(Core)

CBSE-i
CLASS
IX
UNIT-1
MATHEMATICS
Number System and
Number Sense
(Core)
Shiksha Kendra, 2, Community Centre, Preet Vihar,Delhi-110 092 India

The CBSE-International is grateful for permission to reproduce
and/or translate copyright material used in this publication. The
acknowledgements have been included wherever appropriate and
sources from where the material has been taken duly mentioned. In
case anything has been missed out, the Board will be pleased to rectify
the error at the earliest possible opportunity.
All Rights of these documents are reserved. No part of this publication
may be reproduced, printed or transmitted in any form without the
prior permission of the CBSE-i. This material is meant for the use of
schools who are a part of the CBSE-International only.

PREFACE
The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making
the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a
fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the
learning process in harmony with the existing personal, social and cultural ethos.
The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It
has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been
conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain
elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged.
The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in
view.
The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to
nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand,
appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations
and additions wherever and whenever necessary.
The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The
speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink
their approaches for knowledge processing by their learners. In this context, it has become imperative for them to
incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to
upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant
life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of
these requirements.
The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and
creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and
media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all
subject areas to cater to the different pace of learners.
The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now
introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is
to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous
and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some nonevaluative
components in the curriculum which would be commented upon by the teachers and the school. The objective
of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal
knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives,
SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'.
The Core skills are the most significant aspects of a learner's holistic growth and learning curve.
The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework
(NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to
millions of learners, many of whom are now global citizens.
The Board does not interpret this development as an alternative to other curricula existing at the international level, but as
an exercise in providing the much needed Indian leadership for global education at the school level. The International
Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The
Board while addressing the issues of empowerment with the help of the schools' administering this system strongly
recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to
their peers through the interactive platforms provided by the Board.
I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr.
Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the
development and implementation of this material.
The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion
forums provided on the portal. Any further suggestions are welcome.
Vineet Joshi
Chairman

Advisory
Shri Vineet Joshi, Chairman, CBSE
Shri Shashi Bhushan, Director(Academic), CBSE
ACKNOWLEDGEMENTS
Conceptual Framework
Shri G. Balasubramanian, Former Director (Acad), CBSE
Ms. Abha Adams, Consultant, Step-by-Step School, Noida
Dr. Sadhana Parashar, Head (I & R),CBSE
Ideators
Ms. Aditi Misra Ms. Anuradha Sen Ms. Jaishree Srivastava Dr. Rajesh Hassija
Ms. Amita Mishra Ms. Archana Sagar Dr. Kamla Menon Ms. Rupa Chakravarty
Ms. Anita Sharma Ms. Geeta Varshney Dr. Meena Dhami Ms. Sarita Manuja
Ms. Anita Makkar Ms. Guneet Ohri Ms. Neelima Sharma Ms. Himani Asija
Dr. Anju Srivastava Dr. Indu Khetrapal Dr. N. K. Sehgal Dr. Uma Chaudhry
Material Production Group: Classes I-V
Dr. Indu Khetarpal Ms. Rupa Chakravarty Ms. Anita Makkar Ms. Nandita Mathur
Ms. Vandana Kumar Ms. Anuradha Mathur Ms. Kalpana Mattoo Ms. Seema Chowdhary
Ms. Anju Chauhan Ms. Savinder Kaur Rooprai Ms. Monika Thakur Ms. Ruba Chakarvarty
Ms. Deepti Verma Ms. Seema Choudhary Mr. Bijo Thomas Ms. Mahua Bhattacharya
Ms. Ritu Batra
Ms. Kalyani Voleti
Material Production Groups: Classes VI-VIII
English :
Science :
Mathematics :
Geography:
Ms. Rachna Pandit
Dr. Meena Dhami
Ms. Seema Rawat
Ms. Suparna Sharma
Ms. Neha Sharma
Mr. Saroj Kumar
Ms. N. Vidya
Ms. Leela Grewal
Ms. Sonia Jain
Ms. Rashmi Ramsinghaney
Ms. Mamta Goyal
History :
Ms. Dipinder Kaur
Ms. Seema kapoor
Ms. Chhavi Raheja
Ms. Leeza Dutta
Ms. Sarita Ahuja
Ms. Priyanka Sen
Political Science:
Ms. Kalpana Pant
Dr. Kavita Khanna
Ms. Kanu Chopra
Ms. Keya Gupta
Ms. Shilpi Anand
English :
Ms. Sarita Manuja
Ms. Renu Anand
Ms. Gayatri Khanna
Ms. P. Rajeshwary
Ms. Neha Sharma
Ms. Sarabjit Kaur
Ms. Ruchika Sachdev
Geography:
Ms. Deepa Kapoor
Ms. Bharti Dave
Ms. Bhagirathi
Ms. Archana Sagar
Ms. Manjari Rattan
Mathematics :
Dr. K.P. Chinda
Dr. Ram Avtar Gupta
Dr. Mahender Shankar
Mr. J.C. Nijhawan
Ms. Rashmi Kathuria
Ms. Himani Asija
Political Science:
Material Production Groups: Classes IX-X
Ms. Sharmila Bakshi
Ms. Srelekha Mukherjee
Science :
Ms. Charu Maini
Ms. S. Anjum
Ms. Meenambika Menon
Ms. Novita Chopra
Ms. Neeta Rastogi
Ms. Pooja Sareen
Economics:
Ms. Mridula Pant
Mr. Pankaj Bhanwani
Ms. Ambica Gulati
History :
Ms. Jayshree Srivastava
Ms. M. Bose
Ms. A. Venkatachalam
Ms. Smita Bhattacharya
Coordinators:
Dr. Sadhana Parashar, Ms. Sugandh Sharma, Dr. Srijata Das, Dr. Rashmi Sethi,
Head (I and R) E O (Com) E O (Maths) E O (Science)
Shri R. P. Sharma, Consultant Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO
Ms. Seema Lakra, S O Ms. Preeti Hans, Proof Reader

CONTENTs
PREFACE
ACKNOWLEDGEMENTS
1. Syllabus 1
2. Scope document 2
3. Teacher's Support Material 5
♦ Teacher’s Note 6
♦ Activity Skill Matrix 11
♦ Warm up W1 : Recalling Key Terms 12
♦ Warm up W2 : Laws of Exponents 13
♦ Pre Content Worksheet P1 14
• Usefulness of Numbers in our Life
♦ Pre Content Worksheet P2 15
• Exploring Number Facts
♦ Content Worksheet CW1 16
• Definition of Rational Numbers
♦ Content Worksheet CW2 17
• Decimal Representation of Rational Numbers
♦ Content Worksheet CW3 18
• Successive Magnification method to represent
Rational Numbers of Number Line
♦ Content Worksheet CW4 20
• Irrational Numbers and their Representation on Number Line
♦ Content Worksheet CW5 22
• Density of Rational and Irrational Numbers
♦ Content Worksheet CW6 23
• Insertion of Rational Numbers/Irrational Numbers
between Two Rational/Irrational Numbers
♦ Content Worksheet CW7 24
• Real Number System

♦ Content Worksheet CW8 25
• Properties of Irrational Numbers w.r.t. addition and
Multiplication
♦ Content Worksheet CW9 26
• Rationalising the Denominator
♦ Content Worksheet CW10 27
• Properties of Real Numbers
♦ Post Content Worksheet PCW1 27
♦ Post Content Worksheet PCW2 27
♦ Post Content Worksheet PCW3 27
♦ Post Content Worksheet PCW4 28
4. Assessment Guidance Plan 29
5. Study Material 33
6. Students’ Support Material 65
♦ SW1 : Warm up W1 66
• Recalling Key Terms
♦ SW2 : Warm up W2 69
• Laws of Exponents
♦ SW3 : Pre Content Worksheet P1 71
• Usefulness of Numbers in our Life
♦ SW4 : Pre Content Worksheet P2 76
• Exploring Number Facts
♦ SW5 : Content Worksheet CW1 77
• Definition of Rational Numbers
♦ SW6 : Content Worksheet CW2 80
• Decimal Representation of Rational Numbers
♦ SW7 : Content Worksheet CW3 85
• Successive Magnification Method to Represent Rational
Numbers on Number Line

♦ SW8 : Content Worksheet CW4 93
• Irrational Numbers and their Representation on Number Line
♦ SW9 : Content Worksheet CW5 101
• Density of Rational and Irrational Numbers
♦ SW10 : Content Worksheet CW6 106
• Insertion of Rational Numbers/Irrational Numbers between
two Rational/Irrational Numbers
♦ SW11 : Content Worksheet CW7 108
• Real Number System
♦ SW12 : Content Worksheet CW8 111
• Properties of Irrational Numbers w.r.t. Addition
and Multiplication
♦ SW13 : Content Worksheet CW9 114
• Rationalising the Denominator
♦ SW14 : Content Worksheet CW10 116
♦ SW15 : Post Content Worksheet PCW1 119
♦ SW16 : Post Content Worksheet PCW2 120
♦ SW17 : Post Content Worksheet PCW3 122
♦ SW18 : Post Content Worksheet PCW4 123
♦ Suggested Useful Videos/Links/PPT’s 127
♦ Suggested Useful Extra Readings 128

SYLLABUS — UNIT : 1
NUMBER SYSTEM AND NUMBER SENSE (CORE)
Topic - Contemporary Global Challenges
Review & recall
Introduction to
rational numbers
and irrational
numbers
Natural numbers, whole numbers, integers and their representation on
number line. Symbols to represent them as a system N, W, I respectively.
Definition of rational numbers as numbers in the form p/qwhere p & q are
integers and q≠0,
Difference between rational numbers and fractions, representation of
rational numbers on number line.
Symbols Q to represent rational number system
Irrational numbers as numbers which are not rational.
Symbols IR to represent irrational number system
Expression of rational numbers as terminating decimal or non-terminating
recurring decimal
Expression of irrational numbers as non-terminating & non-recurring
decimal.
Introduction to real
numbers
Real numbers as a system containing both rational as well as irrational
numbers
Symbol R to represent real number system
Representation of real numbers on real line
Infiniteness of rational and irrational numbers
Algebra of Real
numbers
Sum and difference of rational numbers,
Product of two rational numbers, division of two rational numbers
Sum and difference of irrational numbers,
Product of two irrational numbers, division of two irrational numbers
Properties of rational numbers w.r.t. addition and multiplication
Properties of real numbers w.r.t. addition and multiplication
1
Rationalization of denominator of irrational number
a+ b
1

SCOPE DOCUMENT
Key concepts:
1. Rational numbers
2. Irrational numbers
3. Rationalization
4. Real line
Learning objectives:
1. To review and recall systems of natural numbers, whole number and integers and their
representation on number line.
p
2. To define rational numbers as numbers in the form , where p, q are integers and q 0.
q
≠
3. To understand that all fractions are rational numbers, but all rational numbers do not
represent fractions.
4. To identify rational numbers and irrational numbers.
5. To express rational numbers as terminating or non-terminating recurring decimals.
6. To express irrational numbers as non-terminating non-recurring decimals.
7. To represent rational numbers on real line.
8. To represent irrational numbers 2, 3, 5 on real line.
9. To understand that there are infinite rational and irrational numbers.
10. To understand that between any two rational numbers infinite rational numbers exist.
11. To understand that between any two rational numbers infinite irrational numbers exist.
12. To determine the required number of rational numbers between two given rational numbers.
13. To determine the required number of irrational numbers between two given rational
numbers.
14. To determine the required number of rational numbers between two given irrational
numbers.
15. To determine the required number of irrational numbers between two given irrational
numbers.
16. To understand that between any two irrational numbers infinite rational numbers exist.
17. To understand that between any two irrational numbers infinite irrational numbers exist.
2

18. To understand that real numbers comprises of rational and irrational numbers.
19. To understand the relation between all number systems, when they are represented as
hierarchy diagram.
20. To understand that numbers in the form a+ b, where a & b are whole numbers, b≠
0 are
irrational numbers.
21. To determine the sum of two irrational numbers.
22. To determine the difference of two irrational numbers.
23. To determine the product of two irrational numbers.
24. To perform the division of two irrational numbers.
25. To determine the sum of a rational number and an irrational number.
26. To determine the difference of a rational number and an irrational number.
27. To determine the product of a rational number and an irrational number.
a
28. To express the division of a rational number and any irrational number as and nationalize
it.
b
1
29. To rationalize the denominator of irrational numbers in the form
a+ b
Extension Activities:
i
i
i
Making number dictionary. Find information about various types of numbers and write about
them with pictorial representation.
Pi Day: Celebrating πday by preparing a πchain several weeks in advance and displaying it
around the school.
Making an irrational number Clock. Express each integral number on clock as some
irrational number.
Life skill Activity
Find the square root of your roll no. up to 20 decimal places (wherever possible) and find
a) How many of you are perfect square?
b) Is the square root of your roll number a rational number or an irrational number?
c) Identify your habits which make you rational as a person.
d) Identify your habits which makes you irrational person.
e) Chalk out a programme to get rid of irrational habits. Follow it religiously.
3

Perspective:
Use the square root values calculated by you to enhance your memory. Play a memory game with
your friends and see how many of you are able to speak accurately upto 20 decimal places.
Afterwards pick any 20 new words from dictionary and make a word chain in following fashion:
Wmwmwmwmwm....
(w=word, m= meaning)
SEWA
Hold a competition to see how many can speak the word chain in order.
Use your knowledge of number system to conduct some interesting activities in junior classes
with the permission of your teachers to create the interest of your junior schoolmates in knowing
more about numbers.
Research
An activity on creating square root spiral is discussed with you. Further it is extended to create a
spiral squares. Can we create such spirals for other regular polygons? You may use Geo-Gebra (an
open source software) to explore the patterns.
4

Teacher's
Support
Material
5

TEACHER’S NOTE
The teaching of Mathematics should enhance the child's resources to think and reason, to visualise
and handle abstractions, to formulate and solve problems. As per NCF 2005, the vision for school
Mathematics include :
1. Children learn to enjoy mathematics rather than fear it.
2. Children see mathematics as something to talk about, to communicate through, to discuss among
themselves, to work together on.
3. Children pose and solve meaningful problems.
4. Children use abstractions to perceive relationships, to see structures, to reason out things, to
argue the truth or falsity of statements.
5. Children understand the basic structure of Mathematics: Arithmetic, algebra, geometry and
trigonometry, the basic content areas of school Mathematics, all offer a methodology for
abstraction, structuration and generalisation.
6. Teachers engage every child in class with the conviction that everyone can learn mathematics.
Students should be encouraged to solve problems through different methods like abstraction,
quantification, analogy, case analysis, reduction to simpler situations, even guess-and-verify
exercises during different stages of school. This will enrich the students and help them to understand
that a problem can be approached by a variety of methods for solving it. School mathematics should
also play an important role in developing the useful skill of estimation of quantities and
approximating solutions. Development of visualisation and representations skills should be integral
to Mathematics teaching. There is also a need to make connections between Mathematics and other
subjects of study. When children learn to draw a graph, they should be encouraged to perceive the
importance of graph in the teaching of Science, Social Science and other areas of study. Mathematics
should help in developing the reasoning skills of students. Proof is a process which encourages
systematic way of argumentation. The aim should be to develop arguments, to evaluate arguments,
to make conjunctures and understand that there are various methods of reasoning. Students should
be made to understand that mathematical communication is precise, employs unambiguous use of
language and rigour in formulation. Children should be encouraged to appreciate its significance.
At the secondary stage students begin to perceive the structure of Mathematics as a discipline. By
this stage they should become familiar with the characteristics of Mathematical communications,
various terms and concepts, the use of symbols, precision of language and systematic arguments in
proving the proposition. At this stage a student should be able to integrate the many concepts and
skills that he/she has learnt in solving problems.
The present unit focuses on lots of MS Excel activities and exploration in order to meet the following
6

learning objectives-
1. To review and recall systems of natural numbers, whole number and integers and their
representation on number line.
2. To define rational numbers as numbers in the form p/q, where p, q are integers and q?0.
3. To understand that all fractions are rational numbers, but all rational numbers do not represent
fractions.
4. To identify rational numbers and irrational numbers.
5. To express rational numbers as terminating or non-terminating recurring decimal.
6. To express irrational numbers as non-terminating non-recurring decimal
7. To represent rational numbers on real line.
8. To represent irrational numbers √2, √3, √5 on real line.
9. To understand that there are infinite rational and irrational numbers.
10. To understand that between any two rational numbers infinite rational numbers exist.
11. To understand that between any two rational numbers infinite irrational numbers exist.
12. To determine the required number of rational numbers between two given rational numbers.
13. To determine the required number of irrational numbers between two given rational numbers.
14. To determine the required number of rational numbers between two given irrational numbers.
15. To understand that between any two irrational numbers infinite rational numbers exist.
16. To determine the required number of irrational numbers between two given irrational numbers.
17. To understand that between any two irrational numbers infinite irrational numbers exist.
18. To understand that real numbers comprises of rational and irrational numbers.
19. To understand the relation between all number systems, when they are represented as hierarchy
diagram.
20. To understand that numbers in the form a+ √b,
where a and b are whole numbers, b?0 are
irrational numbers. To determine the sum of two irrational numbers.
21. To determine the sum of two irrational numbers
22. To determine the difference of two irrational numbers.
23. To determine the product of two irrational numbers.
7

24. To determine the sum of a rational and an irrational number.
25. To determine the difference of a rational and an irrational number.
26. To determine the product of a rational and an irrational number.
27. To rationalize the denominator of irrational numbers in the form 1/(a+ √b)
All the tasks designed to take up the chapter keeping in mind the following pedagogical issues:
To create supportive classroom environment in which learners can think together, learn together,
participate in the discussions and can take intellectual decisions.
To provide enough opportunities for each learner of expression so that teacher can have insight
into the knowledge acquired, knowledge required, refinement required in the knowledge gained
and the thinking process of the learner.
Emphasis on creating a good communicative environment in the class.
To cater various learning styles.
Richard Dedekind, a German Mathematician has stated that 'Numbers are free creations of the human
mind that serve as a medium for the easier and clearer understanding of the diversity of thought.'
Chapter on Number System and Number Sense is important as it laid the foundation of all knowledge
in Mathematics. This chapter in class IX intends to introduce the rational and irrational numbers and
hence the system of real numbers. Essential Pre-requisite to understand these numbers are the
knowledge of Natural numbers, whole numbers, integers, fractions etc.
Warm up activity W1 is a fun activity where the student has to place each number in the box under
suitable headings. Also the students are required to place the given numbers on number line. While
conducting these activities in class, teacher can simultaneously list different type of numbers on
black board and can motivate the students to create the number dictionary. Discussion should be held
in class about various types of numbers known to students like even numbers, odd numbers,
composite numbers, prime numbers, palindromes, twin primes, perfect numbers etc.
Exponential operations on numbers open up new vistas of application of Mathematics in real life.
Warm up activity W2 focuses on laws of exponents. Students can be given the drill of problems based
2n
on laws of exponents. At the same time teacher can talk about Fermat's number (Fn=2 +1),,
P
Messenere numbers (Mp= 2 -1)etc. Students can be motivated to explore more about these numbers
and their presence in nature and our lives.
Warm-up activities can be taken up in class to create interest in study of Number System and to feel
joy of learning Mathematics. For this teacher needs to expand his/ her knowledge bank with number
repository.
8

Pre-content activities P1 gives opportunities to think about the given numbers and to give their own
views about number. This activity will allow the students to speak from their experience and
knowledge and will give the insight to the teachers into students' knowledge and thought process.
The teacher has to learn here in handling the response as vague as '3 is not an interesting number' or
'3.33 is a funny number'. For such responses instead of scolding students teacher can probe them
more in order to help them reach some significant observations. In fact after the warm up W1 almost
all responses may be expected responses. But teachers should be ready for offbeat responses and
must welcome them as opportunity to present some interesting unknown facts in front of students.
Students at this stage must also understand the need of having various types of numbers and their
history. For example negative numbers were used by the Indians in Book-keeping in 7th century.
Positive numbers were used to denote assets while negative numbers we used for debts. There was
lots of controversy on use of negative numbers till 18th century. Mathematicians like Leibnitz
rejected the idea of negative number. In his opinion these numbers can lead to absurd conclusions
and misconceptions.
Gradually the fractions and rational numbers can be introduced. While defining rational numbers
difference between fractions and rational numbers should be clearly demarcated. Fraction
represents the part of whole, while rational number can represent the location of any point on
number line. It is essential to understand that all fractions are rational numbers but not vice-versa.
All Rational numbers represent some or the other point on the number line. Representation of the
rational numbers on the number line also depicts that there are infinite rational numbers and
between two given rational numbers infinite rational numbers lie. Infiniteness of rational numbers is
beautifully explained through successive magnification method of representation of rational
numbers on number line as well as MS Excel sheet. MS Excel or the Spreadsheets can also be used to
introduce the irrational numbers. Lots of activities are explained through MS Excel approach in
Teacher's Support Material. Teachers are suggested to have prior hands on experience for such
activities. It is simple to create Spreadsheets to show that irrational numbers are non-terminating
and non-recurring decimals and to find the rational numbers / irrational numbers between any two
given rational/ irrational numbers. MS Excel approach intends to give hands on experience to
students, but to attempt the problems of finding the rational numbers / irrational numbers between
any two given rational/ irrational numbers teacher must explain the traditional methods. Some fun
activities like colouring the legomen or the activity taken up in post content worksheet PCW1 are
given to reinforce the understanding and recognition.
Representation of irrational numbers on number line can be done with the help of Pythagoras
theorem. A beautiful experiment can be conducted at this stage. Students can generate square roots
spiral using right angle triangles for different irrational numbers. Further, they can construct a spiral
square by representing irrational numbers as diagonal of each square. The pattern generated
regarding the length of each square is a geometric pattern. Further the activity can be taken up as a
9

project and students can observe various patterns if they try to make spiral polygons with different
number of sides. Once the concept of rational and irrational numbers is formed operation on these
numbers can be defined and students can be given a drill in finding the sum, difference, product,
quotient of two rational numbers, two irrational numbers, a rational and an irrational number etc.
Rationalisation of denominator of irrational numbers plays significant role in solving advance
problems. Before explaining the rationalization process students should be given the idea of
conjugate of an irrational number of the form a+ √b.
Properties of rational as well as the irrational numbers with respect to addition and multiplication
can be explored by the students using self exploratory worksheets given in Student Support Material.
Rational and Irrational numbers comprises Real numbers so the properties of Real numbers can be
explored in the same way. This unit can be used to inculcate basic life skills of being rational and
making informed choices in life using the vocabulary of rational and irrational numbers. Also,
students can find the value of √2 that is irrational number and irrational number up to 20 or
30.......... decimal places. There can be a memory game on the values of √2
in order to enhance
thinking skills. Students can apply the technique of memorizing the numbers in an order to memorize
the word chain containing words and meanings in a particular fashion.
Concept of absolute number is also explained to give the physical significance of numbers in
measuring quantities.
Post Content Activities contain some interesting projects and Maths Lab experiments regarding
golden ratio, spiral polygons etc. to give the exposure of unlimited possibilities hidden in numbers
and in every sphere of life.
10

ACTIVITY SKILL MATRIX
Type of Activity
Warm Up (W1)
Warm Up (W2)
Pre-Content (P1)
Pre-Content (P2)
Content (CW1)
Content (CW2)
Content (CW3)
Content (CW4)
Content (CW5)
Content (CW6)
Content (CW7)
Content (CW8)
Content (CW9)
Content (CW10)
Content (CW11)
Post-Content (PCW 1)
Post-Content (PCW 2)
Post-Content (PCW 3)
Post-Content (PCW 4)
Post-Content (PCW 5)
Name of Activity
Recalling key term
Laws of exponents
Usefulness of numbers in our
life
Exploring number facts
Definition of rational
numbers
Decimal representation of
rational numbers
Successive magnification
method to represent
rational numbers on number
line
Irrational numbers and its
representation on number
line
Density of rational and
irrational numbers
Insertion of Rational
numbers/irrational numbers
between two rational /
irrational numbers
Real number system
Properties of irrational
numbers w.r.t. addition and
multiplication
Rationalizing the
denominator
Absolute value of a real
number
Properties of real numbers
Complete the picture
Representation on number
line
Oral assessment questions
Jigsaw-puzzle
Assignment
Skill to be developed
Expression, communication, seeing relation
between concrete and abstract
Problem solving skill, application
Memory, knowledge and creative skill
Expression, thinking skill
Memory, understanding, expression
Observation, analytical skill
Observation and graphical skill
Observation and analytical skills, drawing
skill
Analytical and synthesizing skills
Computational skill
Synthesizing the information gained, thinking
skill, application of knowledge, analytical
skills
Thinking skill, analytical skill exploratory skill
Computational skills, thinking skill
Computational skills, thinking skill
Observation, analytical skill
Analytical & synthesizing skill
Drawing skill, visual skill
Thinking skill, analytical skill
Synthesizing the information gained, thinking
skill, memory
Problem solving skill, computational skill
11

ACTIVITY 1: WARM UP (W1)
Recalling Key Terms
Specific Objective:
To review and recall systems of natural numbers, whole number and integers and their
representation on number line.
Description:
In earlier classes students have learnt about various types of numbers viz. natural numbers,
whole numbers, integers. They know what fractions are. In the first activity (see student's
worksheet 1 (W1)), each student will segregate the given numbers into these four types. In
activity 2 (see student's worksheet 1 (W1)), they will mark the numbers on the number line by
drawing a stick figure at the marked number.
Execution:
Provide a list of numbers to students and ask the students to segregate them according to their
type. It is possible that one number may come in many categories.
Teacher may draw the boxes on the board or provide a working sheet to each student.
Integers Whole Numbers Natural Numbers Fractions
Afterwards, ask the students to draw stick figure at the asked place on the number line.
Students will be asked to assess their knowledge on the basis of self assessment rubric.
Parameters for assessment:
i
i
i
Has knowledge of natural numbers, whole numbers and integers
Has Knowledge of fraction
Has knowledge of plotting above numbers on a number line
Extra reading:
You may suggest this website link to students for extra reading
http://www.purplemath.com/modules/numtypes.htm
12

ACTIVITY 2: WARM UP (W2)
Laws of Exponents
Specific Objective:
To help the learners to realize the importance of numbers around by observing variety of ways in
which they are used.
Description:
Numbers play an important role in daily life. To appreciate the presence of numbers in daily life,
you may use this warm up (W2)(See Student's Worksheet 2 (W2)).
Execution:
Learners would be asked to speak a sentence from their daily life involving the use of numbers.
Further they will be encouraged to list down all uses in prescribed worksheet W2.
Examples:
1. The cost of pair of shoes is Rs 899.
2. Ginny scored 65% marks in grade 7.
3. There are 5 rows having 5 chairs each.
4. The shopkeeper is offering 50% +20% discounts.
8
5. The speed of light is 3x10 m/s.
6. There are 23 pairs of chromosomes in a normal human body.
7. The lily has three petals, buttercups have five of them, the chicory has 21 of them, the daisy
has often 34 or 55 petals, etc.
o
8. Today the temperature is 34.4 C.
9. My mother cooked rice by taking rice and water in the ratio 1:2.
Parameters for assessment:
i
Students are able to give suitable examples on the use of numbers in daily life.
13

ACTIVITY 3: PRE CONTENT (P1)
Usefulness of Numbers in Our Life
Specific Objective:
♦
♦
To test the previous knowledge of students related to number concepts viz. fractions, decimals
and percentages.
To test the knowledge students related to exponents.
Description:
In earlier classes students have learnt about various types of numbers viz. natural numbers,
whole numbers, integers. They have the concepts of fractions, decimals and percentages very
often the students commit error in using the concept of fraction, decimal and percentage and in
understanding the relation between them. This pre content task (P1) has been designed to test
the previous learnt concepts.
Execution:
Ask the students to solve the worksheet (P1) and assess them on the basis of attached self
assessment rubric.
Parameters for assessment:
i
i
i
i
Knowledge of fractions
Knowledge of decimals
Knowledge of percentages
Knowledge of laws of exponents
Extra reading:
You may suggest this website link to students for brushing up of concepts and extra reading.
http://www.mathsisfun.com/decimal-fraction-percentage.html
http://www.bbc.co.uk/skillswise/numbers/fractiondecimalpercentage/
14

ACTIVITY 4: PRE CONTENT (P2)
Exploring Number Facts
Specific Objective:
To explore and write number facts.
Description:
It is quite an interesting activity through which student's get an opportunity to think and write
facts about each number. Consider as an example the number 3.
i. 3 is a natural number.
Execution:
3 is an integer
3 is a whole number
3 is a prime number
3 is a Fibonacci number
Facts about number 3:
Tri- means three. Triangles have three sides, tripods have three legs and the dinosaur
triceratops had three horns.
Students are motivated to think and write about numbers from 1 to 10.
Give one example and encourage the students to think and write number facts. They can take
examples from their surroundings also.
Parameters for assessment:
i
Able to communicate the number facts
Extra reading:
You may suggest this website link to students for exploring number facts.
http://jacob.efinke.com/1-25.html
http://www.richardphillips.org.uk/number/
15

Description:
ACTIVITY 5: CONTENT (CW1)
Definition of Rational Numbers
Specific Objective:
p
To define rational numbers as numbers in the form , where p, q are integers and q≠0.
q
To understand that all fractions are rational numbers, but all rational numbers do not
represent fractions.
Content Worksheet (CW1) is designed to assess the understanding of rational numbers. Also,
students will think and write about fractions and rational numbers.
Execution: Have a discussion on fractions and rational numbers.
Every fraction is a rational number, but not every rational number is a fraction.
Distribute the content worksheet (CW1) (See Student's worksheet 5) and ask the students to
write the examples on the same.
Parameters for assessment:
p
Student's are able to define rational numbers in the form
q≠0.
q
, where p, q are integers and
They understand that all fractions are rational numbers, but all rational numbers do not
represent fractions.
Extra reading:
You may suggest this website link to students for extra reading.
http://www.mathmojo.com/chronicles/2010/07/23/difference-between-fractions-andrational-numbers/
16

Specific Objectives
ACTIVITY 6: CONTENT (CW2)
Decimal Representation of Rational Numbers
i
i
To express rational numbers as terminating or non-terminating recurring decimal.
To express irrational numbers as non-terminating non-recurring decimal.
Description:
During this task students will learn to find the decimal representation of rational numbers as
terminating or non-terminating and repeating using long division method. Further, they will
visualise the decimal representation of irrational numbers through examples.
Execution:
Ask the students to use long division and find the decimal representation of given rational
numbers.
Parameters for assessment:
i
i
Able to express rational numbers as terminating or non-terminating recurring decimal.
Able to express irrational numbers as non-terminating non-recurring decimal.
17

ACTIVITY 7: CONTENT (CW3)
Successive Magnification Method
Specific Objective:
i
i
Plotting rational numbers on a number line
Through successive magnification represent rational numbers (terminating/nonterminating
and recurring decimal) on a number line.
Description:
This activity sheet comprises of two tasks:
1. Video watch http://www.youtube.com/watch?v=1xntK9LE4zQ
2. Plotting rational numbers using successive magnification method
Execution:
i
i
Students will be asked to watch the suggested video and plot the rational numbers on a
number line.
The students will be provided with the blank templates with number lines drawn on it,
marked with two points, further divided into 10 equal parts. Teacher will ask the students to
think of two integers between which the given number will lie and suitably mark the
numbers on the template. The students will be asked to visualize the number up to 1 decimal
place, up to 2 decimal places and up to 3 decimal places.
Example:
Let us take any decimal number 5.37 (say)
A
B
(it will be 5 & 6)
On number line student will think of two integers, between which numbers 5.3 will lie and
mark the alphabets A and B.
To get more accurate visualization of representation we divide the portion between 5 & 6
18

into 10 equal parts
Then first mark to the right of 5 represents 5.1, the second one 5.2 and so on.
Now student will be asked to locate 5.37 on number line. It will lie between 5.3 and 5.4.
Again divide the portion between 5.3 and 5.4 into 10 equal parts.
The first mark to the right of 5.3 will represent 5.31, the second 5.32 and so on.
Then mark the point 5.37.
Parameters for assessment:
Assessment will be done on the following parameters:-
1. Plotting the rational numbers correctly on number line.
2. Represent the number up to 1 decimal place.
3. Represent the number up to 2 decimal places.
Extra reading:
You may suggest this website link to students for extra reading.
http://www.shikshaservices.com/content/getPage/Number%20Systems/539
19

ACTIVITY 8: CONTENT (CW4)
Irrational Numbers and their Representation on Number Line
Specific Objective:
To know about the irrational numbers.
To plot the irrational numbers on a number line.
Description:
The concept of irrational numbers will be introduced through a number bag activity. A bag
containing numbered cards will be given to the students and they will be asked to pick up one
card each from the bag. Some students will have the cards bearing numbers which they are not
aware of e.g. 2.324567801001…., etc.
Execution:
Teacher will start a discussion on new type of numbers which are known as 'irrational numbers'.
Irrational numbers will be introduced as the numbers which cannot be written in the
form
p
where p and q are integers and q≠0.
q
Students will be given a worksheet showing some right angled triangles and they will be asked to
name the various sides of these triangles so as to review the terms - base, height and hypotenuse
in a right angled triangle.
Students will be asked to recall the statement of Pythagoras theorem. Through the worksheet (CW2),
they will learn to plot irrational numbers on a number line.
You may share about: Hippasus of Metapontum
Hippasus of Metapontum
20

Hippasus of Metapontum b. c. 500 B.C. in Magna Graecia, was a Greek philosopher. He was a
disciple of Pythagoras. To Hippasus (or Hippasos) is attributed the discovery of the existence of
irrational numbers. More specifically, he is credited with the discovery that the square root of 2 is
irrational.
Until Hippasus' discovery, the Pythagoreans preached that all numbers could be expressed as the
ratio of integers. Despite the validity of his discovery, the Pythagoreans initially treated it as a
kind of religious heresy and they either exiled or murdered Hippasus. Legend has it that the
discovery was made at sea and that Hippasus' fellow Pythagoreans threw him overboard.
Talk about the value of
1.4142135623……
2
Visit the link http://apod.nasa.gov/htmltest/gifcity/sqrt2.1mil
Parameters for assessment:
i
i
i
i
Identifying the irrational numbers from a set of numbers.
Defining irrational numbers.
Knowledge of the type of decimal representation of irrational numbers
Representation of irrational numbers on a number line
Extra reading:
You may suggest this website link to students for extra reading.
Irrational numbers:
http://www.mathsisfun.com/irrational-numbers.html
Plotting irrational numbers:
http://www.ehow.com/how_4455801_graph-irrational-numbers-number-line.html
Representing irrational number on a number line
http://mykhmsmathclass.blogspot.com/2011/04/representing-square-root-of-irrational.html
21

Specific Objectives
ACTIVITY 9: CONTENT (CW5)
Density of Rational and Irrational Numbers
i
i
i
To understand that there are infinite rational and irrational numbers.
To understand that between any two rational numbers infinite rational numbers exist.
To understand that between any two rational numbers infinite irrational numbers exist.
Description:
This activity sheet is based on observe and tell, think and write strategy.
Execution:
You may ask the student's to read the conversation given in CW4. Ask the students to give
examples of rational numbers and irrational numbers. During the process, students will learn
that there are infinite rational and irrational numbers. Extending the discussion further, ask
them to tell rational numbers and irrational numbers between two given numbers.
Parameters for assessment:
i
i
Able to tell that infinite rational numbers are there between two rational numbers
Able to tell that infinite irrational numbers are there between two rational numbers
Extra reading:
You may suggest this website link to students for extra reading.
Density of rational numbers, http://mykhmsmathclass.blogspot.com/2011/04/density-ofrational-numbers.html
22

Specific Objectives
ACTIVITY 10: CONTENT (CW6)
Rational/Irrational Numbers
Between two Rational/Irrational Numbers
i
i
i
i
To determine the required number of rational numbers between two given rational numbers.
To determine the required number of irrational numbers between two given rational
numbers.
To determine the required number of rational numbers between two given irrational
numbers.
To determine the required number of irrational numbers between two given irrational
numbers.
Description:
This is a practice worksheet. Students will determine the required number of rational number or
irrational numbers between two given rational numbers or irrational numbers.
Execution:
Teacher will demonstrate the method of inserting rational numbers or irrational numbers
between any two rational numbers or irrational numbers by asking questions.
Parameters for assessment:
i
i
i
i
Able to insert rational numbers between two given rational numbers.
Able to insert irrational numbers between two given irrational numbers.
Able to insert irrational numbers between two given rational numbers.
Able to insert rational numbers between two given irrational numbers.
23

ACTIVITY 11: CONTENT (CW7)
Real Number System
Specific Objectives
i
i
To understand that real numbers comprise of rational and irrational numbers.
To understand the relation between all number systems, when they are represented by
hierarchy diagram.
Description:
Through the content worksheet (CW7), students would learn to write about the Real number
system and through examples and hierarchy diagram. they will understand the relation between
all number systems.
Execution:
Talk about rational numbers and irrational numbers and the fact that the number which is
rational cannot be irrational and vice versa. The rational and irrational numbers together forms
Real numbers. Ask the students to express their knowledge in CW7.
Parameters for assessment:
i
i
i
i
Knows that rational numbers and irrational numbers together forms real numbers
Knows that all rational numbers are real numbers but not conversely
Knows that all irrational numbers are real numbers but not conversely
Knows the relation between various number types
24

ACTIVITY 12: CONTENT (CW8)
Properties of Irrational Numbers
Specific Objectives
i
i
i
i
i
i
i
i
i
i
i
i
i
To understand that numbers in the form a+ √b,
where a and b are whole numbers, b>0 are
irrational numbers.
To determine the sum of two rational numbers.
To determine the difference of two rational numbers.
To determine the product of two rational numbers.
To perform the division of two rational numbers.
To determine the sum of two irrational numbers.
To determine the difference of two irrational numbers.
To determine the product of two irrational numbers.
To perform the division of two irrational numbers.
To determine the sum of one rational number and one irrational number.
To determine the difference of one rational number and one irrational number.
To determine the product of one rational number and one irrational number.
To perform the division of one rational number and one irrational number.
Description:
This is a self exploratory task. Students will be asked to explore the addition, subtraction,
multiplication and division using rational numbers and irrational numbers.
Execution:
Firstly have a discussion in classroom on the following:
i
i
i
i
When two rational numbers are added then the result is always a rational number.
When we subtract one rational number from the other, then also the result is a rational
number.
When we multiply two rational numbers, then the product is also a rational number.
When we divide one rational number by another then the result is a rational number.
Ask the students to verify these results and explore addition, subtraction, multiplication and
division for irrational numbers.
Make copies of student's worksheet (CW7) and ask the students to do as directed.
25

Parameters for assessment:
Students are
i
i
i
i
i
i
i
i
i
i
i
i
Able to determine the sum of two rational numbers.
Able to determine the difference of two rational numbers.
Able to determine the product of two rational numbers.
Able to perform the division of two rational numbers.
Able to determine the sum of two irrational numbers.
Able to determine the difference of two irrational numbers.
Able to determine the product of two irrational numbers.
Able to perform the division of two irrational numbers.
Able to determine the sum of one rational number and one irrational number.
Able to determine the difference of one rational number and one irrational number.
Able to determine the product of one rational number and one irrational number.
Able to perform the division of one rational number and one irrational number.
ACTIVITY 13: CONTENT (CW9)
Rationalising the Denominator
Specific Objectives
26
To rationalize the denominator of irrational numbers in the form 1/(a+ √b)
Description:
During this task students will learn to rationalize an irrational number by multiplying with
rationalizing factor.
Execution:
Firstly explain the concept of rationalizing factor of an irrational number. What do we mean by
rationalization? What is the simplest rationalizing factor of a given irrational number?
Explain to them the meaning of rationalizing the denominator.
Make copies of student's worksheet (CW8) and ask the students to practice the rationalization
concept.
Parameters for assessment:
Able to find the rationalizing factor of the given irrational number
Able to rationalize the denominator

Specific Objectives
Execution:
Parameters for assessment:
Post ContentActivities:
ACTIVITY 14: CONTENT (CW10)
Properties of Real Numbers
To express the given rational number in decimal form as
Description: p
q
During this task students will learn to convert a rational numbers given in decimal form
into form.
Demonstrate a few examples and then distribute the student's worksheet.
Able to express rational numbers given in decimal form into p/q form.
p
q
ACTIVITY 15: POST CONTENT (PCW1)
A crossword puzzle for assessing the knowledge of concepts learnt in the chapter
ACTIVITY 16: POST CONTENT (PCW2)
Practice task for assessing plotting a real number on a number line
ACTIVITY 17- POST CONTENT (PCW3)
Oral questions for assessing knowledge of Real number system.
27

ACTIVITY 18-: POST CONTENT (PCW4)
A jigsaw puzzle for assessing knowledge on laws of exponents
Specific objective:
To assess the knowledge of students on law of exponents
Description of task:
It is a jigsaw puzzle kind of an activity. Each student will be given a puzzle template on which
geometrical shapes are drawn. Students will cut and arrange the pieces in the desired shape
(given to them) in such a way that answers on the edges match with the given expressions
correctly.
Materials needed for the activity:
Glue, pair of scissors
Execution:
Students will perform this activity in pair. Then they will be asked to find out all the questions and
solve them. Match the pieces with their respective answers. Get the final shape. Paste the pieces
on desired shape.
Parameters of assessment:
1. Matching of pieces correctly.
2. Getting the final shape.
28

Assessment guidance plan for teachers
ASSESSMENT PLAN
With each task in student support material a self -assessment rubric is attached for students.
Discuss with the students how each rubric can help them to keep in tune their own progress.
These rubrics are meant to develop the learner as the self motivated learner.
To assess the students' progress by teacher two types of rubrics are suggested below, one is for
formative assessment and one is for summative assessment.
Suggestive Rubric for FormativeAssessment (exemplary)
Parameter Mastered Developing Needs
motivation
Plotting of
rational
number on
number
line
Able to represent
m
rational number
n
,n≠0 ,m< n ,on
number line by
dividing the unit
length between
two consecutive
integers into n
equal parts and
marking
m
correctly
n
Able to represent rational
m
number ,n≠
0 ,m< n ,on
n
number line by dividing
the unit length between
two consecutive integers
into n equal parts and
marking m correctly
n
Able to
represent
rational
m
number ,n≠0
n
,m< n ,on
number line by
dividing the
unit length
between two
consecutive
integers into n
equal parts
and marking
correctly
m
n
Needs personal
attention
Not Able to
represent rational
m
number ,n≠0
n
,m< n, on number
line, not able to
divide the unit
length between
two consecutive
integers into n
equal parts and
m
n
marking correctly
Able to represent
mixed rational
numbers
accurately on
number line
Able to represent mixed
rational numbers
accurately on number line
Able to write
the mixed
rational
numbers as sum
of two rational
numbers and
Able to
represent one
of the parts
accurately on
number line
Not Able to write
the mixed rational
numbers as sum of
two rational
numbers
accurately and
hence cannot
represent it on
number line
29

Parameter Mastered Developing Needs
motivation
Able to visualize
the decimal
numbers upto four
decimal places
correctly using
successive
magnification
method
From above rubric it is very clear that
Able to visualize the
decimal numbers upto two
to three decimal places
correctly using successive
magnification method,
feel lost afterwards
Able to
visualize the
decimal
numbers upto
two decimal
places but not
upto four
decimal places
correctly using
successive
magnification
method
Needs personal
attention
Not able to
visualize the
decimal numbers
upto four decimal
places correctly
using successive
magnification
method
i
i
i
i
Learner requiring personal attention is poor in concepts and requires the training of basic
concepts before moving further.
Learner requiring motivation has basic concepts but faces some problem in representation
may be due to lack of spatial ability. He can be trained by showing lots of illustrations, peer
training or by doing remedial worksheets. He can also use geo-gebra templates for practice.
Learner who is developing is able to represent almost all type of rational numbers but needs
more practice to visualize the decimal representation with successive magnification
method.
Learner who has mastered the skill of representing all types of rational numbers.
Teachers' Rubric for Summative Assessment of the Unit
Parameter 5
4
3
2
1
Recognition
of all type
of numbers
i
Able to identify natural nos.,
whole nos., integers, rational
nos. and irrational nos.
accurately
i
Not able to identify natural nos.,
whole nos., integers, rational
nos. and irrational nos.
accurately
i
Able to understand that rational
nos. can be represented as
terminating decimal or non
terminating and recurring
decimal
i
Not able to understand that
rational nos. can be represented
as terminating decimal or non
terminating and recurring
decimal
i
Able to understand that
irrational nos. can be
represented as non terminating
and non-recurring decimals
i
Not able to understand that
irrational nos. can be
represented as non terminating
and non-recurring decimals
30

Parameter 5
4
3
2
1
Representat
ion of real
numbers on
number line
i
Able to represent accurately
rational numbers on number
line by dividing the length
between successive integers
appropriately,
i
Not able to represent accurately
rational numbers on number line
by dividing the length between
successive integers
appropriately,
i
Able to represent accurately
rational numbers on number
line by successive
magnification method
i
Not able to represent accurately
rational numbers on number line
by successive magnification
method
i
Able to represent accurately
irrational numbers on number
line using Pythagoras
Theorem
i
Not able to represent accurately
irrational numbers on number
line using Pythagoras Theorem
Infiniteness
of real
numbers
i
Able to find out the required
number of rational numbers
between the given two
rational numbers.
i
Not able to find out the required
number of rational numbers
between the given two rational
numbers.
i
Able to find out the required
number of rational numbers
between the given two
irrational numbers
i
Not able to find out the
required number of rational
numbers between the given two
irrational numbers
i
Able to find out the required
number of irrational numbers
between the given two
rational numbers
i
Not able to find out the required
number of irrational numbers
between the given two rational
numbers
i
Able to find out the required
number of rational numbers
between the given two
irrational numbers
i
Not able to find out the required
number of rational numbers
between the given two
irrational numbers
Algebra of
rational and
irrational
numbers
i
i
i
Able to perform four basic
operations on rational
numbers correctly
Able to perform addition,
subtraction and multiplication
correctly on irrational
numbers
Able to rationalize the
denominator of 1
correctly a+ b
i
i
i
Not able to perform four basic
operations on rational numbers
correctly
Not able to perform addition,
subtraction and multiplication
correctly on irrational numbers
Not able to rationalize the
denominator of
1
correctly a+ b
31

Parameter 5
4
3
2
1
i
i
Able to simplify the
expressions of the form
a+ b 1 a+ b
, ,
c
3
a c+ d
Able to state and illustrate
that
i
i
Not able to simplify the
expressions of the form
a+ b
c+ d
Not able to state and illustrate
that
a) sum of two rational numbers
is always a rational number
b) difference of two rational
numbers is always a rational
number
c) product of two rational
numbers is always a rational
number
d) quotient of two rational
numbers may or may not be a
rational number
e)sum of two irrational numbers
may or may not be irrational
f) difference of two irrational
numbers may or may not be
irrational
g) product of two irrational
numbers may or may not be
irrational
h) sum of a rational and an
irrational number is always an
irrational number
i) difference of a rational and
an irrational number is always
an irrational number
j) product of a rational and an
irrational number is always an
irrational number
a) sum of two rational numbers is
always a rational number
b) difference of two rational
numbers is always a rational
number
c) product of two rational numbers
is always a rational number
d) quotient of two rational
numbers may or may not be a
rational number
e) sum of two irrational numbers
may or may not be irrational
f) difference of two irrational
numbers may or may not be
irrational
g) product of two irrational
numbers may or may not be
irrational
h) sum of a rational and an
irrational number is always an
irrational number
i) difference of a rational and an
irrational number is always an
irrational number
j) product of a rational and an
irrational number is always an
irrational number
32

Study
Material
33

NUMBER SYSTEM AND NUMBER SENSE
♦
Introduction
You are already familiar with different types of numbers such as natural numbers, whole
numbers, fractions and integers. You are also familiar with their representations on the number
line and their properties with respect to addition and multiplication such as closure,
commutative, associative etc.
In this chapter, we will review and consolidate the study of these types of numbers and extend
the number system by including new types of numbers called rational numbers and irrational
numbers which together form real numbers.
Review of Numbers and Introduction to Rational Numbers
♦
Natural Numbers:
The numbers 1, 2, 3, …… which are believed to be first used by human beings in a natural way for
counting the objects are called natural numbers. The collection of natural numbers is denoted by
'N'.
Whole Numbers: The natural numbers along with the number 0 i.e. 0, 1, 2, 3, 4… are called Whole
numbers. The collection of these numbers is denoted by 'W'.
♦
Integers:
Whole numbers along with the numbers like… -4, -3, -2, -1, form a collection of numbers called
integers. This collection is denoted by 'Z' or 'I'. Letter Z has been taken from a German word
'Zohlen' which means 'to count'.
1, 2, 3….. are positive integers….-4, -3, -2, -1 are negative integer and 0 is neither positive nor
negative.
Can we say that positive numbers are same as non-negative numbers?
Positive numbers are 1, 2, 3, 4………
Non-negative numbers are 0, 1, 2, 3, 4……..
What about non-positive numbers?
♦
Fractional Numbers:
p
A number which can be expressed in the form q where p and q are positive integers, q≠0 for
example,
3 7 8
, , , etc. are all fractional numbers.
4 13 9
34

Let us now define rational numbers.
p
Rational Number: A number which can be expressed in the form
integers, q≠
0.
q
3 7 8
For example - , , , etc. are all rational numbers.
4 13 9
, where p and q are
The integers -5, 8,-11 etc. can be respectively expressed as
-5 8 -11
, , , etc. These are also
rational numbers.
1 1 1
0 0 0
Further 0 can be expressed as , , , etc. So, 0 is also a rational number.
4 13 9
Collection of all rational numbers is denoted by 'Q'. This letter Q is believed to have been taken
from the word 'quotient' and the word rational from ratio.
Number Facts
1. All natural numbers are rational numbers.
2. All fractional numbers are rational numbers.
Brainstorm
What about the converse
of these facts?
3. All whole numbers are rational numbers.
4. All integers are rational numbers.
Equivalent Rational Numbers
1 2 6 100
1
Recall that the fractions , , , represent the same fraction .Hence are called
2 4 12 200
2 a
equivalent fractions. Thus, fractions do not have a unique representation in the form where a
and b are natural numbers.
b
p
Similarly, the rational numbers do not have a unique representation in the form of
q
, where p
and q are integers and q ≠0.
—2 —4 —6
For example , , represents the same rational number . They are called equivalent
3 6 9
rational numbers.
—4
can be obtained from —2 by multiplying its numerator and denominator by number 2.
6
3
—2 —2 × 2 —4
= =
3 3×
2 6
p
In general, if is a rational number, then, we can obtain its equivalent rational number by
q
multiplying its numerator and denominator by the same non-zero number.
35

Think!! What will happen if we multiply or divide the numerator and denominator by 0?
Standard form of a rational number
p
A rational number
q
is said to be expressed in standard form if
(i)
its denominator is positive.
(ii) there is no common factor between p and q (other than 1)
e.g.
Rational numbers in
Standard form
—2
3
18
19
—4
5
10
1
—67
100
17
199
51
13
Rational numbers in nonstandard
form
2
—3
18
—19
—4
6
10
15
—67
—100
17
—199
52
—13
Reason for being in non-standard form
Denominator is negative
Denominator is negative
2 is common factor of numerator as well as
denominator
5 is common factor of numerator as well as
denominator
Denominator is negative
Denominator is negative also 17 is a
common factor of numerator and
denominator
Denominator is negative, also 13 is a
common factor of numerator and
denominator
36

Example 1: State whether the following statements are true or false. Give reasons.
(i)
Every natural number is a rational number.
(ii) Every integer is a rational number.
(iii) Every rational number is a fraction.
(iv) Zero is not a rational number.
—4
(v) is a rational number in standard form.
7
Solution:
n
(i) True. As each natural number n can be written as 1 .
(ii) True. As each integral number p can be written as p .
—52
1
(iii) False, is a rational number but not a fraction
13
(iv) False, 0 is also a rational number as it can be written as
0 0
,
1 10
(v) True, as the denominator is positive and there is no common factor between numerator and
denominator.
Example 2: Is the integer —5 a rational number? If yes, write it in standard form.
etc.
—5
Solution: Yes, —5 is a rational number. Its standard form is .
1
—15
Example 3: Write three rational numbers equivalent to rational number .
10
Solution:
—15 —15 ÷ 5 —3
= =
10 10 ÷ 5 2
—15 —15 × 2 —30
= =
10 10 × 2 20
—15 —15 × 4 —60
= =
10 10 × 4 40
Can you write some more equivalent numbers of ?
—15
10
37

♦
Representation of rational numbers on a number line.
RATIONAL
NUMBERSp/q,
p q,
q≠0
RATIONAL NUMBERS
You have learnt that rational numbers can take integral as well as non-integral values. You have
also learnt the representation of integers on a number line in previous classes.
4
Therefore a rational number like or− 4 can be easily represented in the same manner as shown
below:
1 1
To represent non-integral rational numbers on a number line let us refresh our knowledge of
representing fractional numbers on the number line.
♦ Process of representing fractions in the form m/n , where m < n, n≠
0, m and n
are natural numbers
4
Illustration : To represent
5
on a number line
Steps :
4
Note that 5 lies between integers 0 and 1.
38

Divide the unit length between each pair of consecutive integers into 5 equal parts.
Count 4 marks from zero to its right side.
4
5
0
1
Can you mark
4
5
on a number line?
Process of representing fractions in the form m/n, where m >n, n≠
0, m and n are natural numbers
Illustration - To represent 9/5 on a number line.
Steps 9
Note that 5 lies between integers 0 and 2.
Divide the unit length between each pair of consecutive integer's i.e. 0 and 1 and 1 and 2 into 5
equal parts.
Count 9 marks from zero to its right side or move to 1 unit and then count 4 marks
0 1 2
9
5
Can you mark
− 9 5
on a number line?
Algorithm to represent rational numbers on a number line
a
1. Take the given rational number , a and b are integers and b≠0.
b
a
2. If a < b then divide the number line between 0 and 1 into b equal parts. Locate .
d
b
3. If a > b then express the given rational number as a mixed fraction c . Move c units from 0 and
locate d in the next unit interval.
b
b
39

♦
Decimal Expansion of Rational Numbers
You have already learnt the conversion of fractions to decimals and vice-versa in earlier classes.
We can apply the same process for finding the decimal expansions of rational numbers written in
the form
p
, where p and q are integers and q≠0.
q
Let us explain the process through some examples:
Example 4: Find the decimal expansion of each of the following rational numbers:
212
(i) (ii) − 7 8
—5
1
(iii) (iv) (v)
25
8
3
3
7
Solution: We apply the process of long division and obtain the decimal expansions as shown below:
(i)
25
8.48
212
200
120
100
200
200
0
212
So, = 8.48
25
Hence
212
25
can be represented as 8.48 as decimal.
(ii)
− 7 8
First find the decimal expansion of positive number and then put (—) sign.
8
0.875
7.0
64
60
56
40
40
0
7
So, = 0.875
8
Hence − 7 8
can be represented as —0.875 as a decimal.
40

(iii)
8
3
3
2.666
8.0
6
20
18
20
18
20
18
2..........
Here, the remainder is not coming as 0. Instead, it is 2 and repeating again and again.
Further, the digit 6 is repeating in the quotient.
We represent this situation as follows:
8
= 2.6666………….. = 2.¯6 * A bar (—) has been placed on the repeating digit.
3
(iv)
5
3
3
1.666...
5.0
3
20
18
20
18
20
18
2..........
41

Here, the remainder is not coming as 0. Instead, it is 2 and repeating again and again.
Further, the digit 6 is repeating in the quotient.
—5
So, we write = 1.¯6
—5
3
Hence, = - 1.¯6
3 5
Note that we first find the decimal expansion of positive numbers
sign in the result.
3
and then put a negative
It may also be noted that − 1.¯6 = − (1.¯6 ) = − (1+.¯6 )= −1 −0. ¯6
It is not equal to − 1 + 0. ¯6
42

(v)
1
7
7
0.142857142857142857.....
1.0
7
30
28
20
14
60
56
40
35
50
49
10
07
30
28
20
14
60
56
40
35
50
49
10
07
Find the decimal
expansion of
2 3 4 5 6
, , , , ,
7 7 7 7 7
What do you observe
43

30
28
20
14
60
56
40
35
50
49
Here, again the remainder is not coming as 0. Instead, digits in the remainders are 1, 3, 2, 6, 4,
and 5 and are then repeating again as 1, 3, 2, 6, 4, and 5. Further, a block of digits 142857 is
repeating in the quotient.
1
So, we place a bar ( −) over the whole block of digits (142857) and write = 0. 142857
7
What do you observe in the above decimal expansion?
We observe that in some cases, the decimal expansion is terminating and in some cases, it is not
terminating or non-terminating. What is happening when the decimal expansion is nonterminating?
In this case, one digit or a block of two or more digits is repeating.
The decimal expansion of a rational number
repeating (or recurring)
p
q
is either terminating or non-terminating and
Thus, we see that:
Let us now learn to express decimal form of a number in rational form
integers and q ≠0.
p
q
where p and q are
44
Case I: When the decimal expansion is terminating
p
Example 5: Convert −3.75 in the form , where p and q are integers and q≠
0.
q
Solution: We have:
—3.75 x 100 —375
—3.75 = =
100 100
—15
=
4

p
Example 6: Convert 0.025 in the form of , where p and q are integers and q ≠0.
q
Solution: We have:
0.025
= 0.025 =
25 =
1
1.000 1000 40
Case II: When the decimal expansion is non-terminating repeating
p
Example 7: Convert each of the following into the form , where p and q are integers and q ≠0:
q
—
(i) 0.ˉ5 (ii) 0.093 (iii) — 0.36 (vi) —6. 219
Solution:
(i) Let x = 0.ˉ5
i.e., x = 0.555…….. (1)
Multiplying both sides of (1) by 10, we get
10 x = 5.555………… (2) Here, one digit 5 is
Subtracting (1) from (2), we get
repeating
10 x — x = 5.555…………… — 0.555………..
i.e. 9x = 5
5
Or x =
9
(ii) Let x = 0.093- = 0.093093093…………… (1)
Multiplying both sides of (1) by 1000, we get
Here, three digits
'093' are repeating
1000 x = 93.093093……………….. (2)
Subtracting (1) from (2), we get
999x = 93
93 31
Or x = =
999 333
(iii) For —0.36, we first convert 0.36.
So, let x = 0.36= 0.363636……………. (1)
Multiplying both sides of (1) by 100, we get
Here, two digits '36'
are repeating
100 x = 36.3636……………….. (2)
45

Subtracting (1) From (2), we get
99 x = 36
Or x =
36 4
=
99 11
So,
4
0.36 = 11
− 4 11
Hence, —0.36 =
(iv) For —6.219, we start with —6.219.
So, let x = 6.219 = 6.2191919……………. (1)
Multiplying both sides of (1) by 10, we get
Here, digits are repeating
after one decimal place
10x = 62.191919……………….. (2)
Multiplying both sides of (2) by 100, we get
1000x = 6219.191919………………. (3)
Subtracting (2) from (3), we get
990x = 6219-62
Or 990x = 6157
6157
Or x =
990 —6157
Hence, —6.219=
990 p
In the above discussion, we have seen that a rational number of the form q has either a
terminating decimal expansion or a non - terminating repeating decimal expansion and
conversely that a terminating or a non - terminating and repeating decimal expansion can be
p
converted into the form , where p and q are integers , and q ≠0 .
q
In view of the above, can we say that
A number having a non-terminating and non -repeating
(non - recurring) decimal expansion cannot be a
rational number?
46

Do such numbers exist?
If yes, where and what do we call them?
To examine the answer to all these questions let us first read the following incident.
How Did Mathematicians come to know that
Greeks declared that 2 is a real number.
2 is a real number?
Greeks were having the knowledge of Pythagoras theorem. They also have an idea that (3, 4, 5)
and (5, 12, 13) are Pythagorean triplets. They designed a floor with tiles of sides 1 x 1 unit as
shown below.
They observed that square ABCD is composed of four half tiles which is equal to 2 squares. So, the
area of square ABCD = 2 area of square OARB
Area of square ABCD=2(1)
2
Therefore, Side of square ABCD = 2 .
Since 2 is reflecting the length of the side of the square ABCD, which is existing in a real world.
So, 2 can be a real number.
47

Introduction to Irrational Numbers
We have discussed above that a rational number has either a terminating decimal expansion or a
non - terminating repeating decimal expansion. We have also seen that a terminating or a non -
terminating decimal expansion can always be converted into rational number of the form p .
q
This suggests that a number having a non - terminating, non - repeating decimal expansion
cannot be a rational number. For example, the number having a decimal expansion
2.1010010001………. is not a rational number, because it is a non - terminating non - repeating (or
non- recurring) decimal expansion.
Similarly, the number represented by the non-repeating decimal expansion
3.1011011101111…………….. is not a rational number.
The numbers which are not rational numbers are called irrational numbers.
Thus, numbers such as 2.1010010001…… and 3.1011011101111……. are irrational numbers.
Similarly, the number such as 20.1234567891011121314…… is also an irrational number (why?)
What can we say about numbers like 2, 3, 5,
etc. You have learnt in previous classes the
procedure to find out the square root of natural numbers. Use the same procedure to find the
value of 2 upto some decimal places.
We observe that
2 = 1.41421……..
2 , namely 1.41421….. is non- terminating and non-
Here, we see that decimal expansion of
repeating.
So
2 is an irrational number.
If we find the value of
3
= 1.73205…..
3 , we will observe that
This decimal expansion is also non - terminating and non - repeating.
So, 3 is also an irrational number.
Similarly, by obtaining the values of 5, 7, 11, etc, we can find that are irrational
number.
Clearly, 4 is not an irrational number because 4 = 2, and 2 is a rational number.
Similarly, 9is not an irrational number, because 9 = 3 and 3 is also a rational number. It can be
proved that numbers like ,……… cannot be expressed in the form p
2, 3, 5,
7
, where p
q
and q are integers, and q ≠ 0, i.e they are not rational numbers. Hence, they are irrational
numbers. These proofs shall be discussed in the next class. Numbers like 2, 3, 5,
7 etc.
can also be represented on the number line.
48

Another well known irrational number is π which is the ratio of circumference and diameter of a
circle. Its value is given by π = 3.1415…….
The collection of all irrational numbers is represented by the symbol IR.
Example 8:
Which of the following are rational numbers and which are irrational numbers?
(i) 625 (ii) 0.7 (iii) 0.5796 (iv) 5.478 478 478 .....
(v) 6.14114111411114… (vi) 19 (vii) 47. 83 (viii) 3.2030030003…
Solution:
(i) Rational number, because 625 = 25.
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Rational number, because it is a terminating, decimal expansion.
Irrational number, because it has a non-terminating non-repeating decimal.
Rational number, because it is non-terminating, repeating decimal expansion.
Irrational number, because it is a non-terminating, non repeating decimal expansion.
Irrational number, because it has non-terminating non-repeating decimal expansion.
Rational number, because it is a non-terminating, repeating decimal expansion.
(viii) Irrational number, because it is a non-terminating, non repeating decimal expansion.
l
Representation of irrational numbers on a number line
You know how to represent a rational number on the number line.
Are there points on the number line, which do not represent any rational number? Let us see.
Consider a unit length OA on number line.
Draw a line of unit length BA perpendicular to OA using compass.
Join OB. OAB is a right angled triangle.
OB=
2 2 2 2
(OA + AB ) = 1 +1 = 2
[using Pythagoras Theorem]
Now, with O as centre and OB as radius, draw an to intersect the number line at P.
Clearly OP=OB=
2
49

B
P
-1 0 A 1 3 4 5 6
2
This shows that the point P represents
2 on the number line.
You have already seen that 2is not a rational number as it cannot be expressed as terminating
or non terminating recurring decimal i.e. 2 is an irrational number.
Thus, there are points on the number line which do not correspond to rational numbers.
Such point like
2 is an irrational number.
More examples of irrational numbers are
Let us locate 3 on the number line
C
3, 5 , 6, 7, 10, 11, 13, 14
etc.
3
2
B
-2 -1 0 A P Q 2 3 4
o
In the diagram, construct a right triangle OBC such that BC=OA, ∠BAO=90 (see figure above)
2
Then OC =
( )
2
2 2 2
OB + CB = 2 +1 = 3
With O as centre and OC as radius, draw an arc to intersect the number line at P.
Clearly OC=OQ= 3 . Thus Q represents the irrational number 3 on the number line.
If we continue this procedure, we get representations of more irrational numbers such as
Example 9: Locate 5 on the number line.
5,
7
etc.
50

Solution:
C D E
A P Q R
-1
0 1 2 3 2 5 3
See the picture carefully. Choose your steps and locate
5 on the number line.
Observe the picture
carefully. Imagine the
point A as zero on the
number line
N
K
H
D
F C G J M
A B E I L O
Radius of each
concentric circle cut
the length on number
line representing
either rational
number or irrational
number. Can you find
out the length of AE,
AI, AL and AO?
Can you find out the
length of AD, AH, AK
and AN?
Also identify the
number representing
the length BE, EI, IL,
LO.
51

l
Introduction to real numbers
Taking together all rational numbers and all irrational numbers we obtain a collection of numbers
called the real numbers.
We denote the collection of real numbers by the letter R.
It may be noted that there is no common number between rational numbers and irrational
numbers.
In other words, no rational number can be an irrational number and no irrational number can
be a rational number.
Of course, every rational number is a real number and every irrational number is also a real
number.
As it has been seen that all rational numbers can be represented on the number line, similarly, all
irrational numbers can be represented on the number line. In other words, all real numbers can
be represented on the number line. Due to the above reason, the number line is usually referred
to as the real number line.
Representation of a real number on number line
You have observed through the examples that all rational numbers and all irrational numbers can
be represented on number line. We can also say that each point on number line corresponds to
some or the other real number. Hence we can rename the number line as REAL LINE.
l
Infiniteness of rational and irrational numbers
*
Rational numbers between two rational numbers
Consider the rational numbers—4 and 9.
Let us try to find some rational numbers between —4 and 9.
*
Clearly, —3,—2,—1,0,1,2,3,4,5,6,7and 8 are rational numbers between —4 and 9.
Are these the only rational numbers lying between —4 and 9?
To answer this question, let us write
−40 90
—4 = and 9 =
10 10
−39 −38 −37 −2 −1 0 1 2 88 89
* The rational numbers , , ,... , , , , ,... , lie between
−40 90
10 10 10 10 10 10 10 10 10 10
and and hence between —4 and 9.
10 10
Thus, the number of rational numbers between —4 and 9 is increased.
−400 900
* This number can be increased further by expressing —4 as and 9 as
100 100
Then numbers −399 −398 −397 0 1 898 899
, , ,... , ,... ,
100 100 100 100 100 100 100
will lie between —4 and 9.
52

Thus we can say that
There are infinitely many rational numbers between any two rational numbers.
Alternative method
To find the rational numbers between —4 and 9
First find the average of —4 and 9.
−4+9 5
Step 1- =
2 2
5 5
Clearly > —4 and —4 (Check on the number line)
4
−3
5
And <
4 2
−3
5
So, lies between —4 and and hence between —4 and 9.
4
2
5 +9
Similarly, 2 23
=
2 4
23
5
Clearly, lies between and 9 and hence between —4 and 9.
4
2
Thus we have found rational numbers: − 3 , 5 ,
23 between —4 and 9.
4 2 4
We can continue with this process and see that there are infinitely many rational numbers
between any two rational numbers
*
Irrational numbers between two rational numbers
1 1
Consider two rational numbers say
3
and
2
1
1
= 0.333….., = 0.5
3
2 1 1
In order to find an irrational number between and , we find a number whose decimal
3 2
expansion is non-terminating and non- recurring such as 0.3401001000100001…….
53

1 1
Some other irrational numbers between 3 and 2 may be 0.35509080766…….,
0.3665646278……..etc. We may find infinitely many such numbers.
There are infinitely many irrational numbers between two given rational numbers.
l
Irrational numbers between two irrational numbers
Consider two irrational numbers 2 and 3
2 =1.41421356……
3 =1.73205…
Clearly 1.4242004200042….., 1.51525354………, 1.690701802903………are irrational number (non
terminating and non recurring decimals) between 2 and 3 .
In this way we can find an infinite number of irrational numbers between two irrational
number 2 and 3
We may conclude that
There are infinitely many irrational numbers between two given irrational numbers.
2 5
Example 10 : Find three rational numbers between and
3 18
2 2 x 6 12
Solution: = =
3 3x
6 18
6 7 8
2 5
So, , , are rational numbers between and
18 18 18
3 18
Example 11: Find four rational numbers between —1 and 1
5 5
Solution: —1= — and 1=
5 5
4 3 2
So, − , − , − ,0 are rational numbers between —1 and 1
5 5 5
2 8
Example 12 : Find two irrational numbers between and
5 9
2
Solution: = 0.40
5
8
= 0.888…..
9 2 8
In order to find an irrational number between and we find a number whose decimal
5 9
expansion is non terminating and non recurring. Such numbers are
0.41001000100001…….
0.45900890099001………
54

Example 13: Find three irrational numbers between
3 and 5
Solution: 3 =1.73205080756…..
5 =2.23606797749.…..
Three irrational numbers between 3 and 5 can be 1.91001000100001……..,
1.989080705040………., 1.7451525354…………
l
Algebra of real numbers
Addition of a rational and an irrational
Earlier, you have studied various operations on rational numbers and their properties such as
i
i
Sum of two rational numbers is a rational number.
Product of two rational numbers is a rational number.
Let us examine these properties in case of real numbers also.
Let the number be 3 (rational) and
3 + 2= 3 + 1.41421356…..
= 4.41421356……
2 (irrational)
which is non terminating and non-repeating (non-recurring) decimal.
Hence 3 +
2 is an irrational number.
Similarly, π + 2 = (3.1415…. + 2)
= 5.1415……… which is non - terminating and non - recurring. Hence, π + 2 is an irrational number.
These examples show that:
The sum of one rational and one irrational number is an irrational number.
l
Difference of a rational and an irrational number
Let us find 3—1
As 3 = 1.7320508…….
3 — 1 = 0.7320508……
Which is non terminating and non-repeating.
Hence, ( 3 — 1) is irrational.
Similarly, 5— 2 is also irrational.
55

Thus it can be concluded that
Difference of a rational and an irrational is irrational.
l
Product of one rational and one irrational number
Consider the numbers as 2 (rational) and 3.1010010001 ……….. (Irrational)
So, 2 x (3.1010010001…………)
= 6.20200200002…………
This is non - terminating and non - repeating.
Hence 2 x (3.1010010001…………) is an irrational number.
This shows that
The product of non-zero rational number and an irrational number is irrational.
Think!! What happens if rational number 0 is multiplied by an irrational number?
l
Quotient of a rational and an irrational number
Let us find:
6.40440444044440………… (Irrational number) ÷ 2 = 3.020220222022220………
This is an irrational number.
This shows that:
The quotient of a non-zero rational number and an irrational number is irrational.
On the basis of the above discussion, we may say that
are all irrational numbers.
( ) 5
3+ 2,5— 2, 2 2 or 2 x 2 , 8
Addition and subtraction of two irrational numbers
We explain addition of two or more irrational numbers through examples.
56

Example14: Find the sum:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Solution:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
In the above example,
Note that:
are
but
6
(i)
Sum and difference of two irrational numbers may or may not be irrational.
(ii) Terms such as 2 2,3 2,5 2 are like terms. As all the terms have 2 .
Similarly 4 5,6 5 are like terms because of 5 in both terms and so on.
3 2,5 3 are not like terms as 3 2involves 2 and 5 3 involves 3 which are not the same.
l
Product (and Quotient) of two irrational numbers
Consider the products
and
(rational)[see laws of exponents]
and
=6 15
(iirational) [see laws of exponents]
57

Consider the quotients
(rational)
3x5
(irrational)
5
4x3 4x3
=
3 3
[see laws of exponents]
(rational)
[see laws of exponents]
Note that:
Product (and Quotient) of two irrational numbers may and may not be irrational.
Example 15 :
Classify the following statements as true or false. Justify your answer if false.
(i)
Sum of a rational number and an irrational number is irrational.
(ii) Sum of two irrational numbers is irrational.
(iii) Product of two irrational numbers is irrational.
(iv) Product of one rational number and one irrational number is irrational.
(v) Quotient of a non zero rational number and an irrational number is irrational.
Solution:
(i)
True
(ii) False,
(iii) False,
(iv) False, 0 ×
(v) True
( 3+ 5 ) + ( 3- 5)
3x5 3
= 6,a rational number.
=5×3=15,a rational number.
2 = 0,a is rational number.
Example16: Write any two irrational numbers such that
(i)
Their sum is a rational number.
(ii) Their sum is an irrational number.
(iii) Their difference is rational number.
(iv) Their product is a rational number.
(v) Their quotient is an irrational number.
(vi) Their quotient is a rational number.
58

Solution:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
3 15 3 5
=
5 3 5
12
3
(irrational)
(rational)
(a rational number)
(an irrational number)
(rational)
( rational number)
Rationalization of denominator of an irrational number
If the denominator of an expression is an irrational number of the type
2,3 5, 2+ 2, 3 − 2,3 2 −2 3 etc, then it can be reduced to an expression with rational
denominator. This process is known as rationalizing of the denominator.
To illustrate it, let us consider some examples.
Example 17: Express
1
3
as an expression with rational denominator.
Solution: Here is an irrational number. We also know that , a rational number.
So, =
3
3x 3=3
Why do we rationalize the denominator?
If we wish to locate on the number line, it is easier to do so if the denominator is a
rational number .
In the equivalent form (of ), it is easy to locate on the number line.
Example 18: Rationalize the denominator of
Solution:
= x
5
8
= = =
59

We can also rationalize the denominator as
x
= = =
x
Note that among √8 and √2 √2 , is smaller. So we prefer to multiply by √2 , the smaller one.
We call √2 as the smallest rationalizing factor for √8.
Example 19:
Rationalize the denominator of
Solution:
=
x
(multiply and divide by √3—
1 (Why?))
=
(if the denominator is a+b
x
then multiply and divide by expression a—b
x
=
2 2
[using the indentity (a + b) (a - b) = a - b ]
=
Example20: Rationalize the denominator of
Solution:
1
5+
2
[ multiply and divide by
5−
2
=
To rationalize the denominator of
Example21: Express
multiply and divide the denomintor by
with a rational denominator.
Solution:
=
=
=
=
60

Example22: Rationalize the denominator of
Solution:
=
Multiplying and dividing by 4 + 3 5
=
=
=
=-
+
Example23: If = a + b √5, find the values of a and b.
Solution:
= =
=
=
=
Similarly:
=
x
=
=
So,
+ = +
=
= = 3
61

which is given to be equal to a + b √5
So 3=a+b√5
Or 3+0√5=a+b√5
So,
a=3,b=0
Recall and review
* Laws of Exponents for Real Numbers
Recall laws of exponents, you studied in earlier classes.
Let a, n and m be natural numbers. Then following are the laws of exponents:
(i)
m n m+n
a x a = a
m n
(ii) (a ) = a
m
(iii) a
= a
n
a
mn
m—n
m m m
(iv) a b = (ab)
, m>n
0
Since a = 1, so using (iii),
We get = a ―n
We can extend the laws of exponents to negative exponents also. We can still extend the laws of
exponents where the base is positive real number and exponents are rational numbers which
can further be extended to real numbers also.
Leta>0beareal number and p, q be rational numbers, then
(i) a p a q = a p+q
(ii) (a p )
q = a pq
(iii) = a p—q
(iv) p
b p = (ab) p where b is a rational number
These laws help us to simplify the expressions involving exponents.
Example24: Simplify
(i) 5 2/3 . 5 1/3 (ii) (125) 1/3 (iii) 3 1/4 . 25 1/8 (iv) 7 3/4 ÷49 1/4 (v) 11 1/6 . 3 1/6
62

Solution:
(i) 5 2/3 . 5 1/3 = 5 2/3 + 1/3 = 5 3/3 = 5 1 = 5 (using law (i))
(ii) (125) 1/3 = (5 3 ) 1/3 = 5 3×1/3 = 5 (using law (ii))
(iii) 3 1/4 . 25 1/8 = 3 1/4 . (5 2 ) 1/8 = 3 1/4 . 5 2×1/8 (using law (iii))
= 3 1/4 . 5 1/4
= (3 ×5) 1/4 (using law (iv))
= 15 1/4
(iv) 7 3/4 ÷ 49 1/4
= 7 3/4 ÷ (7 2 ) 1/4 = 7 3/4 ÷ 7 2 x ¼ (using law (ii)
(v)
= 7 3/4 ÷ 7 1/2
= 7 3/4- ½ (using law (iii))
= 7 1/4
11 1/6 . 3 1/6 = (11x3) 1/6 (using law (iv))
= 33 1/6
Example25:
Solution:
Simplify:
16 1/4 = (2 4 ) 1/4 = 2 4x1/4 = 2 1 = 2
(81) 1/4 = (3 4 ) 1/4 = 3 4x ¼ = 3 1 = 3
So,
Alternatively
= 2/3
We can also simplify as follows:
= ( ) 1/4 =[( ) 4 ] 1/4 =
1
2 4
( ) × 4
2
=
3 3
l
Summary
i
i
i
i
i
Collection of real numbers is a collection of rational and irrational numbers.
There are infinite rational and irrational numbers.
Every rational and irrational numbers can be represented on real line.
Every point on real line corresponds to either a rational or an irrational number.
Between any two rational numbers infinite rational numbers exist.
63

i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
Between any two rational numbers infinite irrational numbers exist.
Between any two irrational numbers infinite rational numbers exist.
Between any two irrational numbers infinite irrational numbers exist.
Numbers in the form a+ b , where a and b are whole numbers, b ≠ 0 are irrational numbers.
Sum of two rational numbers is always a rational number.
Difference of two rational numbers is always a rational number.
Product of two rational numbers is always a rational number.
Division of two non zero rational numbers is always a rational number.
Sum of two irrational numbers may or may not be irrational.
Difference of two irrational numbers may or may not be irrational.
Product of two irrational numbers may or may not be irrational.
Quotient between two irrational numbers can be obtained by rationalization.
Sum of a rational number and an irrational number is always irrational.
Difference of one rational number and one irrational number is always irrational.
Product of one rational number and one irrational number is always irrational.
64

Student's
Support
Material

STUDENT'S WORKSHEET 1 (SW1)
Warm Up (W1)
Recalling Key Terms
Name of student -------------------------------------- Date -----------------------------
1. Reema is having a box containing different types of numbers. Help her in segregating the
numbers according to their names. If you think a number can be placed in more than one box,
then please do that.
2 -4 53 -8/4 10/3 -3/4 9 -20
1/5 3/5 7/2 -120 43 -5 0 -2 1
23/4 25/5 12 12/2 34
66

Integers Whole Numbers Natural Numbers Fractions
2. Place the at the asked place on the given number line.
1. -10
2. 7/4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
3. -5/2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
4. ½
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
5. 8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
67

SELF ASSESSMENT RUBRIC -1
WARM UP (W1)
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Has knowledge of various
types of numbers.
Able to segregate different
types of numbers
Correctly represents the
given number on a number
line
68

STUDENT'S WORKSHEET 2 (SW2)
Warm Up (W2)
Laws of Exponents
Name of student -------------------------------------- Date -----------------------------
Warm up 2: Importance of numbers around
A. Write 10 examples where you use numbers.
Yasmine got up at 7 a.m.
B. What would happen if there are no numbers?
69

SELF ASSESSMENT RUBRIC 2 - WARM UP (W2)
Parameter
Always Sometimes Needs clarity Cannot frame
Able to frame examples on
the use of numbers in
daily life
70

STUDENT'S WORKSHEET 3 (SW3)
Pre Content (P1)
Usefulness of Numbers in our life
Name of student -------------------------------------- Date -----------------------------
1. Write six interesting, non-mathematical sentences each using the words portion, whole, half,
a third, a quarter
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
2. Sohail and Shivam are arguing about a headline they read in a newspaper report. Sohail insists
there is a printing error while Shivam is convinced there is nothing wrong with it.
This is what the headline read:
Ardex Company Pvt. Ltd increases sales by 200%
Sohail's argument is "a 'whole' is 100%. Everything else is a fraction of a whole, so has to be less
than 100. So how can anything be 200%? That's ridiculous!"
What arguments would Shivam give to back the headline? Explain in a short paragraph.
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
-----------------------------------------------------------------------------------------------------------------
71

3. You are the teacher. Mark the following as right or wrong. Correct, the incorrect answers.
4. Find {(18 - ).(reciprocal of 86)}.
5. Fill in the missing entries.
Fraction Decimal Percent
95%
0.67%
0.875
55/10000
1.6
72

6. A circle is used to draw pie charts knowing that circle has a central angle of 360°.Is it any coincidence
that the earth has the same number of lines of longitude?
Try to find why 360 is such an important number. Then write short paragraph (100- 200 words) on
your findings.
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------
7. Write 5 fractions equivalent to the given one.
8. Recall the Laws of Exponents for whole numbers
1.
2.
3.
4.
5.
6.
Where and b are real numbers and m and n are positive integers
73

Match the following numbers in column 1with their solutions in column 2
Column 1 Column 2
1.
2.
3.
4.
5. 216
6. 729
9. The same laws of exponents can be extended for m and n as rational numbers
Eg.
1.
2.
3.
4.
5.
6.
Match the following numbers in column 1with their solutions in column 2
Column 1 Column 2
5.
6.
74

SELF ASSESSMENT RUBRIC - 3
PRE CONTENT (P1)
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Knowledge of fractions
Knowledge of decimals
Knowledge of concept of
percentage
Knowledge of laws of
exponents
75

STUDENT'S WORKSHEET 4 (SW4)
Pre Content (P2)
Exploring Number Facts
Name of student -------------------------------------- Date -----------------------------
Exploring Number facts
i. 3 is a natural number.
3 is an integer
3 is a whole number
3 is a prime number
3 is a Fibonacci number
Facts about number 3:
Tri- means three. Triangles have three sides, tripods have three legs and the dinosaur triceratops
had three horns.
ii. Explore and write number facts about numbers from 1 to 10.
Self assessment rubric 4 - Pre Content (P2)
Parameter
Always Sometimes Needs clarity Cannot frame
Relate the presence of
numbers around.
Tell number facts
76

STUDENT'S WORKSHEET 5 (SW5)
Content Worksheet (CW1)
Definition of Rational Numbers
Name of student -------------------------------------- Date -----------------------------
Section 1: Writing
1. What are rational numbers? Define them in your own words. Give 5 examples.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
2. Rational numbers are different from fractions. What do you say? Explain with examples.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
77

3. Am I right, if I say that 2 is a rational number?
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
4. Justify the statement "All integers are rational numbers and not conversely".
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
5. Think and write about the number 0.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
Section 2: Think and write
1. The numbers 15, 30, 45, 60 represents a collection of …………………………….
2. Which collection of numbers does not contain the number 0?
3. 1/18 is an integer. (True/False)
4. 2, 3, 4, 5, 6,… will be placed in which collection of numbers?
5. The numbers
1
10
, 1 1 1
8
, 4
, 8
,
are called ……………………..
6. The numbers of the type ……-3, -2, -1, 0, 1, 2, 3……. are called ………………
7. How many counting numbers are there?
1
5
78

SELF ASSESSMENT RUBRIC -5
CONTENT WORKSHEET (CW1)
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Know about rational
numbers
Able to justify if a number
is rational
Under stand the difference
between rational numbers
and fractions
79

STUDENT'S WORKSHEET 6 (SW6)
Content Worksheet (CW 2)
Decimal Representation of Rational Numbers
(i)
Find the decimal representation of2/9. Is it terminating or non- terminating and repeating?
(ii) Write 5 rational numbers whose decimal representation terminates. Justify your answer.
(iii) Write 5 rational numbers whose decimal representation will be non-terminating and repeating.
80

(iv) I have heard that if the denominator of a rational number is expressed as product of positive
powers of 2 or 5 or 2 and 5 then its decimal expansion will terminate. Do you agree? Explore this
and write your answer.
(v) 0.1010010001…. is non-terminating and non-repeating decimal. We say, it is an irrational
number. Do you agree? Justify your answer. Write 5 irrational numbers.
81

(vi) Which of the following represents rationals numbers? Give reasons.
(a) 0.102020202…..
(b) 0.101010101…..
(c) 0.10100100010000……
(d) 0.10100100010000
(e) 0.232323
(f) 0.231232233......
(g) 0.23223222322223….
(h) 0.23223222322223
(vii)Identifying the following as terminating or non-terminating decimal numbers.
Numbers
Terminating decimal form
Non - terminating but
repeating decimal form
1.652
5.777…..
3.75
9.6767…..
5.372
3.0707…
5.82138213….
1.0101101…..
viii. Using long division method, find the quotient and remainder of the following rational numbers.
Observe the decimal representation of quotient.
82

Rational number Working Space Quotient Remainder
(ix) Write the quotient of the following using long division method and mention whether the quotient
is terminating or non-terminating and repeating.
i.
ii.
iii.
iv.
(x) Which of the following rational numbers have a decimal representation which is non- terminating
and repeating
i.
ii.
iii.
iv.
83

SELF ASSESSMENT RUBRIC - 6
CONTENT WORKSHEET (CW2)
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Know long division method
Identifies terminating or
non - terminating decimal
representation
84

Content Worksheet (CW3)
Successive Magnification Method
Name of student -------------------------------------- Date -----------------------------
3. Watch the given video http://www.youtube.com/watch?v=1xntK9LE4zQ
Plot the following on the number line.
i.
ii.
STUDENT'S WORKSHEET 7 (SW7)
Study the following example and answer the questions following it
To locate the rational number 2.75635 on the number line, we first observe that the number lies
between 2 and 3.
0 2 3 4 5
Further let us locate 2.7
2.7
1
2 3 4
85

And then 2.75 lies between 2.7 and 2.8
Now locate 2.756 between 2.75 and 2.76
And 2.7563 between 2.756 and 2.757
And finally 2.75635 between 2.7563 and 2.7564
86

4. Visualise 6.4 on number line using successive magnification.
A
B
A
B
5. Visualise 2.68 on number line using successive magnification.
A
B
A
B
A
B
6. Visualise 4. 41 on number line upto 4 places of decimal using successive magnification.
A
B
A
B
A
B
A
B
A
B
87

7. Visualise the value of π upto 5 places of decimal using successive magnification.
A
B
A
B
A
B
A
B
A
B
A
B
8. Visualise 2.2 6 on number line up to 4 decimal places using successive magnification.
A
B
A
B
A
B
A
B
A
B
88

9. Visualise 7.746 on number line using successive magnification.
A
B
A
B
A
B
A
B
89

Name of student……………………………….
Do the following:
Content Worksheet (CW 3A)
Successive Magnification Method
1. Convert the following rational numbers in p/q form.
a) 0.457
STUDENT'S WORKSHEET
Date……………………………
b) 0.457457….
c) 0.22222….
d) 0.375
90

e) 1.2353535…..
2. The decimal representation of a rational number is always terminating. Is the given statement
true? If not, then justify with examples.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
3. Geet and Jasmine are having an argument. Geet says 2.3333…..is a rational number and Jasmine
is not agreeing to it. Who is right? Why?
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
91

SELF ASSESSMENT RUBRIC - 7
CONTENT WORKSHEET (CW3)
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Represent number on
number line.
Represent the number
upto 1 decimal place.
Represent the number
upto 2 or more than 2
decimal places.
92

STUDENT'S WORKSHEET 8 (SW8)
Content Worksheet (CW4)
Irrational Number and their Representation on Number Line
Name of student -------------------------------------- Date -----------------------------
Section 1 Knowing Irrational numbers
1. What are irrational numbers? Can they be expressed in the form p/q, q ≠ o, p and q both are
integers?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
________________________________________________________________________________
2. How is the decimal representation of an irrational number? Is it terminating or non-terminating?
If it is non-terminating, then what is so special about it? Justify with examples.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
3. The decimal representation of 2 is non- terminating and non- repeating. Write its value
up to 25 decimal places.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
93

4. The approximate value of is 22/7 or 3.14. Write your comments about the given values. Are they
rational or irrational? Explain in your own words.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
5. Observe the following numbers and classify them as rational or irrational. Justify your answer.
Number Rational/Irrational Justification
.6
√6
5
0
—676
6. Classify the following numbers as rational or irrational numbers on the basis of their decimal
representation. Justify your answer.
Number Rational/Irrational Justification
4.12873873
2.0100100010001….
2.01001000100001
−6.05050505
3.14
7.434434443….
94

Section 2: Relation between various types of numbers
i
i
i
i
i
Natural numbers- Counting numbers 1, 2, 3, 4.....
Whole numbers- Numbers 0, 1, 2, 3, 4......
Integers- Numbers.............. -3,-2, -1, 0, 1, 2, 3,............
Rational numbers- Numbers which can be expressed in the form p/q, q not equal to zero, p
and q both are integers
Irrational numbers- Numbers which cannot be expressed in the form p/q, q not equal to
zero, p and q both are integers
Irrational
Naturals
Wholes
Integers
Rationals
Observe the relation between various types of numbers in the diagram given above. We say,
1. Rational numbers and Irrational numbers together form Real numbers.
2. All natural numbers are rational numbers but not vice versa.
3. All whole numbers are rational numbers but not vice versa.
4. All integers are rational numbers but not vice versa.
Give examples to justify the above observations.
95

Section 3: Activity:-Locating irrational numbers on a number line
Objective: To locate
Trigger: Get ready with
2
on a number line.
1. Unit square OABC of each side of length 1 unit.
2. A number line.
Brainstorming:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Negative Numbers (-) Positive Numbers (+)
i
Can you find the length of diagonal of a square of side 1 unit?
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
i
What do you observe when you place square OABC on the number line so that O coincides with "0"?
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
i
Can you mark a point on the number line equal to length of diagonal of square OABC?
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
i
How would you locate √2 on a number line?
Think of sides of a right triangle, whose hypotenuse is √2.
96

i
How would you locate 3 on a number line?
Think of sides of a right triangle, whose hypotenuse is
3
i
How would you locate 5 on a number line?
Think of sides of a right triangle, whose hypotenuse is
5
Section 4: Observe the given diagrams and write the location of marked points on
the given number line. Justify your answer.
Diagram Working Space Justification
x
B
1 Unit
O 1 Unit A C
97

Diagram Working Space Justification
x
B
2 Unit
O 2 Unit A C
x
B
3 Units
O 3 Units A C
x
Q
1 Unit
O
2 Units
P R
98

Diagram Working Space Justification
x
Q
2 Unit
O
3 Units
P R
E
1 Unit
D
1 Unit
O A B C
1 Unit
99

SELF ASSESSMENT RUBRIC - 8
CONTENT WORKSHEET (CW4)
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Able to Identify the irrational
numbers from a set of
numbers.
Give the proper definition of
irrational numbers.
Knows the type of decimal
representation of irrational
numbers
Able to Plot the irrational
numbers on number line.
100

STUDENT'S WORKSHEET 9 (SW9)
Content Worksheet (CW5)
Density of Rational And Irrational Number
Name of student -------------------------------------- Date -----------------------------
Student's Worksheet
i. Consider the following number line.
Am I right if I say 2.56 lies between 2 and 3?
Am I right if I say 2.566 lies between 2 and 3?
Am I right if I say 2.5666 lies between 2 and 3?
Am I right if I say 2.56666 lies between 2 and 3?
Am I right if I say 2.566666 lies between 2 and 3?
Am I right if I say 2.5666666 lies between 2 and 3?
Am I right if I say 2.567 lies between 2 and 3?
Am I right if I say 2.5677 lies between 2 and 3?
Am I right if I say 2.56777 lies between 2 and 3?
Am I right if I say 2.567777 lies between 2 and 3?
Am I right if I say 2.5677777 lies between 2 and 3?
Am I right if I say 2.56777777 lies between 2 and 3?
What do you infer from this? How many rational numbers will lie between 2 and 3?
How many rational numbers are there?
Between two rational numbers how many rational numbers are there?
ii. Am I right if I say 2.566010010001…. lies between 2 and 3?
Am I right if I say 2.5666010010001…. lies between 2 and 3?
Am I right if I say 2.56666010010001…. lies between 2 and 3?
Am I right if I say 2.566666010010001…. lies between 2 and 3?
101

Am I right if I say 2.5666666010010001…. lies between 2 and 3?
Am I right if I say 2.56666666010010001…. lies between 2 and 3?
Am I right if I say 2.5676010010001…. lies between 2 and 3?
Am I right if I say 2.56776010010001…. lies between 2 and 3?
Am I right if I say 2.567776010010001…. lies between 2 and 3?
Am I right if I say 2.5677776010010001…. lies between 2 and 3?
Am I right if I say 2.56777776010010001…. lies between 2 and 3?
Am I right if I say 2.567777776010010001…. lies between 2 and 3?
What do you infer from this? How many irrational numbers will lie between 2 and 3?
How many irrational numbers are there?
Between two rational numbers how many irrational numbers are there?
iii. Read the conversation between Meena, Neeta and Nafaisa and answer the following questions.
Meena: Can we insert rational numbers between two irrational numbers?
Neeta: Yes, we can insert many rational numbers between any two irrational numbers.
Meena: How come we do that?
Neeta: Take any two irrational numbers, say 1.010010001…. and 1.020020002….
If we want to have a rational number between them, then first tell me what should be the type of
decimal representation?
Meena: Either terminating or Non- terminating and repeating.
Neeta: That's right. So can you think of two such numbers between 1.010010001…. and
1.020020002….?
Meena: Yes, it is so easy. It can be 1.015, 1.016.
Neeta: What do you say Nafisa?
Nafisa: Yes, it can be 1.0151, 1.0156.
Neeta: You see, you can have infinite such numbers.
Meena: Yes, that's true.
Nafisa: Indeed, we can insert infinite rational numbers between two irrational numbers.
102

Q1 Can you insert 2 rational numbers between the two given irrational numbers? Justify with
example.
Q2 Can you insert 20 rational numbers between the two given irrational numbers? Justify with
example.
Q3 How many rational numbers can be inserted between the two given irrational numbers. Explain
in your own words.
iv.
Read the conversation between Meena and Neeta and answer the following questions.
Meena: Can we insert irrational numbers between two irrational numbers?
Neeta: Yes, we can insert many irrational numbers between any two irrational numbers.
Meena: How come we do that?
Neeta: Take any two irrational numbers, say 1.010010001…. and 1.020020002….
If we want to have an irrational number between them, then first tell me what should be the type
of decimal representation?
Meena: Non- terminating and non-repeating.
Neeta: That's right. So can you think of two such numbers between 1.010010001…. and
1.020020002….?
103

Meena: Yes, it is so easy. It can be 1.015010010001….., 1.016010010001….
Neeta: You see, you can have infinite such numbers.
Meena: Yes, Now I can find it.
Q1 Can you insert 2 irrational numbers between the two given irrational numbers? Justify with
example.
Q2 Can you insert 20 irrational numbers between the two given irrational numbers? Justify with
example.
Q3 How many irrational numbers can be inserted between the two given irrational numbers.
Explain.
104

SELF ASSESSMENT RUBRIC - 9
CONTENT WORKSHEET (CW5)
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Able to visualize rational
numbers between two given
rational numbers.
Able to visualise irrational
numbers between two given
rational numbers
Able to tell that infinite
rational numbers are there
between two rational numbers
Able to tell that infinite
irrational numbers are there
between two rational numbers
105

STUDENT'S WORKSHEET 10 (SW10)
Content Worksheet (CW6)
Rational and Irrational Numbers between two Rational/Irrational Numbers
Name of student -------------------------------------- Date -----------------------------
Observe the given diagram
Do the following. For each of the following write the strategy followed by you. Can you find your
answer using some other method? Explain with example.
1. Insert 10 rational numbers between 2.5 and 2.6
2. Insert 20 rational numbers between 1/3 and ½.
3. Insert 30 rational numbers between 1 and 2.
4. Insert 10 irrational numbers between 2.5 and 2.6
5. Insert 20 irrational numbers between 1/3 and ½.
6. Insert 30 irrational numbers between 1 and 2.
7. Insert 10 irrational numbers between 2.010010001….. and 2.020020002……
8. Insert 10 rational numbers between 2.010010001….. and 2.020020002……
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SELF ASSESSMENT RUBRIC - 10
CONTENT WORKSHEET (CW6)
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Able to insert rational
numbers between two given
rational numbers.
Able to insert irrational
numbers between two given
rational numbers
Able to insert rational
numbers between two given
irrational numbers.
Able to insert irrational
numbers between two given
irrational numbers.
107

STUDENT'S WORKSHEET 11 (SW11)
Content Worksheet (CW7)
Real Number System
Name of student -------------------------------------- Date -----------------------------
i. Explain the system of Real numbers in your own words.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
ii.
Fill in the following empty boxes with types of numbers.
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iii. Seeta said to Reeta "All rational numbers and irrational numbers together comprise real
numbers". Do you agree with Seeta? Why? Or Why not?
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
iv.
Observe the given diagram and write your interpretation.
109

SELF ASSESSMENT RUBRIC - 11
CONTENT WORKSHEET (CW7)
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Has knowledge that rational
numbers and irrational nu
mbers together forms real
numbers
Has knowledge that all rational
numbers are real numbers but
not conversely
Has knowledge that all
irrational numbers are real
numbers but not conversely
Has knowledge about the
relation between various
number types
110

STUDENT'S WORKSHEET 12 (SW12)
Content Worksheet (CW8)
Properties of Irrational Numbers
Name of student -------------------------------------- Date -----------------------------
Student's Worksheet
Fill bag A with five rational numbers and bag B with five irrational numbers.
Bag A
Bag B
1. Now take number from each bag and add them.
Solution: ________________ (Rational/Irrational)
2. Take two numbers one from each bag and subtract first from second.
Solution: ________________ (Rational/Irrational)
3. Choose one number from each bag and multiply them.
Solution: ________________ (Rational/Irrational)
4. Take two numbers from bag A and subtract one from another.
Solution: ________________ (Rational/Irrational)
5. Take two numbers from bag A and multiply them.
Solution: ________________ (Rational/Irrational)
6. Take two numbers from bag A and divide one by other.
Solution: ________________ (Rational/Irrational)
7. Take two numbers from bag B and add them.
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Solution: ________________ (Rational/Irrational)
8. Take two numbers from bag B and subtract one from another.
Solution: ________________ (Rational/Irrational)
9. Take two numbers from bag B and multiply them.
Solution: ________________ (Rational/Irrational)
10. Take two numbers from bag B and divide one by another.
Solution: ________________ (Rational/Irrational)
11. Take two number from bag A and divide them.
Solution: ________________ (Rational/Irrational)
12. Take one number from each bag and divide one by another.
Solution: ________________ (Rational/Irrational)
Brainstorming:
i
i
i
i
i
i
Give examples to show that the sum of two irrationals is rational.
Give examples to show that the sum of two irrationals is irrational.
Define a condition when the sum of two irrationals will be rational.
Give examples to show that the product of two irrationals is rational.
Give examples to show that the product of two irrationals is irrational.
Define a condition when the product of two irrationals will be rational.
112

SELF ASSESSMENT RUBRIC - 12
CONTENT WORKSHEET CW8
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Able to find sum and
difference of two rational
numbers
Able to find sum and
difference of two irrational
numbers
Able to find sum and difference
of one rational and one
irrational number
Able to find the product of two
rational numbers
Able to find the product of two
irrational numbers
Able to find the product of one
rational number and one
irrational number
Able to divide two rational
numbers
Able to divide two irrational
numbers
113

STUDENT'S WORKSHEET 13 (SW13)
Content Worksheet (CW9)
Rationalising the Denominators
Name of student -------------------------------------- Date -----------------------------
The number a— √b is called the conjugate of a+ √b and the number a+ √b is called the conjugate of
a— √b
Write the conjugate of 4+ √2______________
Write the conjugate of 4— √2_____________
The reciprocal of any irrational number a+ √b is
Multiplying and dividing a number by its conjugate is called rationalizing.
Note here: If we multiply and divide by the conjugate of the numerator, it is called rationalizing the
numerator and If we multiply and divide by the conjugate of the denominator, it is called rationalizing
the denominator.
Eg. 1. =
Eg. 2.
1. Rationalise the denominator of the following
a)
b)
c)
d)
e)
2. Rationalise the denominator of the following
a)
b)
c)
d)
e)
114

SELF ASSESSMENT RUBRIC - 13
CONTENT WORKSHEET CW9
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Knows what a rationalising
factor is?
Able to rationalise
denominator of the form √x
Able to rationalise
denominator of the form
(a + √b)
115

Student's Worksheet 14 (SW14)
Content Worksheet (CW10)
Name of student -------------------------------------- Date -----------------------------
Do the following:
1. Convert the following rational numbers in p/q form.
a) 0.457
b) 0.457457….
c) 0.22222….
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d) 0.375
e) 1.2353535…..
2. The decimal representation of a rational number is always terminating. Is the given statement
true? If not, then justify with example.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
3. Geet and Jasmine are having an argument. Geet says 2.3333…..is a rational number and Jasmine
is not agreeing to it. Who is right? Why?
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
117

SELF ASSESSMENT RUBRIC - 14
CONTENT WORKSHEET CW10
No Understanding
Understanding of concept but not able to apply
Understanding of concept, can apply
but commit errors in calculation
Understanding of concept, can apply accurately
Parameter
Can convert a given rational
number in decimal form into
p/q form
Identifies terminating or
non-terminating decimal
representation
118

Post Content (PCW1)
Solve the given crossword using the clues given below.
1
2
3
4
5
6
7
8
9
10
Across
4. decimal representation of rational number can be non terminating and......
6. non positive integers and natural numbers
8. neither prime nor composite
9. rational and irrational
10. non terminating and non recurring decimal
119

Down
1. decimal representation of rational number
2. natural number and zero
3. irrational
5. 1,2,3,4....
7. numbers in the form p/q ,q is not zero, p and q are coprime
Post Content (PCW2)
1. Visualise 4.6 on number line using successive magnification method..
A
B
A
B
2. Visualise 6.82 on number line using successive magnification method.
A
B
A
B
A
B
3. Visualise 2.8 6 on number line upto 4 decimal places using successive magnification method.
A
B
A
B
A
B
120

A
B
A
B
4. Visulise 5. 8 on number line upto 4 decimal places using successive magnification method.
A
B
A
B
A
B
A
B
A
B
5. Make a square root spiral using compass and ruler.
I
I
I
I
I
I
6
7
8
5
9
4
10
3
11
I
2
I
I
12 13
14
15
I
16
17
I
I
I
I
I
121

Post Content (PCW3)
Oral assessment questions
For developing thinking skills
1. Is every natural number a rational number?
2. Is every rational number a natural number?
3. Is every natural number a real number?
4. Is every real number a natural number?
For developing analytical thinking skills
1. Can we insert a rational number between 2 and 3?
2. 100 rational numbers can be inserted between 2 and 7
3. Can we insert only 100 rational numbers between 2 and 7?
4. How many rational numbers can be inserted between 2 and 7?
For testing understanding of concept taught
1. Are 2 and 5 co primes?
2. 1.010010001… is an irrational number? True/False
3. What is rationalizing factor of (2+ √3)?
For testing knowledge of concept taught
1. Give an example of an irrational number between 2 and 3.
2. Give an example of a rational number between 2 and 3.
3. What is smallest Prime number?
4. Which whole number is not a natural number?
5. Am I right if I say "4 is the smallest composite number?"
6. Every real number can be represented by a unique point on a number line. (True/False)
122

STUDENT'S WORKSHEET
Post Content (PCW4)
Name--------------------------------
Date---------------------------------------
Instructions: Cut out the pieces given below and arrange them in the puzzle template in such a way
that the answers on the two edges match correctly.
123

124

Puzzle template
125

126
Solution

SUGGESTED USEFUL VIDEOS
1. Real Number System
http://www.youtube.com/watch?v=1bU0uY2XcJs
2. Number System http://www.youtube.com/watch?v=zv1pAhuWL6U&feature=player_embedded
3. Rational Vs Irrational Numbers
fvwrel
http://www.youtube.com/watch?v=T7J-2Kt9WNs&feature=
4. Irrational Numbers http://www.youtube.com/watch?v=oORCAz-V_Bg&feature=related
http://www.youtube.com/watch?v=Nbpyj5UErGs&feature=mfu_in_order&list=UL
6. Expressions involving square root
http://www.youtube.com/watch?v=uVRHnqmM0kQ&feature=BFa&list=UL-FNdk5o3nNc&
index=3
http://mykhmsmathclass.blogspot.com/2011/04/video-
7. Rationalising the denominator
rationalising-denominator.html
http://mykhmsmathclass.blogspot.com/2011/04/video-density-
5. Density of rational numbers
of-rational-numbers.html
8. Rationalising the binomial denominator
http://mykhmsmathclass.blogspot.com/2011/04/video-rationalising-binomial.html
9. Decimal representation of a rational number
http://www.youtube.com/watch?v=UnbE4b8RHr0&feature=related
10. Converting decimals to fraction
http://www.youtube.com/watch?v=yaqc2miG9Qw&feature=related
11. Converting repeating decimals to fractions
http://www.youtube.com/watch?v=q6G9IeEBxEA&feature=related
http://www.youtube.com/watch?v=vIKlammTCsk&feature=related
http://www.youtube.com/watch?v=3q5XdVS0838&feature=related
12. Products and quotients of square roots
http://www.youtube.com/watch?v=uwDQGM7PN5k&feature=mfu_in_order&list=UL
127

SUGGESTED USEFUL EXTRA READINGS
Types of numbers
http://www.purplemath.com/modules/numtypes.htm
Fractions, decimals
and percentages
http://www.mathsisfun.com/decimalfraction percentage.html
http://www.bbc.co.uk/skillswise/numbers/fractiondecimalpercentage/
Interesting number
facts
http://jacob.efinke.com/125.html
http://www.richardphillips.org.uk/number/
Fractions and
rational numbers
http://www.mathmojo.com/chronicles/2010/07/23/diff
erence-between-fractions-and-rational-numbers/
Plotting of rational
numbers using
successive
magnification
http://www.shikshaservices.com/content/getPage/Number%20Systems/
539
Irrational numbers
http://www.mathsisfun.com/irrational-numbers.html
Plotting irrational
numbers
Representing
irrational number
on a number line
http://www.ehow.com/how_4455801_graph-irrationalnumbersnumber-line.html
http://mykhmsmathclass.blogspot.com/2011/04/representingsquare-root-of-irrational.html
http://mykhmsmathclass.blogspot.com/2011/04/representingirrational-number-on.html
Density of rational
numbers
http://mykhmsmathclass.blogspot.com/2011/04/density ofrationalnumbers.html
128

CENTRAL BOARD OF SECONDARY EDUCATION
Shiksha Kendra, 2, Community Centre, Preet Vihar,
Delhi-110 092 India