2)

(viii) (âˆš3 + âˆš2)2

(ix) âˆš5 â€“ 2

(x) âˆš23

(xi) âˆš225

(xii) 0.3796

(xiii) 7.478478â€¦â€¦

(xiv) 1.101001000100001â€¦â€¦

Solution:

(i) âˆš7

Given: âˆš7

Since, it is not a perfect square root,

Therefore, it is an irrational number.

(ii) âˆš4

Given: âˆš4

Since, it is a perfect square root of 2.

Therefore, 2 can be expressed in the form of 2/1, thus it is a rational number.

(iii) 2 + âˆš3

Given: 2 + âˆš3

Here, 2 is a rational number, and âˆš3 i is not a perfect square thus it is an irrational number.

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

Therefore, 2 + âˆš3 is an irrational number.

(iv) âˆš3 + âˆš2

Given: âˆš3 + âˆš2

Here, âˆš3 is not a perfect square thus it is an irrational number.

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

Since, the sum of two irrational numbers is always an irrational number.

Therefore, âˆš3 + âˆš2 is an irrational number.

(v) âˆš3 + âˆš5

Given: âˆš3 + âˆš5

Here, âˆš3 is not a perfect square thus it is an irrational number

Similarly, âˆš5 is not a perfect square thus it is an irrational number.

Since, the sum of two irrational numbers is always an irrational number.

Therefore, âˆš3 + âˆš5 is an irrational number.

(vi) (âˆš2 â€“ 2)2

Given: (âˆš2 â€“ 2)2

(âˆš2 â€“ 2)2 = 2 + 4 â€“ 4 âˆš2

= 6 â€“ 4 âˆš2

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

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

Therefore, (âˆš2 â€“ 2)2 is an irrational number.

(vii) (2 â€“ âˆš2)(2 + âˆš2)

Given: (2 â€“ âˆš2)(2 + âˆš2)

(2 â€“ âˆš2)(2 + âˆš2) = ((2)2 âˆ’ (âˆš2)2) [As, (a + b)(a â€“ b) = a2 â€“ b2]

= 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

Given: (âˆš3 + âˆš2)2

(âˆš3 + âˆš2)2 = (âˆš3)2 + (âˆš2)2 + 2âˆš3 x âˆš2 [ As, (a + b)2 = a2 â€“ 2ab + b2 ]

= 3 + 2 + 2âˆš6

= 5 + 2âˆš6

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

Therefore, (âˆš3 + âˆš2)2 is an irrational number.

(ix) âˆš5 â€“ 2

Given: âˆš5 â€“ 2

Here, âˆš5 is an irrational number but 2 is a rational number.

Since, the difference between an irrational number and a rational number is an irrational number.

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

(x) âˆš23

Given: âˆš23

âˆš23 = 4.795831352331â€¦

Since, the decimal expansion of âˆš23 is non-terminating and non-recurring

Therefore, âˆš23 is an irrational number.

(xi) âˆš225

Given: âˆš225

âˆš225 = 15 or 15/1

Since, âˆš225 can be represented in the form of p/q and q â‰  0.

Therefore, âˆš225 is a rational number

(xii) 0.3796

Given: 0.3796

Since, the decimal expansion is terminating.

Therefore, 0.3796 is a rational number.

(xiii) 7.478478â€¦â€¦

Given: 7.478478â€¦â€¦

Since, the decimal expansion is a non-terminating recurring decimal.

Therefore, 7.478478â€¦â€¦ is a rational number.

(xiv) 1.101001000100001â€¦â€¦

Given: 1.101001000100001â€¦â€¦

Since, the decimal expansion is non-terminating and non-recurring.

Therefore, 1.101001000100001â€¦â€¦ is an irrational number

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

(i) âˆš4

(ii) 3âˆš18

(iii) âˆš1.44

(iv) âˆš9/27

(v) â€“ âˆš64

(vi) âˆš100

Solution:

(i) âˆš4

Given: âˆš4

Since, âˆš4 = 2 = 2/1, it can be written in the form of a/b.

Therefore, âˆš4 is a rational number.

The decimal representation of âˆš4 is 2.0

(ii) 3âˆš18

Given: 3âˆš18

3âˆš18 = 9âˆš2

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

Therefore, 3âˆš18 is an irrational number.

(iii) âˆš1.44

Given: âˆš1.44

Since, âˆš1.44 = 1.2, it is a terminating decimal.

Therefore, âˆš1.44 is a rational number.

The decimal representation of âˆš1.44 is 1.2

(iv) âˆš9/27

Given: âˆš9/27

Since, âˆš9/27 = 1/âˆš3, as the quotient of a rational and an irrational number is an irrational number.

Therefore, âˆš9/27 is an irrational number.

(v) â€“ âˆš64

Given: â€“ âˆš64

Since, â€“ âˆš64 = â€“ 8 or â€“ 8/1, as it can be written in the form of a/b.

Therefore, â€“ âˆš64 is a rational number.

The decimal representation of â€“ âˆš64 is â€“8.0

(vi) âˆš100

Given: âˆš100

Since, âˆš100 = 10 = 10/1, as it can be written in the form of a/b.

Therefore, âˆš100 is a rational number.

The decimal representation of âˆš100 is 10.0

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

(i) x2 = 5

(ii) y2 = 9

(iii) z2 = 0.04

(iv) u2 = 17/4

(v) v2 = 3

(vi) w2 = 27

(vii) t2 = 0.4

Solution:

(i) x2 = 5

Given: x2 = 5

When we take square root on both sides, we get,

x = âˆš5

Since, âˆš5 is not a perfect square root,

Therefore, x is an irrational number.

(ii) y2 = 9

Given: y2 = 9

When we take square root on both sides, we get,

y = 3

Since, 3 = 3/1, as it can be expressed in the form of a/b

Therefore, y is a rational number.

(iii) z2 = 0.04

Given: z2 = 0.04

When we take square root on both sides, we get,

z = 0.2

Since, 0.2 = 2/10, as it can be expressed in the form of a/b and is a terminating decimal.

Therefore, z is a rational number.

(iv) u2 = 17/4

Given: u2 = 17/4

When we take square root on both sides, we get,

u = âˆš17/2

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

Therefore, u is an irrational number.

(v) v2 = 3

Given: v2 = 3

When we take square root on both sides, we get,

v = âˆš3

Since, âˆš3 is not a perfect square root,

Therefore, v is an irrational number.

(vi) w2 = 27

Given: w2 = 27

When we take square root on both sides, we get,

w = 3âˆš3

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

Therefore, w is an irrational number.

(vii) t2 = 0.4

Given: t2 = 0.4

When we take square root on both sides, we get,

t = âˆš(4/10)

t = 2/âˆš10

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

Therefore, t is an irrational number.

### Question 6: Give an example of each, of two irrational numbers whose:

(i) Difference in a rational number

(ii) Difference in an irrational number

(iii) Sum in a rational number

(iv) Sum is an irrational number

(v) Product in a rational number

(vi) Product in an irrational number

(vii) Quotient in a rational number

(viii) Quotient in an irrational number

Solution:

(i) Difference in a rational number

âˆš5 is an irrational number

Since, âˆš5 – âˆš5 = 0

Here, 0 is a rational number.

(ii) Difference in an irrational number

Let the two irrational number be 5âˆš3 and âˆš3

Since, (5âˆš3) – (âˆš3) = 4âˆš3

Here, 4âˆš3 is an irrational number.

(iii) Sum in a rational number

Let the two irrational numbers be âˆš5 and -âˆš5

Since, (âˆš5) + (-âˆš5) = 0

Here, 0 is a rational number.

(iv) Sum is an irrational number

Let the two irrational numbers be 4âˆš5 and âˆš5

Since, 4âˆš5 + âˆš5 = 5âˆš5

Here, 5âˆš5 is an irrational number.

(v) Product in a rational number

Let the two irrational numbers be 2âˆš2 and âˆš2

Since, 2âˆš2 Ã— âˆš2 = 2 Ã— 2 = 4

Here, 4 is a rational number.

(vi) Product in an irrational number

Let the two irrational numbers be âˆš2 and âˆš3

Since, âˆš2 Ã— âˆš3 = âˆš6

Here, âˆš6 is an irrational number.

(vii) Quotient in a rational number

Let the two irrational numbers be 2âˆš2 and âˆš2

Since, 2âˆš2 / âˆš2 = 2

Here, 2 is a rational number.

(viii) Quotient in an irrational number

Let the two irrational numbers be 2âˆš3 and 2âˆš2

Since, 2âˆš3 / 2âˆš2 = âˆš3/âˆš2

Here, âˆš3/âˆš2 is an irrational number.

### Question 7: Give two rational numbers lying between 0.232332333233332 and 0.212112111211112.

Solution:

Let a = 0.212112111211112

Let b = 0.232332333233332

Here a<b as on the second decimal place a has digit 1 and b has digit 3.

If the second decimal place is considered as 2 then it lies between a and b.

Therefore, Let x = 0.22

and y = 0.22112211…

Thus, a < x < y < b

Hence, x and y are the rational numbers required.

### Question 8: Give two rational numbers lying between 0.515115111511115 and 0.5353353335

Solution:

Let a = 0.515115111511115

Let b = 0.5353353335

Here a<b as on the second decimal place a has digit 1 and b has digit 3.

If the second decimal place is considered as 2 then it lies between a and b.

Therefore, Let x = 0.52

and y = 0.520520…

Thus, a < x < y < b

Hence, x and y are the rational numbers required.

### Question 9: Find one irrational number between 0.2101 and 0.2222…

Solution:

Let a = 0.2101

and b = 0.2222…

Here a<b as on the second decimal place a has digit 1 and b has digit 2.

If the third decimal place is considered as 1 then it lies between a and b.

Therefore, Let x = 0.2110110011…

Thus, a < x < b

Hence, x is the irrational number required.

### Question 10: Find a rational number and also an irrational number lying between the numbers 0.3030030003… and 0.3010010001…

Solution:

Let a = 0.3010010001…

and b = 0.3030030003…

Here a<b as on the third decimal place a has digit 1 and b has digit 3.

If the third decimal place is considered as 2 then it lies between a and b.

Therefore, Let x = 0.302

and y = 0.302002000200002…

Thus, a < x < y < b

Hence, x and y are the rational and irrational numbers required respectively.

### Question 11: Find two irrational numbers between 0.5 and 0.55.

Solution:

Let a = 0.5

and b = 0.55

Here a<b as on the second decimal place a has digit 0 and b has digit 5.

If the second decimal place is considered between1 to 4 then it lies between a and b.

Therefore, Let x = 0.510510051000…

and y = 0.53053530…

Thus, a < x < y < b

Hence, x and y are the irrational numbers required.

### Question 12: Find two irrational numbers lying between 0.1 and 0.12.

Solution:

Let a = 0.1

and b = 0.12

Here a<b as on the second decimal place a has digit 0 and b has digit 2.

If the second decimal place is considered 1 then it lies between a and b.

Therefore, Let x = 0.11011011000…

and y = 0.11100010100…

Thus, a < x < y < b

Hence, x and y are the irrational numbers required.

### Question 13: Prove that âˆš3 + âˆš5 is an irrational number.

Solution:

Let âˆš3 + âˆš5 be a rational number equal to x.

Therefore, x = âˆš3 + âˆš5

x2 = (âˆš3 + âˆš5)2

x2 = (âˆš3)2 + (âˆš5)2 + 2 âˆš3 âˆš5

= 3 + 5 + 2âˆš15

= 8 + 2âˆš15

x2 – 8 = 2âˆš15

(x2 – 8)/2 = âˆš15

Here, (x2 – 8)/2 is a rational but âˆš15 is an irrational number.

Therefore, âˆš3 + âˆš5 is an irrational number.

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