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CHAPTER1

POINTS TO REMEMBER

1. Definition of a rational number.A number r is called a rational number, if

it can be written in the formp

q, wherepand qare integers and q0.

Note.We visit that q0 because division by zero is not allowed.

2. Equivalent rational numbers. Rational numbers do not have a unique

representation in the formp

q, wherepand qare integers and q0. For example,

1

2

2

4= =

3

6

4

8

5

10= = = ......, and so on, these are called equivalent rational

numbers.

3. Standard form of a rational number.A rational number r= pq

, q0 is said

to be in its standard form ifpand qare co-prime.

Note:Two integers are said to be co-prime when they have no common

factors other than 1.

4. In general, there lie infinitely many rational numbers between any two given

rational numbers.

5. Definition of an irrational number.A number s is called an irrational number,

if it cannot be written in the formp

q, wherepand qare integers and q0.

6. Collection of real numbers. All rational numbers and all irrational numbers

taken together form the collection of real numbers. It is denoted by R obviously,a real number is either rational or irrational.

7. Corresponding to every real number, there exists a unique point on the number

line. Also, corresponding to every point on the number line, there exists a

unique real number. This is why we call the number line, The real number

line.

8. Decimal expansion of a rational number. The decimal expansion of a

rational number is either terminating or non-terminating recurring. Conversely,

a number whose decimal expansion is terminating or non-terminating recurring

is rational.

NUMBER SYSTEMS

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12 GOLDEN SAMPLE PAPER (MATHEMATICS)IX

9. Decimalexpansion of an irrational number.The decimal expansion of an

irrational number is non-terminating non-recurring. Conversely, a numberwhose decimal expansion is non-terminating non-recurring is irrational.

10. Few facts

(i) The sum or difference of a rational number and an irrational number is

irrational.

(ii) The product or quotient of a non-zero rational number with an irrational

number is irrational.

(iii) If we add, subtract, multiply or divide two irrationals, the result may be

rational or irrational.

11. Radical sign.Let a> 0 be a rational number and nbe a positive integer. Then

an = bmeans bn

= aand b> 0. Here, the symbol is called the radicalsign. In particular, if ais a real number, then a = bmeans b2= aand b> 0.

12. Identities relating to square roots. Let a and b be positive real numbers,

then,

(i) ab a b= (ii)a

b

a

b=

(iii) ( ) ( )a b a b+ = a b

(iv) ( ) ( )a b a b+ = a2 b

(v) ( ) ( )a b c d + + = ac ad bc bd + + +

(vi) ( )a b+ 2 = a+ 2 ab + b

13. Laws of exponents

(i) am. an= am+ n (ii) (am)n= amn

(iii)a

a

m

n = am n, m> n (iv) am bm= (ab)m

Here, a, nand mare natural numbers. a is called the base and mand n are

called the exponents.

14. Definition.Let a> 0 be a real number. Let mand nbe integers such that mand

nhave no common factors other than 1, and n> 0. Then,

a a amn n

m mn= =d i

15. Rationalisation.To rationalise the denominator of1

a b+, we multiply this

bya b

a b

, where aand bare integers.

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NUMBER SYSTEMS 13

NCERT TEXTBOOK EXERCISES WITH SOLUTIONS

EXERCISE 1.1 (Page 5)

Example 1.Is zero a rational number ? Can you write it in the formp

q, where

p and q are integers and q 0 ?

Sol.Yes ! zero is a rational number. We can write zero in the formp

q, wherep

and qare integers and q0 as follows :

0 =0

1=

0

2=

0

3etc.,

denominator qcan also be taken as negative integer.

Example 2.Find six rational numbers between 3 and 4.

Sol.+3 4

2=

7

2

+7

32

2=

13

4

+13

34

2=

25

8

+25

38

2=

49

16

+49

316

2=

97

32

+97

332

2= 193

64.

Thus, six rational numbers between 3 and 4 are7

2,

13

4,

25

8,

49

16,

97

32and

193

64.

Aliter.We write 3 and 4 as rational numbers with denominator 6 + 1 (= 7), i.e.,

3 =3

1=

3 7

1 7=

21

7

and 4 =4

1

=4 7

1 7

=28

7.

Thus, six rational numbers between 3 and 4 are22

7,

23

7,

24

7,

25

7,

26

7and

27

7.

Note.This is known as the method of finding rational numbers in one step.

Example 3.Find five rational numbers between3

5and

4

5.

Sol.3

5=

30

50,

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4

5 =

40

50 .

Therefore, five rational numbers between3

4and

4

5are

31

50,

32

50,

33

50,

34

50,

35

50.

Example 4. State whether the following statements are true or false. Givereasons 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.

Sol.(i) True, since the collection of whole numbers contains all natural numbers.

(ii) False, for example 2 is not a whole number.

(iii) False, for example1

2is a rational number but not a whole number.


Example 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.

Sol.True, since collection of real numbers is made up of rational and irrational

numbers.

(ii) False, because no negative number can be the square root of any natural

number.

(iii) False, for example 2 is real but not irrational.

Example 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.

Sol.No. For example, 4 = 2 is a rational number.

Example 3.Show how 5 can be represented on the number line.Sol.Representation of 5 on the number line

Consider a unit square OABC and transfer it onto the number line making sure

that the vertex O coincides with zero.

Then OB = 2 21 1+ = 2

Construct BD of unit length perpendicular to OB.

Then OD = 2 2( 2) 1+ = 3

Construct DE of unit length perpendicular to OD.

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NUMBER SYSTEMS 15

Then OE =

2 2

( 3) 1+ = 4 = 2Construct EF of unit length perpendicular to OE.

Then OF = 2 22 1+ = 5

Using a compass, with centre O and radius OF, draw an arc which intersects

the number line in the point R. Then R corresponds to 5 .

3 2 1 O 1 2 3

1 1

1

5

3

21

1

4=2

C B

DE

F

R

A

4 4

Representation of 5

Example 4.Classroom activity (Constructing the square root spiral) :

Take a large sheet of paper and construct the square root spiral in the following

fashion. Start with a point O and draw a line segment OP1of unit length. Draw a line

segment P1P

2perpendicular to OP

1of unit length [see figure]. Now draw a line segment

P2P

3 perpendicular to OP

2. Then draw a line segment P

3P

4 perpendicular to OP

3.

Continuing in this manner, we can get the line segment Pn 1

Pnby drawing a line

segment of unit length perpendicular to OPn 1

. In this manner, we will have created

the points : P1, P

2, P

3, ......, P

n,......, and joined them to create a beautiful spiral depicting

2,

3,

4, ... .

1

1

P3 P

2

P1

3

2

O

Pn

Constructing square root spiral.


Example 1.Write the following in decimal form and say what kind of decimal

expansion each has :

(i)36

100(ii)

1

11

(iii) 41

8(iv)

3

13

(v)2

11(vi)

329

400.

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Sol.(i)

36

100 = 0.36

The decimal expansion is terminating.

(ii) 11) 1.000000 ( 0.090909......

99

100

99

100

99

1

1

11= 0.090909...... = 0.09

The decimal expansion is non-terminating repeating.

(iii) 41

8=

4 8 1

8

+=

32 1

8

+=

33

8

8 ) 33.000 ( 4.125

32

10

8

2016

40

40

41

8= 4.125


(iv) 13 ) 3.00000000000 ( 0.230769230769......

26

40

39

100

91

90

78

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NUMBER SYSTEMS 17

120

117

30

26

40

39

100

91

90

78

120

117

3

3

13= 0.230769230769 ...... = 0.230769 .


(v) 11 ) 2.0000 ( 0.1818......

11

90

88

20

11

90

88

2

2

11= 0.1818...... = 0.18 .


(vi) 400 ) 329.0000 ( 0.8225

3200

900

800

1000

800

2000

2000

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329

400 = 0.8225


Example 2.You know that1

7= 0.142857. Can you predict what the decimal

expansions of2

7,

3

7,

4

7,

5

7,

6

7are, without actually doing the long division ? If so,

how ?

[Hint:Study the remainders while finding the value of1

7carefully.]

Sol.Yes ! We can predict the decimal expansions of2

7

,3

7

,4

7

,5

7

,6

7

, without

actually doing the long division as follows :

2

7= 2

1

7= 2 0.142857 = 0.285714

3

7= 3

1

7= 3 0.142857 = 0.428571

4

7= 4

1

7= 4 0.142857 = 0.571428

5

7= 5

1

7= 5 0.142857 = 0.714285

67

= 6 17

= 6 0.142857 = 0.857142 .

Example 3.Express the following in the formp

q, where p and q are integers

and q 0.

(i) 0.6 (ii) 0.47

(iii) 0.001 .

Sol.(i) Let x= 0.6 = 0.6666......

Multiplying both sides by 10 (since one digit is repeating), we get

10x= 6.666......

10x= 6 + 0.6666...... 10x= 6 +x

10xx= 6 9x= 6

x=6

9

x=2

3

Thus, 0.6 =2

3

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NUMBER SYSTEMS 19

Here p= 2

q= 3 (0)

(ii) Let x= 0.47 = 0.47777......


10x= 4.7777......

10x= 4.3 + 0.47777...... 10x= 4.3 +x

10xx= 4.3 9x= 4.3

x=4.3

9=

43

90

Thus, 4.7 =43

90

Here p= 43

q= 90 (0).

(iii) Let x= 0.001 = 0.001001001......

Multiplying both sides by 1000 (since three digits are repeating), we get

1000x= 1.001001......

1000x= 1 + 0.001001001...... 1000x= 1 +x

1000xx= 1 999x= 1

x=1

999

Thus, 0.001 =1

999

Here p= 1

q= 999 (0).

Example 4.Express 0.99999...... in the formp

q. Are you surprised by your

answer ? With your teacher and classmates discuss why the answer makes sense.

Sol.Let x= 0.99999......


10x= 9.9999......

10x= 9 + 0.99999...... 10x= 9 +x 10xx= 9 9x= 9

x=9

9= 1

Thus, 0.99999...... = 1 =1

1

Here p= 1

q= 1.

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Since 0.99999...... goes on for ever, so there is no gap between 1 and 0.99999......

end hence they are equal.

Example 5.What can the maximum number of digits be in the repeating block

of digits in the decimal expansion of1

17? Perform the division to check your answer.

Sol.The maximum number of digits in the repeating block of digits in the

decimal expansion of1

17can be 16.

17 ) 1.000000000000000000000000000000

85 ( 0.05882352941176470588235294117647......

150

136

140

136

40

34

60

51

90

85

50

34

160

153

70

68

20

17

30

17

130

119

110

102

80

68

120

119

100

85

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NUMBER SYSTEMS 21

150

136

140

136

40

34

60

51

90

85

50

34

160

153

70

68

20

17

30

17

130

119

110

102

80

68

120

119

1

Thus,1

17= 0.0588235294117647

By Long Division, the number of digits in the repeating block of digits in the

decimal expansion of1

17= 16.

The answer is verified.

Example 6.Look at several examples of rational numbers in the formp

q(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 ?

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Sol.The property that qmust satisfy in order that the rational numbers in the

formp

q(q0), wherepand qare integers with no common factors other than 1, have

terminating decimal representation (expansions) is that the prime factorisation of q

has only powers of 2 or powers of 5 or both, i.e.,qmust be of the form 2m 5n; m=

0, 1, 2, 3, ......, n= 0, 1, 2, 3, ...... .

Example 7. Write three numbers whose decimal expansions and non-

terminating non-recurring.

Sol. 0.01001 0001 00001......,

0.20 2002 20003 200002......,

0.003000300003......,

Example 8. Find three different irrational numbers between the rational

numbers5

7and

9

11.

Sol. 7 ) 5.000000 ( 0.714285........

49

10

7

30

28

20

14

60

56

40

35

5

Thus,5

7= 0.714285...... = 0.714285

11 ) 9.0000 ( 0.8181......

8820

11

90

88

20

11

9

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NUMBER SYSTEMS 23

Thus,

9

11 = 0.8181...... = 0.81

Three different irrational numbers between the rational numbers5

7and

9

11can be taken as

0.75 075007500075000075......

0.7670767000767......,

0.808008000800008......,

Example 9.Classify the following numbers as rational or irrational :

(i) 23 (ii) 225

(iii) 0.3796 (iv) 7.478478......(v) 1.101001000100001......

Sol.(i) 4.795831523

4 23.00 00 00 00 00 00 00 00 00

16

87 700

609

949 9100

8541

9585 55900

47925

95908 797500

767264

959163 3023600

2877489

9591661 14611100

9591661

95916625 501943900

479583125

959166302 2236077500

1918332604

9591663043 31774489600

28774989129

2999500471

Thus, 23 = 4.795831523......

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... The decimal expansion is non-terminating non-recurring.

23 is an irrational number.

(ii)

15

1 225

1

25 125

125

... 225 = 15 = 151

225 is a rational number.

Here p= 15

and q= 1(0).

(iii) ... The decimal expansion is terminating.

0.3796 is a rational number.

(iv) 7.478478...... = 7.478

... The decimal expansion is non-terminating recurring.

7.478478...... is a rational number.

(v) 1.101001000100001......

... The decimal expansion is non-terminating non-recurring.

1.101001000100001...... is an irrational number.


Example 1.Visualise 3.765 on the number line, using successive magnification.

Sol.

Example 2.Visualize 4.26on the number line, up to 4 decimal places.

Sol. 4.26 = 4.262626......

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NUMBER SYSTEMS 25


Example 1.Classify the following numbers as rational or irrational:

(i) 2 5 (ii) (3 + 23 ) 23

(iii)2 7

7 7(iv)

1

2

(v) 2.

Sol.(i) ... 2 is a rational number and 5 is an irrational number.

2 5 is an irrational number.... The difference of a rational number and

an irrational number is irrational.

(ii) (3 + 23 ) 23 = 3 + 23 23 = 3

which is a rational number.

(iii)2 7

7 7=

2

7

which is a rational number.

(iv) ... 1(0) is a rational number and 2 (0) is an irrational number.

1

2 is an irrational number.... The quotient of a non-zero rational number

with an irrational number is irrational.

(v) ... 2 is a rational number and is an irrational number.

2is an irrational number.... The product of a non-zero rational number

with an irrational number is irrational.

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NUMBER SYSTEMS 27

(iii)

1

5 + 2 (iv)

1

7 2 .

Sol.(i)1

7=

1

7

7

7| Multiplying and dividing by 7

=7

7.

(ii)1

7 6=

1

7 6

7 6

7 6

+

+

| Multiplying and dividing by 7 + 6

= 6

7 6

7

+

= 7 + 6 .

(iii)1

5 2+=

1

5 2+

5 2

5 2

| Multiplying and dividing by 5 2

=5 2

5 2

=

5 2

3

.

(iv)1

7 2=

1

7 2

7 2

7 2

+

+

| Multiplying and dividing by 7 + 2

=7 2

7 4

+

=

7 2

3

+.


Example 1.Find:

(i) 641/2 (ii) 321/5

(iii) 1251/3.

Sol.(i) (64)1/2= (82)1/2

= 82 1/2= 81= 8.

(ii) 321/5= (25)1/5

= 25 1/5= 21= 2.

(iii) 1251/3= (53)1/3

= 53 1/3= 51= 5.

Example 2.Find:

(i) 93/2 (ii) 322/5

(iii) 163/4 (iv) 1251/3.

Sol.(i) 93/2= (91/2)3= 33= 27.

(ii) 322/5= (25)2/5= 25 2/5= 22= 4.

(iii) 163/4= (24)3/4= 24 3/4= 23= 8.

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(iv) 1251/3= (53)1/3

= 53 (1/3)= 51=1

5.

Example 3.Simplify:

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

7

3

1

3

(iii)

1/2

1/4

11

11(iv) 71/2. 81/2.

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

10 3

152

+

= 213/15.

(ii)7

3

1

3

=7

3 7

1

(3 )= 21

1

3= 321.

(iii)1/ 2

1/ 4

11

11= 111/2 1/4= 111/4.

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

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