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Number Systems

NCERT Textual Exercise (Solved)

EXERCISE 1.1

1. Is zero a rational number? Can you write it in the form pq

, where p and q are integers and q ≠ 0.

2. Find six rational numbers between 3 and 4.

3. Find five rational numbers between 35

45

and .

4. State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number. (ii) Every integer is a whole number. (iii) Every rational number is a whole number.

Test Yourself – NS 1

1. Is a negative integer a rational number? Can you write it in the form pq

, where p and q are integers and q ≠ 0.

2. Find four rational numbers between – 1 and 32 .

3. Find 10 rational numbers between −311

811

and .

4. Are the following statements true or false. Give reasons for your answers. (i) Every rational number is an integer. (ii) Every whole number is an integer. (iii) Every natural number is a rational number.

5. Find three rational numbers between − −25

15

and .

EXERCISE 1.2 1. State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number. (ii) Every point on the number line is of the form m , where m is a

natural number. (iii) Every real number is an irrational number. 2. Are the square roots of all positive integers irrational? If not, give an example

of the square root of a number that is a rational number.

15

Number Systems

NCERT Textual Exercise (Solved)

3. Show how 5 can be represented on the number line.

Test Youself – NS 2 1. Are the following statements true or false? Justify your answer. (i) Every irrational number is a real number. (ii) Every point on the number line is of the form n , where n is a

perfect square. (iii) Every real number is a rational number. (iv) Every rational number is a real number. 2. Are the square roots of all positive integers rational? If no, give an example

of the square root of a number that is an irrational number. 3. Represent 13 using the fact that 3² + 2² = 13. 4. Represent − −3 5and on the same number line. 5. Which of the following square roots are irrational? (i) 100 (ii) 14

(iii) 72 (iv) 1

144 (v)

507867

EXERCISE 1.3 1. Write the following in decimal form and say what kind of decimal expansion

each has:

(i) 36

100 (ii)

111

(iii) 4 18

(iv) 3

13

(v) 2

11 (vi) 329400

2. You know that 17

0142857= . . Can you predict what the decimal expansions

of 27

37

47

57

67

, , , , are, without actually doing the long division? If so, how?

16

Number Systems

NCERT Textual Exercise (Solved)

3. Express the following in the form pq

, where p and q are integers and q ≠ 0.

(i) 0 6. (ii) 0 47. (iii) 0 001.

4. Express 0.99999 ..... in the form pq

. Are you surprised by your answer?

With your teacher and classmates discuss why the answer makes sense. 5. What can the maximum number of digits be in the repeating block of digits

in the decimal expansion of 1

17? Perform the division to check your answer.

6. Look at several examples of rational numbers in the form pq

(q ≠ 0),

where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

7. Write three numbers whose decimal expansions are non-terminating non - recurring.

8. Find three different irrational numbers between the rational numbers 57

911

and .

9. Classify the following numbers as rational or irrational :

(i) 23 (ii) 225 (iii) 0.3796 (iv) 7.478478..... (v) 1.101001000100001....

Test Yourself – NS 3 1. Write the following in decimal form and say what kind of decimal expansion

each has:

(i) 42

100 (ii) 17

(iii) 2

13 (iv) 3 38

(v) 327500

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Number Systems

NCERT Textual Exercise (Solved)

2. Express the following in the form pq

, where p and q are integers and q ≠ 0.

(i) 0 35. (ii) 0 585. (iii) 5 2. (iv) 15 712. (v) 23 43. 3. What is the maximum number of digits in the repeating block of digits in

the quotient while computing?

(i) 3326 (ii)

1117

4. Which of the following rational numbers can be represented only as non-terminating repeating decimals?

(i) 1

12 (ii) 3635

(iii) 771640 (iv)

427

(v) 3

125 5. Find three different irrational numbers between 3

11 and

67

.

6. Classify the following numbers as rational or irrational.

(i) 17 (ii) 144 (iii) 0.4896 (iv) 6.4848..... (v) 2.2020020002.......... 7. Write five numbers whose decimal expansions are non-terminating, non-

recurring. 8. Find two irrational numbers between 2 and 2.5. 9. Find two irrational numbers between 2 3and .

10. Find two irrational numbers between 17

27

and .

EXERCISE 1.4 1. Visualise 3.765 on the number line using successive magnification. 2. Visualise 4.26 on the number line up to 4 decimal places.

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Number Systems

NCERT Textual Exercise (Solved)

Test Yourself – NS 4 Visualise the following on the number line by using successive magnification. (1) 5.778 (2) 6.23 (3) 3.47

EXERCISE 1.5 1. Classify the following numbers as rational or irrational : (i) 2 5− (ii) ( )3 23 23+ −

(iii) 2 77 7

(iv) 12

(v) 2p 2. Simplify each of the following expressions : (i) ( ) ( )3 3 2 2+ + (ii) ( ) ( )3 3 3 3+ −

(iii) ( )5 2 2+ (iv) ( ) ( )5 2 5 2− + 3. Recall, p is defined as the ratio of the circumference (say c) of a circle to its

diameter (say d). That is, p = cd

. This seems to contradict the fact that p is irrational. How will you resolve this contradiction?

4. Represent 9 3. on the number line. 5. Rationalise the denominators of the following :

(i) 17

(ii) 1

7 6−

(iii) 1

5 2+ (iv)

17 2−

Test Yourself – NS 5 1. Classify the following numbers as rational or irrational.

(i) 2 3+ (ii) ( )2 3 3− +

(iii) 2 59 5

(iv) 13

(v) 125

(vi) 27

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Number Systems

NCERT Textual Exercise (Solved)

(vii) 3 8 (viii) 4 + 2p

(ix) 6 36− (x) 644

2. Rationalize the denominator of the following:

(i) 112 (ii)

18 3−

(iii) 1

7 2+ (iv) 1

5 3−

(v) 5

6 5− (vi)

34 3

(vii) 7

5 5 (viii)

68 27×

(ix) 3 13 1−+

(x) 1

3 2 1− −

3. Simplify:

(i) ( ) ( )2 3 2 3+ − (ii) ( ) ( )4 2 4 2+ −

(iii) ( )3 5 2+ (iv) ( ) ( )7 3 7 3− + (v) ( ) ( )5 7 2 7+ + (vi) ( ) ( )5 5 5 5+ −

(vii) ( ) ( )2 5 5 7+ − (viii) ( )7 2 2−

(ix) ( ) ( )13 8 13 8− + (x) ( ) ( )5 3 3 5− −

4. Represent the following numbers on a number line. (i) 2 6. (ii) 8 7.

EXERCISE 1.6 1. Find: (i) 641/2 (ii) 321/5

(iii) 1251/3

2. Find: (i) 93/2 (ii) 322/5

(iii) 163/4 (iv) 125–1/3

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Number Systems

NCERT Textual Exercise (Solved)

3. Simplify:

(i) 22/3 . 21/5 (ii) 133

7

(iii) 11

11

12

14

(iv) 71/2 . 81/2

Test Yourself – NS 6 1. Find the following: (i) (81)1/2 (ii) (216)1/3 (iii) (625)1/4

2. Find the following: (i) 2432/5 (ii) (512)–1/9

(iii) 343–1/3

3. Find the following: (i) 32/3 . 31/5 (ii) (21/2)3

(iii) 13

13

14

12

(iv) 42/3 . 82/3

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NCERT Exercises and Assignments

Exercise 1.1 1. Yes. Zero is a rational number. Zero can be written in any of the following forms:

01

02

03

07

, , ,− − etc.

Thus, 0 can be written as pq

where p = 0 and q in any non - zero integer.

Hence, 0 is a rational number. 2. Since we require 6 rational numbers between 3 and 4, so we write

31

= 31

× 77

= 217

and41

= 41

× 77

= 287

Also 21 < 22 < 23 < 24 < 25 < 26 < 27 < 28

217

< 227

< 237

< 247

< 257

< 267

< 277

< 287

Hence, six rational numbers between 3 and 4 are 227

, 237

, 247

, 257

, 267

and 277

3. Since we require 5 rational numbers between 35

45

and , so we write

35

= 35

× 66

= 1830

and 45

= 45

× 66

= 2430

Also 18 < 19 < 20 < 21 < 22 < 23 < 24

Hence, 5 rational numbers between 35

and 45

are 1930

, 2030

, 2130

2230

and 2330

,

4. (i) TRUE: Every natural number lies in the collection of whole numbers. (ii) FALSE: –5 is not a whole number.

(iii) FALSE: 79

is not a whole number.

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25

Test Yourself – NS 1

1. Yes. E.g. –2 can be written as −

21

of form pq

2. −38

14

916

78

, , ,

3. − −211

111

011

111

211

311

411

511

611

711

, , , , , , , , ,

4. (i) False (ii) True (iii) True

5. − − −720

620

520

, ,

Exercise 1.2 1. (i) TRUE, A real number is either rational or irrational.

(ii) FALSE, Numbers of other type like − − −5 4 12

78

, , , also lie on the number line. (iii) FALSE, Rational numbers are also real numbers. 2. No. Square roots of all positive integers are not irrational. e.g., 4, 9, 16, 25 ..... etc. are positive

integers but their square roots are rational numbers, i.e. 4 2 9 3 16 4 25 5= = = = ……, , , ,

3. ( )5 2 12 2 2= + We construct right angled DOAB, right angled at A such that OA = 2 units and AB = 1 unit. \ By Pythagoras Theorem,

OB OA AB= + = + =2 2 2 22 1 5

Now, cut off a length OC = OB = 5 on the number line. \ Point C represents the irrational number 5 .

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Test Yourself – NS 2 1. (i) True (ii) False (iii) False (iv) True 2. No. 2 is irrational 5. (ii) and (iii)

Exercise 1.3

1. (i) 36

1000 36= . , terminating decimal

(ii) 111

\ 111

0 090909 0 09= …=. . , non-terminating and repeating

(iii) 4 18

338

33 1258 125

41251000

4 125= =××

= = . , terminating decimal

(iv) 3

13

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27

1

\ 313

0 2307692307 0 230769= …=. . , non-terminating and repeating

(v) 211

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28

\ 2

110 181818 018= ……=. . , non-terminating and repeating

(vi) 329400

329 4100

=÷

=82 25100

.

= 0.8225, terminating decimal 2. Yes. It can be done as follows:

27

2 17

2 0142857 0 285714= × = × =. .

37

3 17

3 0142857 0 428571= × = × =. .

47

4 17

4 0142857 0 571428= × = × =. .

57

5 17

5 0142857 0 714285= × = × =. .

67

5 17

6 0142857 0 857142= × = × =. .

3. (i) Let x = = …0 6 0 666. . ...(1) Multiplying both sides by 10, we get 10x = 6.666.... ...(2) Subtracting (1) from (2), we get 10x – x = (6.666....) – (0.666...) 9x = 6

\ x =69

\ x =23

\ 0.6 = 23

(ii) Let x = = …0 47 0 477. . ...(i) Multiplying both sides of eq. (i) by 10, we get

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29

10x = 4.777... ...(ii) Multiplying both sides of eq. (ii) by 10, we get 100x = 47.777... ...(iii) Subtracting (ii) from (iii), we get 100x – 10x = (47.777...) – (4.777....) 90x = 43

\ x =4390

\ 0 47 4390

. =

Shortcut method: Let x = 0 47. Multiplying both sides by 10, we get 10 4 7x = .

= 4 79

+

10 36 79

x = +

x =

4390

\ 0 47 4390

. =

(iii) Let x = = …0 001 0 001001001. . ...(1) Multiplying both sides by 1000, we get 1000x = 1.001001001.... ...(2) Subtracting (1) from (2), we get 1000x – x = (1.001001001....) – (0.001001001....) 999x = 1

\ x =1

999

\ 0 001 1999

. =

4. Let x = 0.9999 ............. = 0 9. ...(1) Multiplying both sides by 10, we get 10x = 9.9999........ ...(2) Subtracting (1) from (2), we get 9x = (9.9999.......) – (0.9999........)

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9x = 9 x = 1

\ 0 9999 1. …=

Yes, at a glance we are surprised at our answer. But the answer makes sense when we observe that 0.9999.... goes on forever. So there is no gap between 1 and 0.9999....... and hence they are equal.

5.

\ 17

0 0588235294117647= .

\ The maximum number of digits in the quotient while computing 117

16are .

6. Rational number whose denominators when multiplied by a suitable integer produces a power of 10 which can be expressed in the finite decimal form. But this can always be done only when the denominator of the given rational number has either 2 or 5 or both of them as the only prime factors.

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31

E.g. 12

1 52 5

510

0 5=××

= = . 38

3 1258 125

3751000

0 375=××

= = .

1320

13 520 5

65100

0 65=××

= = . and so on

Thus, we obtain the following property: If the denominator of a rational number in its standard form has no prime factors other

than 2 or 5, then and only then it can be represented as a terminating decimal. 7. Three numbers whose decimal representations are non-terminating and non-repeating are

2 3 5, , , ..... etc. OR 0.1010010001 ............., –1.2323423452....... and 0.4040040004 8.

\ 57

0 714285= .

911

0 81= .

Thus, three different irrational numbers between 57

and 911

are 0.727207200.....,

0.7676676667........ and 0.8080080008....... 9. (i) 23 is an irrational number as 23 is not a perfect square. (ii) 225 15 15 15= × = which is rational. \ 225 is a rational number. (iii) 0.3796 is a rational number as it is terminating decimal. (iv) 7.478478 ....... is non-terminating but repeating. So, it is a rational number. (v) 1.1010010001..... is non-terminating and non - repeating. So, it is an irrational number.

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Test Yourself – NS 3 1. (i) 0.42, terminating (ii) 0142857. , non-terminating and repeating (iii) 0153846. , non-terminating and repeating (iv) 3.375, terminating (v) 0.654, terminating

2. (i) 3599 (ii)

585999

(iii) 479 (iv)

5185330

(v) 232099

3. (i) 1 26922307 8. , digits

(ii) 0 647588235294117 15. , digits 4. (i), (ii) & (iv)

5. 311

0 27 67

0 877142= =. ; . ; Any 3 answers

6. (i) irrational (ii) rational (iii) rational (iv) rational (v) irrational 7. 2.1010010001...... and 2.2020020002....., etc. 9. 2 1 4142 3 1 7320= … = …. , . , 1.505005..... and 1.6161161116....... etc.

10. 17

0142857 27

0 285714= =. .and OR 0.15015001500015...... and 0.2020020002.......

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Exercise – 1.4 1.

We will proceed by successive magnification process: 3.765 lies between 3 and 4. [i.e. in the interval (3, 4)]. Divide interval [3, 4] into 10 equal parts and look at (3.7, 3.8) through a magnifying glass. [see

fig.(i)] Now divide (3.7, 3.8) into 10 equal parts and look at (3.76, 3.77) through the magnifying glass.

[see fig. (ii)] Further divide [3.76, 3.77] into 10 equal parts and look at [3.765, 3.766] through magnifying

glass that 3.765 lies in the interval [3.76, 3.77] [see fig. (iii)]. 2. We will proceed by successive magnification process: 4.2626 lies between 4 and 5 i.e. in the interval [4, 5]. Divide [4, 5] into 10 equal parts and look at [4.2, 4.3] through magnification glass [see fig. (i)]. Now divide [4.2, 4.3] into 10 equal parts and look at [4.26, 4.27] through magnification glass

[see fig. (ii)]. Further divide [4.26, 4.27] into 10 equal parts. Look at [4.262, 4.263] through magnification

glass [see fig. (iii)]. Finally divide [4.262, 4.263] into 10 equal parts and look at [4.262, 4.263] through magnification

glass that 4.2626 lie in the interval [4.262, 4.263] [see fig. (iv)].

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Exercise – 1.5 1. (i) 2 5− is an irrational number being a difference between a rational and an irrational. (ii) ( )3 23 23 3 23 23 3+ − = + − = , which is a rational number.

(iii) 2 77 7

27

= , which is a rational number.

(iv) 12 is irrational being the quotient of a rational and an irrational.

(v) 2p is irrational being the product of rational and irrational.

2. (i) ( ) ( ) ( ) ( )3 3 2 2 3 2 2 3 2 2+ + = + + + = 6 + 3 2 + 2 3 + 6 (ii) ( ) ( ) ( ) ( )3 3 3 3 3 3 9 32 2+ − = − = − = 6 (iii) ( ) ( ) ( )( ) ( )5 2 5 2 5 2 2 5 2 10 22 2 2+ = + + = + +

= 7 + 2 10

(iv) ( ) ( ) ( ) ( )5 2 5 2 5 22 2− + = − = 3 3. There is no contradiction as either c or d are irrational and hence p is an irrational number. 4. Mark the distance 9.3 units from a fixed point A on a given line to obtain a point B such that AB

= 9.3 units. From B, mark a distance of 1 unit and mark the new point as C. Find the midpoint of AC and mark that point as O. Draw a semicircle with centre O and radius OC. Draw a line

Number Systems

35

perpendicular to AC passing through B and intersecting the semicircle at D. Then BD = 9 3. . To represent 9 3. on the number line. Let us treat the line BC as the number line with B as

Zero, C as 1 and so on. Draw an arc with centre B and radius BD, which intersects the number line in E. Then E represents 9 3. .

5. (i) 17

17

77

= × = 77

(ii) 1

7 6 1

7 6 × 7 + 6

7 + 67 + 67 6−

=−

=( ) − ( )

=−

7 + 6

7 62 2

= 7 6+

(iii) 15 2

15 2

× 5 25 2

5 25 2+

=+

−−

=−

( ) − ( )=

−−

5 2

5 22 2

= 5 23−

(iv) 1

7 21

7 2× 7 2

7 27 2

7 4−=

−++

=+

( ) − ( )=

+−

7 2

7 22 2 = 7 2

3+

Test Yourself – NS 5 1. (i) Irrational (ii) Rational (iii) Rational (iv) Irrational (v) Rational (vi) Irrational (vii)Irrational (viii) Irrational (ix) Rational (x) Irrational

2. (i) 36

(ii) 8 35+

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36

(iii) 7 25− (iv) − +( )5 3

4

(v) 5 6 5( )+ (vi) 3

4

(vii) 35

25 (viii) 6

6

(ix) 2 3− (x) − + +( )2 6 24

3. (i) 1 (ii) 14 (iii) 8 2 15+ (iv) 4 (v) 17 7 7+ (vi) 20 (vii) 10 2 7 5 5 35− + − (viii) 9 2 14− (ix) 5 (x) 15 5 5 3 3 15− − +

Exercise – 1.6 1. (i) 641/2 = (82)1/2 (ii) 321/5 = (25)1/5

= 82×1/2 = 8 = 25×1/5 = 2 (iii) 1251/3 = (53)1/3 = 53×1/3 = 5 2. (i) 93/2 = (32)3/2 (ii) 322/5 = (25)2/5

= 32×3/2 = 25×2/5

= 3³ = 27 = 2² = 4 (iii) 163/4 = (24)3/4 (iv) 125–1/3 = (53)–1/3

= 24×3/4 = 53×–1/3

= 2³ = 8 = 5 – 1 = 1/5

3. (i) 22/3 . 21/5 = 22/3 + 1/5 (ii) 13

133

7 7

3 7

=

( )( )

= 210 3

15+

= 21315 =

133 7× =

3211

(iii) 1111

111 2

1 4

12

14

/

/ ( )=−

= 111/4 (iv) 71/2 . 81/2 = (7 × 8)1/2 = 561/2

Test Yourself – NS 6 1. (i) 9 (ii) 6 (ii) 5