8-th math - Kar

TEXT BOOK COMMITTEE

Chairman

Dr. B. J. Venkatachala, Professor, H.B.C.S.E(T.I.F.R, Mumbai), Department of Mathematics, Indian Institute of Science, Bengaluru.

Members

1. Dr. G. Sheela, Asst. Professor, Department of Education, Manasagangotri, Mysore University, Mysuru. 2. Sri T. K. Raghavendra, Lecturer, D.I.E.T., Chickballapur. 3. Sri A. Ramaswamy, Asst. Master, Govt. Empress High School, Tumkuru.

d 4. Sri Vinay A. Joseph, Asst. Master and P.R.O., St. Xavier's High School, e Shivajinagar, Bengaluru.

5. Smt. Vasanthi Rao, Retired Teacher, Rajajinagar, Bengaluru.

h 6. Sri G. M. Jangi, Artist, D.S.E.R.T., Bengaluru. BS lis Scrutinizers

1. Dr. Ashok M. Limkar, Subject Inspector of Mathematics, D.D.P.I., Office,

b Vijayapura. T 2. Sri A. S. Hanuman, Subject Inspector of Mathematics, D.D.P.I., Office, u Shivamogga. K p Editorial Committee Members e 1. Dr. K. S. Sameera Simha, Joint Secretary, Vijaya Educational Institutions, ? r Jayanagar, Bengaluru.

2.Dr. S. Shiva Kumar, Professor, R.V. Engineering College, Bengaluru.

e Chief advisors b 1. Sri Nagendra Kumar, Managing Director, Karnataka Text Book Soci-ety,

Bengaluru-85.

to 2. Sri Panduranga, Deputy Direcor(in-charge), Karnataka Text Book Society,

Bengaluru-85.

t Chief Co-ordinator o Prof. G. S. Mudambadithaya, Curriculum review and Text book preparation, N Karnataka Text Book Society, Bengaluru-85.

Programme Co-ordinator

Smt. R. N. Shashikala, Asst. Director, Karnataka Text Book Society, Bengaluru-85.

Foreword

The Government of India through NCERT have brought out NCF-2005 to revise the curriculum of schools and suggested all the states to introduce revised textbooks in the schools based on the new curriculum. Accordingly state Governments took up the work and requested respective DSERTs to start introducing new curriculum and texts. Karnataka Government has suggested to its DSERT to take up the challenge to fulfil the vision of NCF-2005. DSERT, Karnataka started the process: constituted

d committees to revise the syllabi, identified the writers and requested these

people to write texts books based on the new syllabi incorporating the ex-

e pectations of NCF-2005. Karnataka Text Book Society, took the initiative h and coordinated the whole programme of writing these text books. BS lis The current work, a text book in mathematics for 8-th standard, is a

step taken in this direction. An effort has been made here to look at the mathematics needed at 8-th standard through a different lens. At first glance, this may look a totally unconventional approach. Some may feel

b that it is hard on the part of 8-th standard students. On the other hand T u that is the correct age for the students to learn new concepts and ideas.

Students are receptive to new intellectual challenges. It is the onus of

K p the teachers to teach new things to the students and prepare them to the e challenges of the ever changing world. This text book is also an effort ? r to integrate our students with the national mainstream where CBSE has

surged forward and parents think that their wards will be better off by learning CBSE texts.

e We have tried here to tell something new about numbers and number b system. Similarly, some thing new about graphs, postulates of geometry

and congruency of triangles are also introduced with more expectations. Quadrilaterals have been introduced now itself. There are optional prob-

to lems at the end to challenge the students. It is my earnest request to all my teacher friends to take up the new

t challenge. Let the parents of our students not feel that their wards are

always in the back seats.

No B. J. Venkatachala Homi Bhabha Centre for Science Education, TIFR, Mumbai

Table of Contents

Indian Mathematics - A brief introduction

(i)

Chapter 1.

Unit 1. Playing with numbers

1

Unit 2. Squares, square-roots, cubes

d and cube-roots

27

e Unit 3. Rational numbers

53

h Unit 4. Commercial arithmetic

80

BS lis Unit 5. Statistics

102

Chapter 2.

T b Unit 1. Algebraic Expressions

132

K pu Unit 2. Factorisation

148

e Unit 3. Linear equations in one variable

156

? r Unit 4. Exponents

172

e Unit 5. Introduction to graphs

192

b Chapter 3.

to Unit 1. Axioms, postulates and theorems 219

Unit 2. Theorems on triangles

250

tUnit 3. Congruency of triangles

266

NoUnit 4. Construction of triangles

291

Unit 5. Quadrilaterals

306

Chapter 4.

Unit 1. Mensuration

330

Optional problems

341

INDIAN MATHEMATICS - A BRIEF INTRODUCTION

Indian Mathematics dates back to the Vedic times. The first

significant mathematical texts of Vedic times are Shulva Sutras. Shulva is a sanskrit word for chord. These contain the details of

d construction of sacrificial altars. These ancient texts introduce e surds of the type 2, 3, etc. (In fact most of the ancient matheh matics was developed because of the interest in Yajna and Yaga BS lis and astrology.) BaudhayanaSutra and Apastamba Sutra give a

very good approximation to 2 in the form

T b 1

+

1 3

+

3

1 ?

4

-

3

?

1 4?

34

,

K pu which is correct up to 5 decimal places. The classical Pythagoras' theorem is stated in the above su-

e tras, far earlier than the Greeks discovered it. Another ancient ? r unsolved problem known as squaring a circle finds its place in e Shulva sutras. One is required to construct, using only a ruler

and a compass, a square whose area is equal to that of the given

b circle. Shulva sutras give approximate methods for construct-

ing such a square. This remained unsolved over two thousand

to years, and only in 18th century it was proved that such a con-

struction is impossible.

t Indian mathematicians are credited with being the first to o give an approximate value for . Aryabhata I (476 AD) gave an N approximate value for as 3.1416; he mentions that a circle of

diameter 20000 units has circumference approximately equal

to 62832 units. It is interesting to note that Aryabhata I clearly

mentions, in the fifth century itself, that is not rational and he

is using its approximate value. Only in 1761, Lambart proved

that is an irrational number, and in 1882, Lindeman proved

that is, in fact, a transcendental number.

(ii)

Indian Mathematics

The most remarkable contribution for which the entire world

still salutes India is the invention of the decimal system by in-

troducing zero and infinity. The use of decimal system is so

easy that even children can grasp it and use it in their calcula-

tions. If you really want to appreciate the simplicity of the deci-

mal system and the concept of place value, you must first study the prevalent Roman system of representing numbers. Accord-

d ing to Florian Cajori, an eminent historian of great repute " of e all the mathematical discoveries, no one has contributed

more to the general progress of intelligence than Zero." Us-

h ing base 10, Indians were able to grasp very large numbers(See BS lis Chapter 2, Unit 4, Exponents for more details).

Ancient Jain contribution to mathematics is another important milestone in the history of Indian Mathematics. Their find-

T b ings are recorded in famous Jain texts, dating back to 500 BC u to 200 BC. Here again, you see an approximate value for as K p 10 and it is calculated up to 13 decimal places. e Ancient India has vastly contributed in the areas: the meth? r ods of Arithmetic called Vyakthaganita; The method of Algebra

called Avyakthaganita. Ancient Indian Mathematicians had in-

e troduced all the four operations: addition, multiplication, subb traction and division. They also knew how to operate with frac-

tions, solving simple equations, finding square and square-root,

to finding cube and cube-root, and also knew about permutation

and combination.

t Mahavira(9th century AD), a great Jain mathematician from

Karnataka, gave the well known formula

No n

C

r

=

(n

n! - r)!r!

for the first time in the history of mathematics, in his Ganita Sara Sangraha. Aryabhata I is one of our greatest mathematicians and astronomers of all times. He is the one who systematically developed mathematics and is called, justifiably, the father of Algebra. He gave tables to trigonometric ratio Sine,

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