MathsModuleVol I

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Distance Education Programme Sarva Shiksha Abhiyan

(DEP-SSA)

(An IGNOU-MHRD, Govt. of India Project)

Maidan Garhi, New Delhi 110 068

Teaching of Mathematics at Upper Primary

Level

Volume I

3

2

AC

Dividend = Divisor Quotient +Remainder

43210 3 2 1 4

5 > 4 > 3 > 2 > 1 or 5 < 4 < 3 < 2 < 1

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4 km/hr. YX6 km/hr.

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bXuw

Teaching of Mathematics atUpper Primary Level

Volume I

Distance Education Programme Sarva Shiksha Abhiyan(DEP-SSA)

(An IGNOU-MHRD, Govt. of India project)Maidan Garhi, New Delhi 110 068


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Distance Education Programme- Sarva Shiksha Abhiyan, IGNOU, 2009

Printed at: Laxmi Printindia, 556, G.T. Road, Shahdra, Delhi-110 032.

ALL RIGHTS RESERVED

No part of this publication may be reproduced, stored in a retrieval system or

transmitted in any form or by any other means electronics, mechanical,

photocopying, recording or otherwise without the prior permission from the publishers.

This book is an unpriced publication and shall not be sold, hired out or otherwise

disposed of without the publishers consent, in any form of binding or cover other than

that in which it is published.


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Teaching of Mathematics atUpper Primary Level

Volume I

Guidance

Prof. K.R. Srivathsan Dr. S.S. JenaPro-Vice Chancellor, IGNOU Former Project DirectorIn-charge, Director DEP-SSA DEP-SSA, IGNOU

Academic Coordination

Dr. Sarat Kumar Rout Dr. Pradeep Kumar Programme Officer Former Programme Officer

DEP-SSA, IGNOU DEP-SSA, IGNOU

Distance Education Programme Sarva Shiksha Abhiyan(DEP-SSA)

(An IGNOU-MHRD, Govt. of India project)Maidan Garhi, New Delhi 110 068


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Expert CommitteeProf. Provin Sinclair

PVC, IGNOU

Dr. S. S. Jena

Former Project

Director, DEP-SSA

Prof. M.L.Koul

Former In-charge

Project Director,

DEP-SSA

Prof. P.R Ramanujam

Director, STRIDE,

IGNOU

Dr. Mohit Mohan

MohantyAddl. Director (Retd.)

OPEPA, Orissa

Shri Madan Mohan

MohantyDeputy Director

(Retd.), BSE, Orissa

Ms. Jai Chandiram

Media AdvisorDEP-SSA, IGNOU

Ms. Avantika Dam

Asst. Teacher,CIE Basic School,

University of Delhi

Unit WriterDr. Vijay S. Patel

Lecturer, SCERT,

Gujurat

Shri Thimmaraju,

Lecturer, DIET,

Gadog, Karnataka

Shri Manoj Kumar

Shukla, Lecturer,

SCERT, Uttarakhand

Shri C.P. Mantri

Udaipur

Rajasthan

Shri B.B.P Gupta

lecturer, SCERT,

M.P

Shri Sachindananda

Mishra

Lecturer, DIET,

Cuttack, Orissa

Shri Tapas Kr. Nayak

Lecturer, SCERT,

Orissa

Shri P. S. Rawat

Lecturer, SCERT,

Haryana

Ms. Pankaj Lohani

SISE, Allahabad

Uttar Pradesh

Shri Avtar Singh

DIET Fatehgarh

Punjab

Shri Ashok Kr. Sharma

DIET Fatehgarh

Punjab

Shri Sanjay Kr. Gupta

SCERT Solan

Himachal Pradesh

Dr. C. Saroja

DIET Chennai

Tamil Nadu

Dr. S. Suresh Babu

SCERT

Andhra Pradesh

Editorial TeamContent Editing Language Editing Unit Design, Format

Editing, Course

Coordination

Proof Reading

Dr. Mohit Mohan

Mohanty, Addl.

Director (Retd.),

OPEPA, Orissa

Prof. C.B. Sharma

School of Education,

IGNOU

Dr. Sarat Kumar Rout

Programme Officer

DEP-SSA, IGNOU

Dr. Sarat Kumar Rout

Programme Officer

DEP-SSA, IGNOU

Shri Madan Mohan

Mohanty

Deputy Director

(Retd.), BSE, Orissa

Dr. Eisha Kannadi

Sr. Lecturer, School of

Education, IGNOU

Graphic Designer Cover Page EditingMr. S.S. Chauhan SOS,

IGNOU

Mr. Mitrarun Haldar

M/s Pink Chilli

Communications, Dwarka, New

Delhi-110078

Mrs. Kashish Thakkar

Computer Programmer,

DEP-SSA, IGNOU

Secretarial SupportAll Support Staff

DEP-SSA, IGNOU

ProductionSh. Deepak Israni

AFO, DEP-SSA


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Teaching of Mathematics at Upper Primary Level

Volume I

Block1 Page No.NUMBER SYSTEM

UNIT 1

Numbers and Numerals 1

UNIT 2

Number Line and Operations on Numbers 49

UNIT 3

Data and Its Graphical Representation 69

Block2MATHEMATICS IN DAILY LIFE

UNIT 4

Percentage and Its Applications 99

UNIT 5

Simple and Compound Interest 121

UNIT 6

Ratio and Proportion 145

UNIT 7

Time and Distance 165


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ACKNOWLEDGEMENTS

Sarva Shiksha Abhiyan (SSA) is a flagship programme of Govement of India to providequality elementary education to all. The Distance Education Programme (DEP) is a

national component, created by the MHRD, Govt. of India on July 1, 2003 covering all

the States and UTs. Indira Gandhi National Open University (IGNOU) has been

entrusted with the responsibility as a national apex institution for open and distance

education to implement distance education activities across the country to meet the

educational and training needs of the states at elementary level. The DEP-SSA at

national level aims at the capacity building of functionaries such as Master Trainers,

Coordinators of BRCs/CRCs, faculty of DIETS and SCERTs and evolve a sustainable

training system for elementary school teachers through Open and Distance Learning

(ODL) mode.

The National Council of Educational Research and Training (NCERT) has came out

with National Curriculum Frame Work (NCF)-2005 which emphasizes on constructivist

pedagogy for transaction of learning experiences. Subsequently, NCERT has revised the

text books adopting the principles of constructivist pedagogy. Further this new

pedagogy demands that teacher in the class room should display the role of facilitator

instead of playing the role of knowledge distributor which is also very typical and

complex. In this context, the teacher must know how to adopt the new paradigm for

effective transaction of learning experiences in the classrooms? Secondly, learning

outcome is an important indicator of teaching-learning process and quality education at

all level. Hence, learning achievement of students are not up to the level of expectation

particularly in Language (English), Mathematics, Science at national level. Therefore,

Government of India and various state governments are working on the proposition of

Learning Enhancement Programme (LEP) focusing on Language (English),

Mathematics, and Science through SSA. It is quite essential to improve the teaching

competencies of the teachers in Mathematics at elementary level with appropriate

interventional strategies. The present module has made a small effort in this direction.

The specific objectives of this module are:

To improve teaching skills of teachers in Mathematics at elementary level.

To help the teachers to follow the principles of constructivist pedagogy for effective

transaction of mathematical concepts, facts and principles in the class room by

To enhance the professional competencies of teachers in developing interest and

curiosity of children towards learning Mathematics at elementary level.

I hope that at the end of training programmes, this module would enable the teachers to

absorb necessary skills and competencies for better transaction of teaching learning

experiences in Mathematics at upper primary level.


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I would like to express my gratitude to Department of School Education Literacy,

MHRD, Govt. of India for sponsoring DEP-SSA to improve the professional

competency of functionaries associated with SSA.

I take this opportunity to express my thankfulness and gratitude to Prof. V.N.

Rajasekharan Pillai, Vice Chancellor, IGNOU & Chairman, Advisory Committee, DEP-

SSA for his constant support, encouragement and able guidance throughout the year to

carry out project activities for accomplishment of its goals and objectives.

I also take this opportunity to express my gratitude to the unit writers and the experts

involved in preparation of this module. My sincere gratitude goes to Dr. Mohit Mohan

Mohanty Ex-Reader, SCERT, Orissa and Sh. Madan Mohan Mohanty for their hard

labour in designing, developing and editing the module. I express my heartful thanks

Prof. C.B. Sharma and Dr. Eisha Kannadi, School of Education, IGNOU for language

editing of the present document.

I am also especially thankful to my colleagues of DEP-SSA for their coordination in the

development of this training module and my thanks are also due to all the support staff

working in this project towards the completion of this assignment.

I look forward to receive constructive suggestion for the improvement of this training

module.

December 2009 Project Director

DEP-SSA, IGNOU


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Teaching of Mathematics at

Upper Primary LevelVolume I

Block 1 Number System

Unit 1 Numbers and Numerals

Unit 2 Number Line and Operations on Numbers

Unit 3 Data and Its Graphical Representation

Block 2 Mathematics In Daily Life

Unit 4 Percentage and Its Applications

Unit 5 Simple and Compound Interest

Unit 6 Ratio and Proportion

Unit 7 Time and Distance

Volume II

Block 3 Introduction to Algebra

Unit 8 Algebraic Expression and Operations

Unit 9 Factorization

Unit 10 Algebraic Equations

Block 4 Geometrical Shapes and Figures

Unit 11 Introduction to Geometrical Figures and Shapes

Unit 12 Construction of Geometrical Figures

Unit 13 Perimeter, Area and Volume

Unit 14 Symmetry


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ABOUT THE VOLUME

The main focus of the module is to equip the teachers with required skills andcompetencies to apply principles of constructivist pedagogy while transacting learning

experience in Mathematics with the students. The first and foremost step towards the

achievement of this objective is to acquaint the teachers and resource persons with

transactional strategies of mathematical concepts and principles in conformity to

principles of constructivist pedagogy. Secondly, this module is intended to empower the

teachers to be free from the tyranny of traditional approach of teaching mathematics in

which abstract concepts are usually presented to the students in an authoritarian way and

to adopt activity approach, play way method and creating social situation which stresses

the presentation of concrete experiences. This volume comprises two blocks 1 and 2. The

briefs of each unit have been presented below:

In Unit 1 we define number and numerals and discuss how Roman Numerals and Hindu-

Arabic Numerals are written, and discuss about natural number system and its properties

and operations followed by elaboration of regular fractions and decimal fractions. We

define whole numbers, rational numbers, integers and their operations.

Unit 2 explains how abstract number can be taught to the young students by associating

with concrete objects available in their surrounding. The unit begins with number line

and representation of numbers on it. Subsequently unit elaborates how different number

system can be represented on number line and ordering of the numbers. The unit

concludes with addition and subtraction of whole numbers and integers.

Unit 3 discusses about the concept and importance of data, methods of their

presentations both in tabular and graphical forms and basic descriptive statistics like

measures of central tendency i.e. mean, median and mode have been discussed.

Unit 4 In this unit an attempt has been made to clarify the basics of understanding and

calculating percentages along with its application in several areas including calculating

profit and loss. The unit open up with the brief explanation of explain the concept of

percentage and followed by conversion fractions and decimals into percentage. We ended

the unit by focusing on how to solve daily life problems by using percentage.

Unit 5 defines concepts of simple & compound interest and subsequently their

application have been discussed to enable the teachers to deal effectively in the

classroom transactions.

Unit 6 discusses the different methods of comparison of two quatities to develop the

concept of ratio in the minds of the children. Further unit defines the concept of ratio and

compare two ratios. The unit concludes with the description of proportion as equality of

two ratios and apply the concepts of ratio and proportion in real life-situations.


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Unit 7 explains the concept of speed and calculate any one of time, distance or speed of a

journey when other two are given. The second part of the unit explain the concept of

relative speed of two moving bodies; and calculate the time required for two movingbodies having considerable length.


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Teaching of Mathematics at

Upper Primary Level

Volume I

Block1 Page No.NUMBER SYSTEM

UNIT 1

Numbers and Numerals 1

UNIT 2

Number Line and Operations on Numbers 49

UNIT 3

Data and Its Graphical Representation 69


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UNIT 1 NUMBERS AND NUMERALS

Structure

1.1 Introduction

1.2 Objectives

1.3 Numbers and Numerals

1.3.1 Counting Numbers and Natural Numbers

1.3.2 Natural Numbers: Properties and their Operations

1.3.3 Concept of Zero and the Place Value System

1.3.4 Whole Numbers: Properties and their Operations

1.3.5 Fractions and Decimal1.3.6 Integers: Properties and their Operations

1.3.7 Rational Numbers

1.3.8 Rational Numbers: Their Properties and Operations

1.3.9 Decimal Equivalences of Rational Numbers

1.4 Unit Summary

1.5 Glossary

1.6 Answers to Check Your Progress

1.7 Assignments

1.8 References

1.1 INTRODUCTION

Let us ask the children, Which scientific inventions have been extremely beneficial?

Answers may include petrol engines, cell phones, TV etc. Then we may ask Which of

the inventions could be possible without the involvement of numbers? "None of them

could be possible", would be the answer.

In ancient times, people felt the necessity of counting their belongings, such as the trees

planted, animals reared and such other things. They used pebbles, sticks, lines drawn onthe wall etc. to count their belongings by using the concept of one-to-one

correspondence for counting.

People had different collections of pebbles for representing different objects they had.

The number of pebbles in different collections might have confused them. Hence, they

thought of evolving some symbols representing quantities in different collections. Thus

perhaps the numerals have evolved.

In counting collection of objects, one has to use numbers signifying the quantity in the

collection like-six cows, ten pebbles, five fingers, fifteen trees. In these examples six,


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ten, five and fifteen are names of the numbers signifying the quantity of objects in

different collections. While the quantity in a collection remains unchanged, the number

names may be different in different languages and societies for the same quantity.

For writing numbers, we use words or number names which had its own difficulties,

particularly when there are large numbers to be written. Therefore, symbols

representing numbers were evolved for writing numbers. These symbols representing

number names are called numerals. 1, 5, 9, 13, and 15 are examples of numerals

representing numbers one, five, nine, thirteen and fifteen respectively. Like number

names, the numerals representing same numbers were different in different cultures.

Civilizations developed simultaneously in different parts of the world. Everywhere the

problem of counting must have been felt. They must have solved the problems by

developing numerals. Some examples of numeral used in different civilizations aregiven below.

(The symbols of numerals of Babylonian and Mayan civilizations are to be given

below)

i) In Babylonian Civilization :

1 2 3 10 100 1000

ii) In Maya Civilization :

1 2 3 4 5 6 7 8 9 10

iii) In Roman Civilization :

1 5 10 50 100 500 1000

I V X L C D M

As the objects to be counted became large, large number of numerals were necessary. It

was difficult to remember them. Further, it was difficult to go for operations like

addition, subtraction etc. with those numerals.

This problem was solved by Indian mathematicians by inventing zero and developing

the place value system. These two inventions have helped us in writing as big a numberas we need using only ten numerals, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 called digits.

The important fact about the place value system is that the location or position of a

numeral in the number expresses its value. This system of writing numbers developed in

India reached the Western land through the Arabians. Hence, the above numerals are

named as Hindu-Arabic numerals. Today we are proud of our ancestors.

1.2 OBJECTIVES


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After studying this unit, you will be able to:

differentiate counting numbers and natural numbers;

explain the operations on natural numbers and their properties;

explain regular fractions and decimal fractions; appreciate the utility of zero;

know the whole numbers and the properties of operations on them;

know the integers and the operations on them; and

define rational numbers and apply various operations on them.

1.3 NUMBERS AND NUMERALS

We have already discussed about the difference between numbers and numerals. In

brief, the symbols used for writing numbers are numerals.

As we have seen that the Hindu-Arabic system of numerals used only ten symbols or

digits for writing all possible numbers by using the concepts of place value which was

nearly absent in other systems. Thus, while same digits are used in 10, 100, 1000 they

differed in the place value of 1.

The place value system followed in writing the numerals we use is as follows:

Ten Thousands

104

Thousands

103

Hundreds

102

Tens

101

Ones

100

Because of this, the Hindu-Arabic numerals are known as ten-base numerals and the

numbers are said to be in the decimal system (deci in Latin means ten). You can try to

write numbers using Roman Numerals and Hindu-Arabic Numerals and can easily

realize how the latter makes it convenient to write any number.

The numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0 are now known as the digits when placed inthe places shown above to build a number.

Thus, in the number 29, 2 and 9 are the digits that occupy the tens place and the units

place respectively. Such as:

2 assumes the value 2 tens i.e. 20

and 9 assumes the value 9 ones i.e. 9

Thus, a number having several digits can be expanded as shown below.


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375 = 3 hundreds + 7 tens + 5 ones

= 300 + 70 + 5

5039 = 5 thousands + zero hundreds + 3 tens + 9 ones

= 5000 + 0 + 30 + 9

= 5000 + 30 + 9

1.3.1 Counting Numbers and Natural Numbers

Numbers help us to count objects. The numbers which we use for counting are 1, 2, 3

Numbers created for counting did not solve all problems of daily life relating to

calculation unless operations like addition, subtraction, multiplication and division wereassociated with the numbers. Counting numbers with the association of the above

operations on them give us the natural number system.

The Natural numbers are based upon the following principles:

i) The smallest natural number is 1.

ii) Each natural number has a successor which is 1 more than the other.

iii) Except 1, each natural number has a predecessor which is 1 less than the other.

The second and the third principles help us in developing the process of addition and

subtraction.

Next comes the operation of multiplication. Multiplication is nothing but repeated

addition of the same number. For instances, such as 5 + 5 + 5 is represented as 5 3.

Therefore, 5 3 = 5 + 5 + 5 = 15.

Continuous subtraction is division. Let us consider the following example.

Example 1: There are 9 children in a group and the teacher wants to form smaller

groups with 2 children in a group. The teacher sends away two of them to form the first

group.

After sending 2 children, the number of remaining children is 7. Again she sent 2

children to form the second group.

After this, the number of children left is 5. Likewise she sent 2 children to form the

third group and another two children to form the fourth group. Now she finds that she

cant get two more children to form the fifth group, as only one child is left.

Thus, it is found that 2 children could be taken away four times from a group of 9

children and one child remained at the end.


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This lengthy operation could be made short by the process of division.

Thus, 9 2 = 7

7 2 = 5

5 2 = 3

3 2 = 1

This long operation is written in a short form as given below:

In this case of division

9 is the Dividend,

2 is the Divisor,

4 is the Quotient, and

1 is the Remainder.

The relation connecting them is:

1.3.2 Natural Numbers: Properties and their Operations

A. Addition

Example 2: To add 5 and 4, we write:

5 + 4 = 5 + 1 + 1 + 1 + 1

= (5 + 1) + 1 + 1 + 1

= 6 + 1 + 1 + 1 (Since, successor of 5 = 5 + 1 = 6)

= (6 + 1) + 1 + 1

= 7 + 1 + 1 (Since, successor of 6 = 6 + 1 = 7)

= (7 + 1) + 1

= 8 + 1 (Since, successor of 7 = 7 + 1 = 8)

= 9 (Since, successor of 8 = 8 + 1 = 9)

Thus, to add 4 to 5, we stretch open our four fingers and go on counting 6, 7,

8, 9 and get the result 9.

9 2 = 4 (Quotient) + 1 (Remainder)

Dividend = Divisor Quotient +Remainder


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Properties of Addition

Let us assign each student in the class to add two natural numbers. The

numbers given for addition to different child may be different.

As they find the result, we may ask them to observe: What kind of numbers

they got as a result? They will be unanimous in saying that the result is a

natural number.

Thus we arrive at the conclusion that the sum of two natural numbers is a

natural number.

This property is called closure property.

A child can be asked to state any two natural numbers as she liked. Twochildren may be asked to add the numbers in the reverse order.

As an example, 23 and 39 are separately added in two different orders

i.e. 23 + 39, 39 + 23.

Likewise, several pairs of children may be asked to add different pair of

natural numbers. One child may be asked to add in a particular order while

the other in the pair may be asked to add the two numbers in the reverse

order of the first child.

Each pair of children may now be asked to compare the results they got. Itwill be found that every pair of children say that the results of both of them

was equal.

This will lead the children to conclude that the sum of any two natural

numbers added in the direct or reverse orders will get the same result.

i.e., the natural numbers can be added in any order.

Thus we say addition is commutative.

Let us go for another activity. Consider the numbers 3, 2 and 5.

Adding 3, 2 and 5 in different ways as: 3 + (2 + 5) and (3 + 2) + 5, we get 10

as the answer in both the cases. This shows that addition of natural numbers

is associative. Thus we see that adding any two of them first and adding the

third number with the result gives the same result. Let us try the other

combination also (3 + 5) + 2 = 8 + 2 = 10. Again the same result is obtained.

This property is known as associative property of addition.

B. Subtraction

Example 3: To subtract 3 from 7


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7 3 = 7 1 1 1

= (7 1) 1 1

= 6 1 1 (since, the predecessor of 7 is 6)

= (6 1) 1

= 5 1 (since, the predecessor of 6 is 5)

= 4 (since, the predecessor of 5 is 4)

Thus, the operation of subtraction has been done using the principle of

predecessors in natural numbers.

Properties of Subtraction

Let us now consider some cases of subtractions in natural numbers.

i) 5 3 = 2; The answer is a natural number.

ii) 2 1 = 1; The answer is a natural number.

iii) 1 4 = ; Not defined in natural numbers.

Hence, natural numbers are not closed under subtraction.

Recalling the commutativity of the operation of addition, we can say that

subtraction is not commutative. We may check this property with any two

numbers, say 5 & 7.

5 7 7 5

Hence, subtraction is not commutative in natural numbers.

Thirdly, we can also say that subtraction is neither associative nor

distributive.

C. Multiplication

Repeated addition of a number is multiplication.

Consider the addition: 3 + 3 + 3 + 3

We briefly express it as 4 3

Therefore, 4 3 = 3 + 3 + 3 + 3 = 12

Properties of Multiplication

Consider 6 7 = 42

We see that the product of natural numbers 6 and 7 is 42 and it is also a

natural number.


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This is known as closure property of multiplication.

Commutative property

We see that

Thus we observe: 5 3 = 3 5.

Hence, multiplication is commutative.

1 5 = 5

3 1 = 3

When we multiply any number with 1, we get the same number as

the product.

Hence, we say that 1 is the multiplicative identity.

Consider Natural numbers 5, 9 and 7. Observe the two orders of multiplying

the three numbers:

i) (5 9) 7 = 45 7 = 315

ii) 5 (9 7) = 5 63 = 315

Thus, we see that:

(5 9) 7 = 5 (9 7)

3 + 3 + 3 + 3 + 3 = 15

i.e. 5 3 = 15

5 + 5 + 5 + = 15

i.e. 3 5 = 15

Check Your Progress

Notes : a) Space is given below for your answer.

b) Compare your answer with the one given at the end of thisunit.

E1. Is there an additive identity in natural numbers? Give reasons for your

answer.

................................................................................................................

................................................................................................................

................................................................................................................


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The grouping of numbers differently does not affect the product of natural

numbers. Thus we say, multiplication is associative.

Let us consider the price of articles in a book shop. The price board shows:

1 work book costs Rs. 8.00.

1 pencil costs Rs. 3.00.

A child wants to buy 4 work books and 4 pencils. What should be the total

cost?

He calculates as follows:

Price of 4 work books = Rs. 8 4 = Rs. 32

Price of 4 pencils = Rs. 3 4 = Rs. 12Total Price = Rs. 32 + Rs. 12 = Rs. 44.

The shopkeeper calculates as follows:

Cost of 1 book and 1 pencil = Rs. 8 + Rs. 3 = Rs. 11.

Cost of 4 sets of 1 book & 1 pencil = Rs. 11 4 = Rs. 44

Both the answers tally with each other.

Thus, we find:

4 8 + 4 8 = 32 + 12 = 444 (8 + 3) = 4 11 = 44

That is,

4 (8+3) = 48 + 43

Hence, we say that multiplication distributes over addition.

D. Division

The operation has already been discussed as continued subtraction of a

certain number from another.

The rule that follows in division is as we have seen in earlier example:

It can be seen that

division in many cases leaves a remainder. Such as 14 4 = quotient 3 and

remainder 2. We say 14 is not divisible by 4.

So we say that division is not closed on natural numbers.

Dividend = Divisor Quotient + Remainder


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It is very clear that 8 4 does not give the same result as 4 8. There we see

that Division is not Commutative on Natural Numbers.

Let us check if:

(32 8) 2 and 32 (8 2) give us the same result.

(32 8) 2 = 4 2 = 2

32 (8 2) = 32 4 = 8

Thus we find that (32 8) 2 = 32 (8 2).

So we say division is not associative on natural numbers.

Let us recall the properties of different operations on natural numbers and put

them in the table below:

Properties

Closure Commutativity AssociativityExistence

of Identity

Addition Yes Yes Yes No

Subtraction No No No No

Multiplication Yes Yes Yes Yes

Division No No No No

Distributive property does not find place in the table as it relates to multiplication and

addition jointly.

1.3.3 Concept of Zero and the Place Value System

For subtractions like 2 2, 5 5, 12 12 etc. no result could be assigned so long as we

remain confined to natural number. We could only say that the result is Nothing.

It was felt necessary that a numeral should be developed to represent nothing. Thus the

Indians conceived zero (0). Counting numbers helped the primitive people to count

things but they had no means to measure a part of an object; that necessity forced them

to generate half (1/2) , quarter (1/4) and so on so that parts of an object can also be

measured.

1.3.4 Whole Numbers: Properties and their Operations

The collection of numbers, now available i.e. 0, 1, 2, 3, 4, are whole numbers and is

denoted by W.


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The only additional member which W has over N is zero (0).

Properties of Operations on Whole Numbers

A. Addition

The operation of addition is the same as discussed in natural numbers. Hence

all the properties of addition in N, also the same in case of W. Thus:

Addition on W is closed.

Addition on W is commutative.

Addition on W is associative.

Because of the existence of 0 in W, we can see that:

5 + 0 = 5

12 + 0 = 12

347 + 0 = 347

Hence, 0 is the additive identity in W.

B. Subtraction

The operation of subtraction in W is similar to the operation on N. But in

N we could only subtract a smaller number from a bigger one.

But now even a number can be subtracted from itself.

Thus 3 3 = 0

5 5 = 0

9 9 = 0

Hence in W, we can subtract a number from a number equal to or greater than

the first.

C. Multiplication

The multiplication operation in W is similar to that in N. The properties of

operations in W are also the same as in N.

A special case of multiplication can be seen in W which is:

3 0 = 0

5 0 = 0

9 0 = 0

0 0 = 0


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2

1

2

1

Figure 1.1

3 parts of 5 equal parts; 2 parts out of 5 equal parts as follows in Figure 1.2.

5

3

5

2

Figure 1.2

A fractions provides us with 2 information:

i) The whole is divided into how many equal parts; it is known as Denominator.

ii) How many parts are taken, it is known as Numerator.

Similarly, if the denominator is 7 and the numerator is 5, then the fraction is written as

7

5.

A. Various Kinds of Fractions

I) Unit Fractions:2

1,

3

1,

4

1,

7

1etc. are the fraction each of which has 1 as the

numerator.

II) Equivalent Fractions: The paper sheet AB is divided into

2 equal parts of a whole in Figure 1.3,

Figure 1.3


Figure 1.4


Figure 1.5

A P B

A P B

5

1

5

1

5

1

5

1

5

1

A P B


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AP represents2

1in Figure 1.3.

AP represents4

2in Figure 1.4 (2 parts out of 4 equal parts).

AP represents8

4in Figure 1.5 (4 parts out of 8 equal parts).

Thus,2

1=

4

2=

8

4are known as equivalent fractions.

If

b

ais a given fraction, the equivalent fraction of it can be obtained by

multiplying the numerator and denominator of both by the same number.

Thus,b

a= ...

3

3

2

2=

=

b

a

b

a

It is evident from above that infinite number of equivalent fractions are

available for any given fraction.

B. Categorization of Fractions

I. Proper Fraction and Improper Fraction

Numerator and denominator of a fraction are the two constituents of

a fraction.

Depending upon the relative sizes of them, a fraction can be put into

2 categories.

i) If numerator < denominator, then the fraction is known as proper

fraction.

32 ,

74 ,

169 are proper fractions.

ii) If numerator > denominator, then the fraction is known as an

improper fraction.4

5,

7

11,

3

16are improper fractions. What does

the improper fraction4

5mean?


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Does it mean 5 part taken out of 4 equal parts of a whole? It cannot be. So

then what is it?

Activity for you:

Take 2 square sheets of card board of equal sizes as in Figures 1.6.

Figure 1.6

Cut each of them along their diagonals. You get 8 triangular pieces as

shown in below mentioned Figure 1.7.

Figure 1.7

Each is a quarter (4

1) of the original square-sheet.

Now put 4 of them together to give rise to the original square shape. You will

now have the shapes as shown in Figure 1.8. Thus you get 1 whole and a quarter (41 ).

Figure 1.8

Thus we find that4

5= 1

4

1. 1

4

1is known as a mixed fraction (or a mixed

number).

Conversion of an improper fraction into a mixed number

4

5: 5 4 = quotient 1 and remainder 1. Thus

4

5= 1

4

1

Think of an activity to demonstrate this concept of conversion.

Alternative method of conversion

As shown in the diagram along with3

1

3

1

3

1i.e. 1 =

3

3


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1 =3

3, 1 =

4

4

Similarly 1 =5

5, 1 =

6

6, 1 =

7

7etc.

4

1

4

1

4

1

4

1i.e. 1 =

4

4

Meaning of the above statements are as follows:

1 = 3 numbers of one thirds together.

1 = 5 numbers of one fifths together and so on.

Similarly,

2 = 2 1 = 2 3 numbers of one thirds together,

= 6 numbers of one third together.

Thus 13

3= 1 +

3

1=

3

3+

3

1=

3

13+=

3

4

15

3= 1 +

5

3=

5

5+

5

3=

5

35+=

5

8

Again3

8=

3

233 ++=

3

3+

3

3+

3

2= 1+1+

3

2= 2 +

3

2= 2

3

2

II. Like Fraction and Unlike Fraction

If a bread is cut into 12 equal slices and we take:

1 slice, then it is12

1of the bread,

5 slices, then it is12

5of the bread,

7 slices, then it is12

7of the bread.

The 3 fractions that we see above are12

1,

12

5,

12

7and those have equal

denominators. Such fractions which have equal denominators are known aslike fractions. Why like?

12

1of the bread is 1 slice of it.

12

5of the bread consists of 5 slices of it.


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12

7of the bread consists of 7 slices of it.

Thus, unit contained in the fraction i.e. the unit fractions contained in them

are all alike. Therefore, the term like fractions having unequal denominators are

unlike fractions.

7

2,

5

3,

11

6are unlike fractions.

III. Fraction of a Collection of Objects

Consider a collection of 12 breads. The whole here is the entire collection.

The collection is divided into 2 equal parts as shown in Figure 1.9.

Figure 1.9

Each part is of a half i.e.2

1of the whole collection. It can be seen that

2

1of 12

= 6.

i)

2

1of 12 i.e. 1 part out of 2 equal parts of 12 = 12 2 = 6.

Similarly:

ii)3

1of 12 i.e. 12 3 = 4.

iii)4

1of 12 i.e. 12 4 = 3.

iv) What is5

2of a collection of 15?

5

2of a collection of 15 means 2 parts out of 5 equal parts of 15.

We divide the collection of 15 into 5 equal parts. Each part (is5

1of the

collection) = 15 5 = 3.

Hence, 2 such parts = 2 3 = 6.

Thus, we see that:


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5

2of 15 {2 (

5

1of 15)}= 2 (15 5) = 2 3 = 6.

Check Your Progress


b) Compare your answer with the one given at the end of this

unit.

E2. Design a diagram to represent the fraction5

1.

................................................................................................................

................................................................................................................

................................................................................................................

................................................................................................................


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E3. Write 3 unit fractions.

................................................................................................................

................................................................................................................

................................................................................................................

E4. Write a fraction equivalent to3

2(i) with numerator 12 (ii) with

denominator 15.

................................................................................................................

................................................................................................................

................................................................................................................

E5. Write 3 like fractions.

................................................................................................................

................................................................................................................

................................................................................................................

E6. Determine7

3of 28.

................................................................................................................

................................................................................................................

................................................................................................................

E7. Express as mixed numbers:3

8,

4

17,

8

21.

................................................................................................................

................................................................................................................

................................................................................................................

E8. Express as improper fractions: 23

2, 4

5

3.

................................................................................................................

................................................................................................................

................................................................................................................


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IV. Decimal Fractions

The fractions which have been discussed so far are known as vulgar fractions.

Some of the vulgar fractions have their denominators as 10, 100, 1000 (i.e.

101, 10

2, 10

3,)

Such fractions are:10

3,

100

23, 1

10

7etc.

This kind of fractions are known as decimal fractions.

The decimal fractions are also written in a different pattern by following the

place value system.

To accommodate decimal fractions in the place value system, the places usedin writing numerals are extended to the right as shown below:

Thousands

1000

Hundreds

100

Tens

10

Ones

1*

Tenths

10

1

Hundredth

100

1

In this ten-base system, every place carries the value equal to one-tenth of value

of the place to the left of it.

Hence, a place developed to the right of the ones place has the value 10

1

and

known as tenths place.

The place to the right of the tenths place carries the value equal to100

1and

known as the hundredths place and so it continues to the right.

The diagram showing the places and their values has a gap between the ones

place and the tenths place (indicated by a star * mark). A dot ( . ) is marked there toseparate the places carrying fractional values.

Writing decimal fractions in the form of ten-base numerals

Thus, 310

1is 3 +

10

1and is written as 3.1.

10

2is written as 0.2 (0

makes it distinct that there is nothing in the units place).

100

53=

100

350+=

100

50+

100

3=

10

5+

100

3= 5 tenths and 3 hundreds =

0.53


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2100

37= 2 +

100

37= 2 +

100

30+

100

7= 2+

10

3+

100

7= 2 + 3 tenths + 7

hundredths = 2.37

Writing ten-base numerals with decimal point in the form of decimal

fractions

2.4 = 2 ones + 4 tenths = 2 +10

4= 2

10

4[2

5

2on further simplification]

3.57 = 3 ones + 5 tenths + 7 hundreds

= 3 +10

5+

100

7= 3 +

100

50+

100

7= 3 +

100

57

= 310057

C. Operations on Fractions

I. Reducing a Fraction into a Fraction of Lowest Order

A series of equivalent fractions are written below:

3

2=

6

4=

9

6=

12

8=

15

10

It can be seen that the numerator and denominator of each of the fractionsexcept the first one have a common factor whereas the numerator and

denominator of the first fraction has no common factor. This is the reason

why we say the first fraction as the fraction of lowest order.

Let us take each of the other fractions.

6

4=

23

22

=

3

2we say numerator and denominator are cancelled

log 2.

96 =

3332

=

32 , thus

32 is fraction of lowest order of

96 and

64 .

II. Changing Fractions into Like Fractions

5

3and

7

4to be written as like fractions : let us write some equivalent

fractions for each of the above fractions.

5

3=

10

6=

15

9=

20

12=

25

15=

30

18=

35

21=

40

24=

45

27


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7

4=

14

8=

21

12=

28

16=

35

20

The moment we got the fraction35

20equivalent to

7

4we stopped, because we

found it denominator to be the same as the denominator of one of fractions equivalent

to5

3(if

35

21).

Now we pick up those two fractions from 2 series of equivalent fractions

which have the same denominator.

Thus

5

3=

35

21,

And7

4=

35

20.

Now5

3and

7

4both have been converted into two like fractions.

But the process followed is quite long.

A short-cut process is as follows :

5

3and

7

4are the given fractions.

L.C.M. of their denominators 5 and 7 is 5 7 = 35.

35 5 = 7 (L.C.M. is divided by the first denominator)

5

3=

75

73

=

35

21.

And 35 7 = 5 (L.C.M. is divided by the second denominator),

7

4= 57

54

= 35

20.

Why is this exercise done?

This process of changing fractions into like fractions helps us in addition and

subtraction of fraction.

III. Addition of Fractions


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3

2+

5

4= ?

The two fractions are changed into like fractions.

L.C.M. of the denominators 3 and 5 is 3 5 = 15

153 = 5 (LCM divided by the first denominator)

3

2=

53

52

=

15

10,

15 5 = 3 (LCM divided by the second denominator).

5

4

= 35

34

= 15

12

.

Now,3

2+

5

4

=15

10+

15

12=

15

1210 +

=15

22=

15

715+

= 15

15+ 15

7= 1 + 15

7

= 115

7

A short process is as follows:

The process discussed above develops the understanding. To be brief, we will

omit some of the steps shown above.

3

2+

5

4=

15

)515(4)315(2 +

=15

3452 +

=15

1210 +

=15

22


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= 115

7

Gradually we make a habit of doing the first step mentally. For instance,

We divide the LCM by the first denominator mentally and multiply the result

obtained to the first numerator mentally too. Thus we get (153) 2 = 5 2 = 10.

Similarly, we divide the LCM by the second denominator and multiply the

result with the second denominator and get (15 5) 4 = 3 4 = 12.

Then the second step is directly given. Thus we write:

3

2+

5

4=

15

1210 +=

15

22= 1

5

7

But initially the child should be made to work in the expanded form for better

understanding of the process.

IV. Subtraction

The process is exactly the same as addition with the difference that minus sign

() is taken in place of the plus sign (+).

V. Multiplication

We want to multiply 3

2

by 7

5

.

Let us first try to give a physical representation of3

2

7

5

Two rectangles of the same size are shown in Figure 1.10 (i&ii).

Figure 1.10(i) Figure 1.10(ii)

Each is divided into 3 rows and 7 columns.

(

7

5

32


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In the first rectangle7

5is taken which includes 5columns out of 7 columns of

the rectangles.

The7

5of the rectangle 10 (i) contains 3 rows, taking

3

2of 3 rows it comes to

over 2 rows.

Thus, two rows out of the 5 columns are shaded in rectangle 10 (ii).

Thus, we see3

2of

7

5of the rectangle (containing 21 units) = 10 units.

The part containing 10 units out of 21 units of the rectangle =21

10.

3

2

7

5=

73

52

.

10 is nothing but equal to 2 5.

And 21 is nothing but equal to 37.

3

2

7

5=

73

52

.

This product of 2 fractions =atorsmindeno2theofproduct

numerators2theofproduct

VI. Division

We describe4

1as 1 part out of 4 equal parts of a whole (object). This

otherwise means 1 divided by 4 gives the result4

1.

So we write 1 4 =41 .

During the discussion of multiplication we have seen that

1 4

1=

4

1.

Thus we see that: 1 4 = 1 4

1.

Let us see another example:

(II)


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3

2 4 =?

Rectangle shown in Figure 1.11(i) is divided into 3 rows and 4 columns.

3

2of rectangle shown in Figure 1.11(i)) includes 2 parts out of 3 equal parts

of it = 2 rows out of 3 of its rows.

Thus,3

2of rectangle in Figure 1.11(i) is shown in Figure 1.11(ii).

Rectangle shown in Figure 1.11(ii) is divided by 4.

The result is 1 rows as rectangle in Figure 1.11(ii) has 4 columns.

Thus,3

2of the original rectangle 4,

= 2 small unit out of 12 small units of rectangle in Figure 1.11(i)

=12

2of rectangle (I).

Thus,3

2 4 =

12

2

12

2=

3

2

4

1(according to the rule of multiplication).

3

2 4 =

3

2

4

1

Figure 1.11(ii)

Figure 1.11(i)


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Observing the 2 examples discussed above, we see that:

1 4 = 1 4

1= 1 (reciprocal of 4).

3

2 4 =

3

2

4

1=

3

2 (reciprocal of 4).

Thus we can say that:

This rule can also be extended to fractional divisors.

Note: (i) If mixed numbers are there for addition, subtraction, multiplication ordivision, those are first to be changed into improper fractions and then the

operation is to be undertaken, and (ii) in a fraction division, no remainder

is left.

Activities may be designed by the teacher in accordance with the discussions

made earlier during multiplication and division in whole numbers.

Dividend Divisor = DividendDivisor

1

Check Your Progress


b) Compare your answer with the one given at the end of thisunit.

E9. Change the mixed numbers into improper fractions:

(i) 25

3(ii)

9

74

................................................................................................................

................................................................................................................


40/219

D. Operations on Decimal Numbers

I. Addition and Subtraction

While adding or subtracting whole numbers, we add the digits contained in

the respective places.

E10. Change the improper fractions into mixed numbers:

(i)5

23(ii)

25

107

................................................................................................................

................................................................................................................

E11. Find the sum of5

3, 1

3

2and 2

2

1.

................................................................................................................

................................................................................................................

E12. Subtract158 from 1

32 .

................................................................................................................

................................................................................................................ E13. Express the decimal fractions into decimal numbers (10 base

numbers):

i) 10

23

ii) 210

7

iii) 3100

9

iv) 21000

41

................................................................................................................

................................................................................................................

................................................................................................................

................................................................................................................ Can you make a general conclusion about the number of decimal

digits that you get on changing decimal fraction into decimal

numbers?


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To add 357, 36 and 128, we write one below the other in such a way

that the digits in ones place of all the 3 numbers remain one below the

others and similarly for other places, so write:

H T O

3 5 7

3 6

1 2 8

Similarly during subtraction we write:

H T O

5 9 4

1 5 7

While adding or subtracting decimal numbers same principle is followed while

writing them one below the other. As such the decimal points i.e. one below the

others. Thus to add 12.8 and 7.25 we write:

12.80

+ 7.25

20.05

Zero is taken at the tenth place so that both the numbers have digits up to the

tenth place.

To subtract 3.02 from 5.7 we write

5.70

3.02

2.68

Note:

i) Since digit at the tenth place of a number has to be added or subtracted

from the tenth digit of the other, therefore, a zero was taken at the tenthplace of the first number in both the examples above.

ii) One zero or several zeros following the last digit after the decimal point in a

decimal number makes no change in value in the number. That if 2.3 = 2.30

= 2.300 and so on try to reason out.

iii) We add or subtract the decimal numbers in the same way as we do with

whole numbers.

After writing them one below the other in the proper manner (decimal

points remaining one below the other).


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II. Multiplication

We know that: 10

2

10

3

= 10

6

Thus 0.2 0.3 = 0.6 (as we see above).

Let us analyze the result obtained.

The result contains 6 (excluding the decimal point).

But 6 = 2 3.

There are 2 digits after the decimal point in the result (i.e. 0 at the tenth place

and 6 at the hundredths place.

We had 1 digit after the decimal point in the first number taken for

multiplication i.e. 0.2 and also 1 digit after the decimal point in the second

number i.e. 0.3.

So we observe:

Number of digits after the decimal point in the first number + number of digits

after the decimal point in the second number = number of digits after the decimal

point in the product.

So the process of multiplication of decimal numbers includes the followingsteps:

Step I: Ignore the decimal points in the numbers to be multiplied and

multiply the resulting numbers and write the result.

Step II: Add the number of digits after the decimal point in both the numbers that

are multiplied. Whatsoever is the result, the number of digits after the

decimal print in the product will be equal to that.

Example 4: 1.2 0.3.

12 3 = 36 (multiplying the numbers ignoring the decimal points).

There is only one digit after the decimal point in each of the two digits. Thus,

the total number of digits after the decimal point in the two numbers is 2.

In the product, the decimal point is placed after two digits from the right of 36

i.e. 0.36.

Hence, 1.2 0.3 = .36 or 0.36.

Some special multiplications:


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We know that:10

3 10 = 3

=> 0.3 10 = 3

Similarly,100

27 10 =

10

27= 2.7

=> 0.27 10 = 2.7

Thus, we conclude that:

When a decimal number is multiplied by 10, units digit changes to tens digit,

tenths digit changes to units digit.

In other words, the decimal point is shifted to the right by one place.

Like-wise: 3.542 100 = 354.2.

While a decimal number is multiplied by 100, the decimal point is shifted to the

right by 2 places.

III. Division

How would we divide 12.56 by 4? The process of division by a whole

number is very much similar to the process of division of whole numbers. The

only difference is that in a whole number division, a remainder may be left. But adecimal number division, no remainder will be left. The division may end giving an

exact quotient.

At times the division never comes to an end. In such cases we continue to any

number of decimal digits and get an appropriate result correct to a desired number of

decimal places.

Example 5: 2.56 to be divided by 4.

Step I: Left most digit of the dividend cannot be divided by 4. So we

take 2 digits from the left i.e. 12 and divide it by 4. We get the

quotient as 3.

Step II: 43 = 12 and 12 is subtracted from 13 and remainder left is 0.

Step III:Next digit with the decimal point is brought down. Thus we get

0.5 as dividend for the second division.

Step IV: As the decimal digit is to be divided by 4, so decimal digit will be

obtained as the quotient. A decimal point is taken in the quotient.


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Further, 5 being divided by 4, the quotient is 1. Thus 1 is taken at the

tenths.

Place of the quotient:

Divisor 4 is multiplied by the quotient 1 taken at this phase and

product 4 is subtracted from 5 leaving the remainder 1. Therefore 6

is brought down from the hundredth place of the dividend.

Step V: Now 16 is to be divided by 4. Actually 16 hundredths are to be

divided by 4 which gives 4 hundredths as the quotient

Note: In practice after the decimal point is marked in the quotient, we ignore the

decimal point during the subsequent work.

Thus 12.56 4 = 3.14

Example 6: Let us divide 23.39 by 7. We proceed in the same manner as in the

previous example.

Note: One zero (or more than one zeros) being placed after a decimal then, there

is no change in its value. Hence, we can consider there to be as many zerosafter 9 in the dividend 23.39. Hence, whenever we need we can bring a

zero down.

Though we have continued with the division till 6 digits after the decimal point

in the quotient, yet the division has not come to an end.

In such a case we take an approximate result correct to some places of

decimal. In this case correct to 2 places of decimal, the approximate answer will

be 3.34 Correct to 3 places of decimal, the approximate answer will be 3.341.

3.14

_______

4 12.56

12

______

0.5

0.4

______0.16

0.16

______

0


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Now let us take an example of decimal number as divisor.

Example 7: Let us divide 48.52 by 0.6.

The decimal divisor should be changed into a whole number

48.5 0.6 i.e.6.0

52.48

To make the denominator decimal free, it is necessary to multiply by 10.

But to keep the value of the fraction unchanged, we multiply the numerator

and the denominator both by 10.

So we get106.0

1052.48 =

62.485 .

Now it can be see that the division would never come to an end and 6 will

continue to be obtained in all the successive places.

The quotient correct to 2 places of decimal = 80.87

If the first digit to be deleted is 5 or more, then the previous digit is increased by

1.


46/219

Check Your Progress



unit.

E14. Complete the following table:

FOR WHOLE NUMBERS

Operation PropertiesClosure Commutativity Associativity Distributivity Identity

Addition

Subtraction

Multiplication

Division

................................................................................................................

................................................................................................................

................................................................................................................

E15. Find the product of 2.37 31.4.

................................................................................................................

................................................................................................................

................................................................................................................ E16. (i) Divide 302.48 by 8.

(ii) Divide 457.35 by 0.7 and get the result correct to 2 decimal

figures.

................................................................................................................

................................................................................................................

................................................................................................................ E17. Simplify 2.3 + 1.2 0.4 1.7.

................................................................................................................

................................................................................................................

................................................................................................................


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1.3.6 Integers: Properties and their Operations

Integers

In our day to day life situations, we often come across the need to write

numbers to represent the measures having opposite characters. Some examples are

given below:

i) Height above the ground, depth below the ground.

ii) Going forward, coming backward.

iii) Ascending, descending.

iv) Gain in a business, loss in the business.

In respect of the situation (i) Ground level is between height and depth.

In respect of (ii) the Standing place lies between forward movement and

backward movement.

In respect of (iii) it is the ground level which lies between the height to be

ascended and the depth to be descended.

In respect of (iv) neither gain nor loss i.e. getting back the capital lies between the

gain and the loss.

To express the measures of two opposite characters we make use of opposite

signs like + and with numbers such as +1 and 1 which are numbers with

opposite sign.

If +1 represents a height of 1m above the ground, 1 would represent a depth

of 1m below the ground.

Similarly other numbers with opposite signs are:

+ 2 and 2

E18. Divide 35

2by 1

4

3.

................................................................................................................

................................................................................................................

................................................................................................................


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+ 3 and 3 and so on.

Thus every whole number other than 0 has an opposite number.

+ 1, + 2, + 3, + 4 etc. are named as positive numbers and 1, 2, 3 etc. are

named as negative numbers.

Zero (0) is neither +ve nor ve.

Ordering of the Positive, Negative Numbers and 0

We know about the ordering of all the whole numbers. What about the ordering of

negative numbers?

Consider three businessmen Haris, Rohit and Manjit starting a business with

say Rs. 10,000.00 each. After a month both Haris and Rohit incurred loss in theirbusiness with a loss of Rs. 500.00 and Rs. 100.00 respectively. But Manjit neither

gained nor lost any thing. Then, at that point of time who had better financial

status?

It is evident that Haris with a loss of 500 Rupees has a worse financial status than

Rohit whose loss is Rs.100.

So we say, 500 < 100

Similarly, a person having no loss and no gain has a better financial position in

the business than the one who has made a loss of Rs. 100.00, i.e. Manjit had betterfinancial position after one month of business than either Rohit or Haris.

So we say, 0 > 100 > 500

Hence the ordering of +ve numbers, ve numbers and 0 is as follows:

5, 4, 3, 2, 1, 0, +1, +2, +3, +4, +5, ..

i.e. .


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(+2) + (2) = 0

(+200) + (200) = 0 and so on.

Thus, sum of 2 opposite numbers is zero.

Integers and their Operations

i) Addition: +5 + 3 = + 8

which has already been discussed in natural numbers.

Consider (+5) + (3)

(+5) + (3) = (+1) + (+1) + (+1) + (+1) +(+1) + (1) + (1) + (1)

= + 1 + 1 + 1 + 1 + 1 1 1 1

= (+11) + (+11) + (+11) + (+1) + (+1)

= 0 + 0 + 0 + 1 + 1

= 0 + 2

= 2

Alternatively,

(+5) + (3) = (+3+2) + (3)

= (+3) + (+2) + (3)

= (+3) + (3) + (+2)

= 0 + 2

= 2

ii) Subtraction

Subtraction of an integer is the same as addition of its opposite number.

Consider the examples below:

Example 8: 5 (+9)

(+5) (+9) = (+5) (9) = (+5) + (5) + [(5) + ( 4)]

= [(+5) (5)] + ( 4)

= (+5 5) 4

= 0 4 = 4

Here the subtraction of +ve number is treated an addition of the negative of the

number.

Hence we get,


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(+5) (+9) = (+5) + (9)

= 5 9 = 4

Example 9: (+5) (3) = ?

+5 (3) is to be defined as + 5 + (+3), because subtraction of a negative

number is the addition of its opposite number.

Therefore, we get, +5 (3) = +5 + 3 = +8.

iii) Multiplication

Consider

Example 10:

a) (+3) (+5)

Already done in natural numbers and whole numbers.

Therefore, (+3) (+5) = + 15 (positive number)

b) 3 (5)

3 (5) = (5) + (5) + (5)

= (10) + (5)

= 15 (The product is a negative number).

c) (3) (+5)

We know that:

(+3) (5) = 15

(+2) (5) = 10 =15+5 = 15 (5)

[5 is subtracted from the previous product].

(+1) (5) = 5 = 10 + 5 = 10 (5)

0 (5) = 5 (5) = 5 + 5 = 0

(1) (5) = 0 (5) = 0+5 = +5

(2) (5) = (+5) = + 5 + 5 = +10

(3) (5) = (+10) (5) = +10 + 5 = +15

Two important results we have got


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i) 0 (an integer) = 0.

ii) (ve integer) (ve integer) = +ve integer.

iv) Division on integers

a) We know that

15 3 = 5

b) Consider (8) (2)

(8) (2) = 8 + 2 = 6

6 (2) = 6 + 2 = 4

4 (2) = 4 + 2 = 2

2 (2) = 2 + 2 = 0

Therefore, (8) (2) = +4

Hence, we can say:

Division of a negative integer by another negative integer gives

a positive integer as quotient.

c) Consider

15 3 = 5 (done in whole numbers)

We know that division is the reverse process of multiplication.

From 3 5 = 15, we get 15 3 = 5 & 15 5 = 3

Similarly, 8 2 = 16 gives 16 2 = 8 and 16 8 = 2

For each multiplication statement of numbers, there are two

division statements.

Let us take (+2) (6) = 12.

The corresponding division statements are

i) (12) (+2) = 6, and

ii) (12) (6) = +2.

We also know that (2) (6) = + 12

(+12) (2) = 6.

And (+12) (2) = 6.

Sign rule of division


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Now we can say that when we divide a ve integer by a +ve integer or a

+ve integer by a ve integer, we get the ve quotient.

When a ve integer is divided by a ve integer, the quotient we

get is +ve integer.

Properties of Operations on Integers

Properties of Addition and Subtraction Operations:

i) Closure property

Let us observe and complete the following table:

Statements Observations

i) 12 + 8 = 20

ii) 5 + 3 = 2

iii) +7 4 = + 3

iv) 3 9 = 6

v) 7 + 0 = 7

Result is an integer

-do-

-do-

-do-

-do-

Thus we see that: the sum of two integers is always an integer.

Therefore, we say: Integers are closed under addition.

Statements Observations

i) 12 8 = + 4

ii) 5 3 = 8

iii) +7 (4) = 11

iv) 1 (5) = 4

v) 3 9 = 6vi) 7 0 = 7

vii) 0 2 = 2

Result in an integer

-do-

-do-

-do-

-do--do-

-do-

On seeing these results, we can say that integers are closed under

subtraction.

ii) Commutative property

Consider,


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5 + (6) = 1 and

(6) + 5 = 1

Therefore, 5 + (6) = 6 + 5

Activity 1

Let us try this with five other pairs of integers assigned to 10 pupils individually

with one pair each to two students. For example, one student A is asked to add

(+7) + (9) and B is asked to add (9) + (+7). Then we can ask them to draw

conclusion from these examples.

Thus, we conclude: addition of integers is commutative.

Now, we can say: subtraction is not commutative for integers.

iii) Associative property of addition

Let us consider the integers,

3, 2 and 5.

(3) + (2) + (5) = 15 = 6

3 + (2 5) = 33 = 6

Therefore, (3 + 2) + (5) = (3) + (2 5).

Even though the grouping is not the same, the answer is the same.

This shows: addition is associative for integers.

iv) Additive identity

Let us fill in the blanks of the following:

Check Your Progress



unit.

E19. Take the same pair of numbers, do the subtraction in both the orders.

Are they equal?

................................................................................................................

................................................................................................................

................................................................................................................


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i) 3 + __ = 3

ii) 5 + 0 = ____

iii) 0 9 = ____

iv) 55 + 0 = ___

v) 0 + ____ = 17

vi) ___+ 0 = 2.5

We see that when 0 is added with any integer, we get the same integer.

Therefore, we can say: zero is the additive identity for integers.

v) Existence of additive inverse

We know that the sum of the opposite numbers is 0. That is

(+1) + C D = 0, (+5) + (5) = 0.

Hence, of a pair of opposite numbers, each is the additive universe of the other,

such as: +3 is the additive inverse of 3 and 3 is the additive inverse of +3.

Zero (0) is the additive universe of itself.

Properties of Multiplication on Integers

i) Integers are closed under multiplication i.e. if for any two integers a and

b, ab = c, then c is also an integer.

ii) Multiplication is commutative for integers i.e., ab = ba for any integers a and

b.

iii) The integer 1 is the multiplicative identity i.e., 1a = a1 = a for any integer a.

iv) Multiplication is associative for integers i.e., (ab) c = a (bc) for any three

integers a, b & c.

v) Multiplication is distributive over addition for integers i.e., a (b+c) = ab +

ac for any three integers a, b & c.

Activity 2

Take, at least five different values for each of a, b and c and verify the above

properties of multiplication.

Properties of Division on Integers

i) Consider (6) (2) = 3. 3 is an integer.

(2) (6) =3

1, and

3

1is not an integer.

Therefore, Integers are not closed under division.


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ii) Let us consider the integers +7 and +9. We see that 7 9 is not defined in

integers. Hence, division is not commutative for integers.

iii) We have seen that (+5) 1 = +5, (7) 1 7, 0 1 = 0

Thus any integer divided by 1 gives the same integer. Hence, 1 is the

identity of division in integers.

1.3.7 Rational Numbers

Let us recall the numbers we have already been familiar with operations. Examples are

Natural Numbers,

Whole Numbers,

Fraction Numbers,

Decimal Fractions, and

Integers.

Necessity begets invention of new sets of numbers. After invention of integers, it was

seen that it could not fulfill all the needs. So it was further extended to include all the

numbers which can be expressed in the form of p/q where p, and q are integers and q =

0. These numbers are known as the rational numbers. Set of rational numbers is denoted

as Q.

Note: All the integers can be written in the form ofp/q.

For example: 3 can be written as 3/1.

2 can be written as 2/1.

We can observe that all the five categories of numbers discussed earlier are instances of

rational numbers.

1.3.8 Rational Numbers: Their Properties and Operations

We have already seen basic operations on integers which are part and parcel of rational

numbers. Hence, we try to discuss the operations on non-integral section of RationalNumbers.

Ordering of Rational Numbers

Consider

2/3, 3/4 and 2/5. We have already discussed with fractions as to how to

change them into like fractions and place in order.

Properties


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i) Closure

Since we have seen all the operations with integers, integers are not

closed under division.

Let us take any two rational numbers and divide one by the other.

Consider 2/3 1/5 = 10/3. It is a rational number.

7 5/3 = 7 5

3=

5

21

Then answer is a rational number.

9 & 0 are rational numbers.

9 0 is not defined in rational numbers.

Therefore, rational numbers are not closed under division.

However, if we exclude zero then the collection of all other rational

numbers is closed under division.

Let us sum up the closure property under all the operation for all the

set of numbers studied.

Closed UnderNumbers

Addition Subtraction Multiplication Division

Rational numbers Yes Yes Yes No

Integers Yes Yes Yes No

Whole numbers Yes No Yes No

Natural numbers Yes No Yes No

ii) Existence of additive identity

Consider the examples below:

3 + 0 = 0 + 3 = 3

7 + 0 = 0 + (7) = 7

3/8 + 0 = 0 + (3/8) = 3/8

Therefore, 0 is the additive identity in the rational numbers also.


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iii) Existence of multiplicative identity

The role of 1 in Rational Numbers.

We have,

6 1 = 6 = 1 x 6

Similarly, 3/8 1 = 3/8 = 1 (3/8).

Therefore, we can say that

1 is the multiplicative identity for rational numbers.

iv) Commutative property of operations

Try to complete the table:

Commutative forNumbers


Rational numbers Yes No

Integers No

Whole numbers

Natural numbers

v) Associative property

Complete the following table:

Associative forNumbers


Rational numbers Yes No Yes No

Integers No

Whole numbers

Natural numbers

vi) Existence of additive inverse

We have seen with integers that

1 + (1) = (1) + 1 = 0


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Similarly, we can see that 2/5 + (2/5) = 0

Hence, between 5

2

and 5

2

each is the additive inverse of the other.

vii) Existence of multiplicative inverse

2

1 2 = 1

Similarly7

3

3

7= 1.

Thus, between7

3and

3

7each is the multiplicative inverse of the other.

We also say each of them to be the reciprocal of the other.

Thus, 5/9 is the reciprocal of 9/5

or 5/9 is the multiplicative inverse of 9/5.

Zero has no multiplicative inverse in rational numbers, because 1/0 is not

defined in rational numbers.

viii) Distributive property over addition in rational numbers

To explain this, consider 3/5, 2/3 and 5/7

Check Your Progress



unit.

E20. What is the multiplicative inverse of 1? Answer is 1. Why?

................................................................................................................

................................................................................................................

................................................................................................................

E21. What is the multiplicative inverse of 0?

................................................................................................................

................................................................................................................

................................................................................................................


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3/5 [2/3 + (5/7)] = 3/5 {(14/21) + (15)/21}

= 3/5 {(1)/21}

=5

3

21

1

= 1/35.

Now, let us consider

[(3/5) (2/3)] + [(3/5) (5/7)]

= (2/5) + (3/7) =35

1514 += 1/35.

Thus, (3/5) [2/3 + (5/7)] = (3/5) (2/3) + (3/5) (5/7).

Hence, we can say that rational numbers have the property of distributivity of

multiplication over addition.

ix) Existence of rational numbers between two given rational numbers

Between any two integers we have a definite number of integers.

Eg.: i) Between 1 & 5, we have 3 numbers. They are 2, 3 & 4.

ii) Between 5 and 4, how many integers are there? 8. They are

4, 3, 2, 1, 0, 1, 2, 3.

iii) How many integers are there between 1 and + 1

Only one. It is 0.

Now, let us see how many rational numbers are there between 3/10 and

7/10. Let us try to list them out.

10

4,

10

5and

10

6.

Is that all? Let us see.

We can also write

10

3=

100

3and

10

7=

100

7

Now the numbers

31/100, 32/100, 68/100, 69/100, lie between 3/10 and 7/10.

Now we have 39 more numbers in between 3/10 and 7/10.


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There is no end of the process.

3/10 can also be expressed as 3000/10000,

and 7/10 as 7000/10000.

Now we see that the rational numbers that occur between 3/10 and 7/10

are 3001/10000, 3002/10000, 6998/10000, 6999/10000.

These are 3999 were numbers. In this way we can go on finding more

and more rational numbers between 3/10 and 7/10.

Therefore, the number of rational numbers between two given rational numbers

is not limited. i.e. we can find countless rational numbers between any two

rational numbers.

1.3.9 Decimal Equivalences of Rational Numbers

Let us to convert the following fractions into equivalent fractions as indicated.

Convert2

1,

3

1and

5

1to equivalent fractions with denominator 10.

2

1=

52

51

=

10

5

Can we convert3

1to an equivalent fraction with denominator 10?

No, as 10 is not a multiple of 3.

3

1=

25

21

=

10

2

Thus we see that:

2

1= 0.5 and

5

1= 0.2

But we can divide 1 by 3 and (as we have seen in case of fractions) find a not ending

result like:

1 3 = 0.333 briefly written as 0.3

So3

1gives us a non-terminating decimal number which is also a recurring decimal

number.

Similarly,


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4

1=

254

251

=

100

25= 0.25 (The results are terminating decimals).

251 =

42541

=100

4 = 0.04

But14

1= 0.0714285714285

= 0.0714285 (Recurring decimal number).

Summing up we find that:

i) Rational numbers give terminating decimals and others give non-terminating and

recurring decimals.

ii) 0 = 0.00, 1 = 1.0, 7 = 7.0 (terminating decimals).

iii) The rational numbers of which the denominators do not have a factor other than 2 or

5 gives terminating decimals. That is the rational numbers whose denominators are

2, 22, 2

3, 5, 5

2, 5

3 25, 2

n5

n(nN) give terminating decimals.


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Check Your Progress



unit.

E22. In which sets of numbers additive identity does not exist?

................................................................................................................

................................................................................................................

................................................................................................................

................................................................................................................

E23. In which of the sets of numbers additive inverse does not occur?

................................................................................................................

................................................................................................................

................................................................................................................

E24. In which of the sets of numbers, multiplicative inverse does not occur?

................................................................................................................

................................................................................................................

................................................................................................................

................................................................................................................

E25. Which of the following rational number do not give terminating

decimals?

8

1,

15

1,

32

3,

35

11,

40

17,

48

23

................................................................................................................

................................................................................................................

................................................................................................................ E26. Determine the decimal equivalence of (i)

32

5(ii)

12

7.

................................................................................................................

................................................................................................................

................................................................................................................

................................................................................................................


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1.4 UNIT SUMMARY

Various sets of numbers are:

Natural Numbers (N): 1, 2, 3, 4,

Whole Number (W): 0, 1, 2, 3, 4,

Integers (I or Z) : 5, 4, 3, 2, 1, 0, +1, +2, +3,

Rational Numbers (Q): Numbers of the formq

pwhere p, q Z and q # 0.

Properties of Operations:

N, W, Z and Q are closed under addition.

N, W, Z and Q are closed under multiplication.

Addition is commutative in N, W, Z and Q.

Multiplication is commutative in N, W, Z and Q.

Addition is associative in N, W, Z and Q.

Multiplication is associative in N, W, Z and Q.

Multiplication distributes over addition in N, W, Z and Q.

Additive identity exists in W, Z and Q Zero (0) is the additive identity.

Multiplicative identity (1) exists in N, W, Z and Q.

Additive inverse exists in Z and Q.

Multiplicative inverse exists only in Q.

1.5 GLOSSARY

Additive Identity : The number which being added to a number a gives the

result a. Thus, 0 is the additive identity.

Additive Inverse : If the sum of two numbers is 0, each of them is known as

the additive inverse of the other (also known as opposite

number).


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Multiplicative Inverse : If the product of two numbers is 1, then each of them is

known as the multiplicative inverse (or reciprocal) of the

other.

1.6 ANSWERS TO CHECK YOUR PROGRESS

E2.3

1,

7

1,

12

1

E3. (i)18

12(ii)

15

10

E4.

17

1,

17

5,

17

14

E5. 12

E6.3

8= 2

3

2,

4

17= 4

4

1,

8

21= 2

8

5

E7. 23

2=

3

8, 4

5

3=

5

23

E8. (i) 23

2=

5

13(ii) 4

9

7=

9

43

E9. (i) 45

3(ii) 4

25

7

E10. 430

23

E11. 115

2

E12. (i) 2.3 (ii) 2.7 (iii) 3.09 (iv) 2.04

E13.

Operations in W Closure Commutative Associative Identity

Addition Yes Yes Yes Yes

Subtraction No No No Yes

7/30/2019 MathsModuleVol

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