Mathematics Quiz Book

P U S T A K

M A H A l !

MATHEMATICS QUIZ BOOK

Rajeev Garg,

M.SC.,

M.tech.

PUSTfiR=MKHAL

Publishers

Pustak Mahal

© Pustak Mahal, N e w Delhi ISBN 978-81-223-036.3-6 Edition: 2 0 1 0 T h e C o p y r i g h t of rhis b o o k , as well as all m a t t e r c o n t a i n e d herein (including illustrations) rests w i t h t h e Publishers. N o person shall cop}' the n a m e of t h e b o o k , its title design, m a t t e r a n d illustrations in any f o r m a n d in a n y language, totally or partially or in any distorted f o r m . A n y b o d y d o i n g so shall face legal action a n d will be responsible for damages.

Preface T h e study of mathematics has dealt with the ideas and assumptions about those concepts with which mathematics starts. Mathematics, commonly abbreviated to Maths or Math—has now become the interdisciplinary tool to all science. The subject was earlier regarded as an intricate one that could be mastered only by the elite of the scholarly world. But, since 1900 — foundational investigations have come to include an inquiry into the nature of mathematical theories and scope of mathematical methods. Now, contrary to the earlier notion, anyone with a taste of figures and interest in reading can find a vast enjoyment in basic mathematics in solving mathematical problems related to every branch of science. This book tells all about. The present volume has now been revised and restructured keeping in view of the latest trends involving the adoption of user-friendly computers in solving complex mathematical problems including the wide spectrum of related areas. It is particularly formulated with an aim to make mathematics interesting for younger generation and children. The book has been divided under 28 chapters namely, the branches of mathematics, history of mathematics, ancient and modern numerals, set theory, arithmatics, algebra, geometry (plane, solid and analytical) trigonometry, vector analysis, famous mathematicians, contribution of Indian mathematicians, computer and mathematics, units measurement and common mathematical formulae etc. The new edition has been re-designed in a simple question-answer form accompanied with several illustrations to make the matter reader-friendly, easy and interesting. A l t h o u g h written f o r the benefit of teenagers and children, we h o p e this b o o k will also be useful for other interested readers on the subject.

— Publishers

\

c

Contents

V.

1.

Branches of Mathematics

9

2.

The History of Numerals

15

3.

Modern Numerals

23

4.

Binary Numerals

33

5.

Set Theory

47

6.

Arithmetic

54

7.

Algebra

58

8.

Plane Geometry

9.

Solid Geometry

79

10.

Analytical and Non-Euclidean Geometry

87

11.

Vector Analysis

94

12.

Trigonometry

100

13.

Calculus

107

14.

Statistics

113

„

66

)

V

15.

Probability

121

16.

Logic and Game Theory

127

17.

Mathematical Tools and Instruments

132

18.

Computer and Mathematics

141

19.

Famous People in Mathematics

154

20.

Mathematicians of India

163

21.

Uses of Mathematics

167

22.

Miscellany

172

23.

Mathematical Brain Twisters

184

24.

Units and Measurement System

194

25.

Fun with Numbers

202

26.

Mathematical Signs and Symbols

207

27.

Measure for Measure

209

28.

Common Formulae

212

•

1 .

Branches of Mathematics What is mathematics? The word mathematics comes from the Greek word "mathematika", meaning "things that are learned". For ancient Greeks, mathematics included not only the study of numbers and space but also astronomy and music. Nowadays, astronomy and music are not included in mathematics. In fact, it is the interdisciplinary tool of science. What are the two main branches of mathematics? Mathematics is divided into two major branches: Pure mathematics and Applied mathematics. Pure mathematics is the study of quantities and their relationship, while applied mathematics is the use of pure mathematics to solve practical problems. What are the main branches of pure and applied mathematics? The main branches of pure mathematics are arithmetic, algebra, plane geometry, solid geometry, analytical geometry, Non-Euclidean geometry, trigonometry, calculus, etc. The main branches of applied mathematics are statics, computers, dynamics, hydrostatics, statistics, optics and atomic studies etc. What is arithmetic? Arithmetic is used to solve problems using numbers. It comes from a Greek word — "arithmos" meaning the science of number. It is the oldest and simplest branch of mathematics. The fundamental operations of arithmetic are addition, subtraction, multiplication and division.

9

What do we study in algebra? Algebra deals with the whole group of numbers by means of symbols. Letters are used as symbols for numbers. Algebra uses equations and inequalities in solving problems. Sometimes algebra is also known as "Generalized Arithmatics." What is fluid dynamics? It deals with the motion of bodies in liquids and gases. What is geometry? It is a branch of mathematics, concerned with the properties of space, usually in terms of plane (two dimensional) and solid (three dimensional) figures. (Fig. 1.1) CUBE CYLINDER

Fig. 1.1 Some geometrical figures What do we study in trigonometry? Trignometry is concerned with the triangle measurement. It makes use of the ratios of the sides of the triangle. It is of practical importance in navigating surveying and simple harmonic motion in physics. What is analytical geometry? It deals with generalized numbers and space relationship. In analytical geometry we study plane and solid shapes with the help of coordinates. In analytical geometry problems are solved using algebraic methods. It is also called coordinate geometry.

10

What is calculus? It deals with the study of different functions. It requires a knowledge of algebra, trigonometry and geometry. The two main branches of calculus are: Integral calculus and Differential calculus. Define integral calculus and differential calculus. Integral calculus deals with the method of summation or adding together the effects of continuously varying quantities, while differential calculus deals in a similar way with rates of change. What do we study in biometry? Biometry literally is the measurement of living things, but generally used to mean the application of mathematics to biology. The term is now largely obsolete, since mathematical and statistical works are the integral parts of most biological disciplines. What is demography? The study related to population statistics is called demography. What is statistics? Statistics is a branch of mathematics concerned with the manipulation of numerical information. It has two branches: descriptive statistics, dealing with the classification and presentation of data and analytical statistics, which studies the ways of collecting data, its analysis and interpretation. Sampling is the fundamental to statistics. What is statics? Statics is an applied branch of mathematics which deals with the mathematics and physics of the bodies at rest. It deals with the forces acting on structures. What is dynamics? Dynamics is the mathematical and physical study of the behaviour of bodies under the action of forces that produce changes of motion in them. What is hydrostatics? It deals with the properties and behaviour of liquids, specially the forces in liquids at rest.

11

What is aerodynamics? It deals with the motion of bodies in air. It is closely related to aeronautics because it studies the flight of aeroplanes and other machines that are heavier than air. (Fig. 1.2). LIFT

DRAG

THRUST WEIGH!

Fig. 1.2 Aerodynamics What is hydrodynamics? It deals with the motion of bodies in liquids. Hydro stands for waier and dynamics for motion. What is dimensional analysis? Dimensional analysis deals with the dimensions of physical quantities such as mass (M), length (L) and time (T) and the derived units are obtainable by multiplication or division from such quantities. With which branch of mathematics the system analysis is associated ? The system analysis is associated with computer science. What is econometrics? The application of mathematics and statistics to solve the problems of economics is called econometrics. How do we define logic and theory of games? The logic and the theory of games are two fields of mathematics concerned with the study of decision making. Logic and game theory are closely related and, in some circumstances, a combination of both may be required.

12

What is topology? Topology is a branch of geometry concerned with general transformation of shapes in which certain correspondence between points is preserved. Topology mainly deals with surfaces. (Fig. 1.3). CUP

TORUS

Fig. 1.3 Examples of Topology What is numerical analysis? Numerical analysis is the general study of methods for solving complicated problems using basic operations of arithmetic. Development of digital computers has made numerical analysis a very important area of applied mathematics. What is Linear programming? Linear programming is a mathematical modelling technique useful for guiding quantitative decisions in business planning, industrial engineering and in the social and physical sciences. It is of recent origin but has become an important part of applied mathematics. What is operational research? It is a branch of mathematics which makes use of scientific methods to the management and administration of organised military, governmental, commercial and industrial systems. What is number theory? It is the branch of mathematics concerned with the abstract study of the structure of number systems and the properties of positive integers of natural numbers such as 1,2, 3, 4. What is mathematical theory of optimisation? Mathematical theory of optimisation is a technique of improving or increasing the value of some numerical quantity that in practice may take 13

the form of temperature, air flow, speed, pay off in a game, information, monetary profit and the like., How do we define Probability? The branch of mathematics which expresses chance in number statements is called probability. For example, if a person tosses a coin, there are two ways it can fall, head or tail. So probability or chance of getting a tail or head in a toss of a coin is one half. Which branch of science belongs to both physics and mathematics? Computer science belongs to both physics and mathematics. This deals with the structure and operation of computer systems, their underlying design and programming principles and techniques for the practical implementation of computer hardware and software in various areas. Which branch of mathematics deals with informations? Information theory. It is the branch of mathematics which deals with the transmission and processing of information. It was developed by an American engineer Cloude E. Shannon in 1948. It has been found very useful in automation of communication systems. What is matrix in mathematics? Matrix in mathematics, is a square or rectangular array of elements. They are a means of condensing information about mathematical systems and can be used for among other things, solving of simultaneous linear equations. o o o

14

2. The History of Numerals How did our ancestors count the things? The people first used pebbles or knots or marks of ten fingers of hands for counting the number of animals in a herd. When did people use symbols to represent numbers? The man created symbols to represent numbers as soon as he learned to write. What is the meaning of word numeral? The symbol that is used to write a number is called numeral. How did our ancestors keep count of their sheep? Every morning as they let the sheep out, they made marks on a tree — one mark for each sheep. In the evening, when they brought the sheep back, they matched each sheep with the mark on the tree. In this way, they could tell if there was any change in the number of sheep (Fig. 2.1).

Fig. 2.1 Ancient method of counting the sheep 15

What familiar things were used for counting by the ancient man? Several familiar things were used for counting by the ancient man. Lion's head was used to indicate one, wings of an eagle to indicate two, leaves of a clover to indicate three and so on (Fig. 2.2). How did the Babylonians represent numbers? They used to write numbers on flat bricks of wet clay with sharpedged sticks. Their number representations from 1 to 10,100,1000 are shown in Fig. 2.3.

Fig. 2.2 Use offamiliar things for counting by the ancient man

Fig. 2.3 Babylonian numerals When did the Babylonians develop cuneiform numerals? The Babylonians developed cuneiform or wedge-shaped writing for numbers some 5000 years ago. How can we write 243 in Babylonian numerals? We write symbol of 100 twice, symbol of 10 four times and then the symbol of 3 (Fig. 2.4).

Fig. 2.4 Representation of 243 in Babylonian cuneiform writing 16

What was the system of numerals developed by the ancient Egyptians? Hieroglyphic system. They used this system for decorative purpose on stone monuments. How were the numbers represented in the hieroglyphic system? See Fig. 2.5.

Fig. 2.5 Number representation in hieroglyphic system Was there any symbol for zero in the ancient Chinese system? No, there was no symbol for zero, and a gap had to be left in to indicate it. For example, the number 7004 was written as

What was the other numeral system of the ancient Egyptians? Another number system used by the ancient Egyptians was called hieratic number system. It was more efficient than the hieroglyphic system. It served them in their day-to-day computations. How did the Greeks write numbers? The Greeks used all the letters of their alphabet plus three additional symbols for writing numbers. The first nine symbols represented the numbers from one to nine; the next nine, the tens from ten to ninety; the last nine, the hundreds from one hundred to nine hundred (Fig. 2.6). They had no symbol for zero. To represent thousands, the Greeks added a bar to the left of the first nine letters.

17

1

2

3

4

5

6

7

8

60

70

80

90

100

200

300

400

9

10

500 600

20

30

40

700

800

900

50

Fig. 2.6 Ancient Greeks used letters of their alphabet to write numbers What was the method used by Hebrews for writing numbers? The Hebrews also used their alphabet for writing numbers (Fig. 2.7). 1

60

2

3

4

5

70

80

90

100

6

200

7

8

9

300

400

500

10

600

20

700

30

40

800

50

900

Fig. 2.7 Herbrews' way of representing numbers What was the system tor writing numbers used by the ancient Chinese? How did the Chinese write numbers more than ninety? The ancient Chinese used rod like symbols to represent numbers (Fig. 2.8). The hundreds were written in the same way as the units. For example, the symbol I I would stand for either two or two hundred depending upon its position in the number. Thousands were written in the same way as tens, the ten thousand in the same way «s the units and so on. The number 7684 was written as d f - L mi

Fig. 2.8 Ancient Chinese used rod like symbols for writing numbers How did the ancient Japanese write numbers? The ancient Japanese used short wooden sticks to represent numbers as shown in Fig. 2.9.

Fig. 2.9 Ancient Japanese numerals

1

2

3

6

4

7

13

5

8

9

10

During the thirteenth and fourteenth century what was the form of receipt in former USSR for taxes collected by officers? The form of receipts of the taxes was as shown in Fig. 2.10. This receipt shows an amount of 3674 rubles and 46 kopecks (1 ruble equals 100 kopecks).

denoted

1,000 RUBLES

denoted

100 RUBieS

denoted

10 RUBlfS

denoted

1 RUB16

denoted

10 KOP6CKS

denoted

1 HOPfECH

Fig. 2.10 Form of a receipt of taxes What is the Roman system of numerals? The Roman system of numbers was probably derived from Etruscans, the earlier inhabitants of Italy. Letters were used for numerals. I stood for one, V stood for five, X for ten, L for fifty, C for a hundred, D for five hundred and M for a thousand (Fig. 2.11). I

II

1

2

X 10

III 3

XX 20

IV 4

L 50

V

VI

5

VII

6

VIII

7

C

D

100

500

8

M

9

CMC 1000

Fig. 2.11 Roman system of numerals

19

IX

10000

How do we write different numbers in the Roman system of numerals? In the Roman system two is represented as II, three as III. An I is put before V to write four. If the symbol of a smaller number preceeded the symbol of a larger number, the smaller number is to be subtracted from the larger (subtractive principle). For example, the symbol for nine is IX. The symbol for six is VI, for seven is VII, for eight is VIII. From these, we see that if a number is smaller than the number that follows it, it is subtracted from the second number; if it is larger than the following number, the second number is added to it. Thus, LX stands for sixty but XL stands for forty. Where did the Hindu-Arabic system of numerals originate? The Hindu-Arabic system of numerals originated in India and was later adopted by the Arabs. The Hindu-Arabic numerals of tenth century are shown in Fig. 2.12.

1

2

5

3

6

7

4

8

~9

Fig. 2.12 Hindu-Arabic system of numerals of 10th century Which was the system of numbers used by the Mayans of Central America? See Fig. 2.13. Their system of numbers also in. . eludedJ zero.

* 0

**

1

•••

2

3

4

5

6

"T

—

=

500

1000

#

— !J 7n

o

a8

9

1i n 0

50

100

Fig. 2.13 Mayans' system of numerals 20

eiQ

Where was zero invented? The idea of zero was originally invented in India. It was introduced to Europe and took the form we know it today. As a matter of fact, zero was used in 628 A.D. by well known Indian mathematician Brahmagupta for carrying out six operations of mathematics. The word 'zero' evolved from the Sanskrit word shunya meaning nothing. It became sifer in Arabic from which the English word 'cipher' is derived. Leonardo Fibonacci latinized it to 'zephirum', and finally, a Florentine treatise, De Arithmetica Opusculum' established the word zero once for all in 1491. What are the unique properties of zero? The unique properties of zero are: a. When zero is placed to the right of 1, the number becomes ten. By continuing to put zeros to the right of 1, we make the value of 1 ten times greater for every zero. b. If zero is put to the left of 1 with a decimal point before it, the value becomes one tenth. Two zeros, make it one hundredth and so on. c. Any number, except zero, multiplied by zero the result is always zero. d. Any number, except zero, divided by zero, the result is undefined. e. Addition or subtraction of zero does not alter the result. f. Zero divided by any number, except zero, the result is always zero. How did the Spanish write numbers around 1000 AD? See Fig. 2.14.

11

22

33

44

55

66

77

88 9

Fig. 2.14 Spanish numerals What is the modern system of numerals? It is the modified Hindu-Arabic system in which w wee write nnumbers u m b e r s from from one to nine and zero as 11,2,3,4,5,6,7,8,9, , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , and 0. T The h e base of our system is ten and so, it is called a decimal system (decern means ten in Latin). It is a place value system in which the position of a symbol indicates its particular value. Any number, no matter hhow o w large, can be written with the help of these ten digits. 21

How did the Italians write numerals around 1400 AD? See Fig. 2.15.

1

2

3 4

5

6

7

8

9

0

Fig. 2.15 Italian numerals Hindu-Arabic and Roman equivalents Hindu- Roman How we Arabic pronounce

Hindu- Roman How we pronounce Arabic 0

nil

I 2

I II III IV V VI VII

3 4 5 6 7

8 9

10 11 12 13 14 15 16

VIII IX X XI XII XIII XIV XV XVI

nought, nothing, zero one

17 18 19

seventeen XVII XVIII eighteen XIX nineteen

two three four

20 21 30 40 50

XX XXI XXX

five six seven eight

60 70 80

nine ten eleven

90 100 101

twelve (a dozen) thirteen

twenty twentyone

thirty XL forty L fifty LX sixty LXX seventy LXXX eighty ninety XC C one hundred CI one hundred and one CXLIV one gross

fourteen

144

fifteen sixteen

CCLXXXVIII two gross 288 1000 M one thousand OOO

22

3. Modern Numerals Which numeral system is in common use today? The Hindu-Arabic numerals are in common use today. These numerals are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This is a decimal system (Fig. 3.1).

1

2

3

4

5

6

7

8

9

0

Fig. 3.1 Hindu-Arabic numerals of today What is the decimal presentation? The decimal presentation is done by a dot. For example, 0.3 is read as zero point three. The dot is called decimal point. 0.3 is equal to 3/10. Similarly, 0.01 is 1/100 and 0.001 is 1/1000 and so on. The place next to decimal point is the tenths position, the one next to it is the hundredths position and so on. The numbers such as 2.3 are called mixed decimals. The numbers which have one number before decimal point and some after it are called mixed decimals. What are the even natural numbers? Any number which is exactly divisible by 2 is called an even number. For example, 2, 4, 6, 8, etc., are the even natural numbers. What are the odd natural numbers? The numbers which are not divisible by 2 are called the odd numbers, such as 1, 3, 5, 7, 9 ... etc. 23

What is meant by the infinite set of numbers? The series of numbers 1,2,3, ••• is infinite. That is, it has no end. What is a fraction, and how is it represented? Fraction in mathematics is a number that indicates one or more equal parts of a whole. The numerator of a fraction is written above the dividing line and the denominator below the dividing line. For example, 3/8 is a fraction. 3 is numerator and 8 is denominator. It shows that a whole number is divided into 8 parts of which 3 are taken. What are proper and improper fractions? When the numerator is less than the denominator, the fraction is called proper fraction. An improper fraction has a numerator that is larger than the denominator, for example, 3/2. What is a decimal fraction? Decimal fraction has as its denominator a power of 10, and these are omitted by use of decimal point and notation for example 0.04 which is 4/100. What are factors? The factors of a number are all the numbers which will divide it exactly without leaving a remainder. For example, 18 has six factors: 1, 2, 3, 6, 9 and 18. What is the rule of division of powers? The numbers written in their index forms can be divided by subtracting their indices. For example, 5 8 -=- 5 3 = 5 5 . If a number is raised to some power and then the whole is raised to some other power what would be the result? If a n u m b e r say, 7 is raised to the p o w e r 2, we write 7 2 . If 7 2 is raised to the p o w e r 3 then we write (7 2 ) 3 . T h e result would be 7 6 , i .e. when a power is raised by some other power, the indices get multiplied.

What are perfect numbers? The perfect n u m b e r is a whole n u m b e r which equals the sum of all its factors other than the number itself. Perfect numbers are rare. For example, 6 and 28 are perfect numbers. (See next page). 24

1 +2 +3

1 +2 +4 +7 +14

6

28 What are powers or indices? When a number is multiplied by itself it is said to have been raised to the power of two. For example, 5x5=5 2 , 5x5x5=5 3 . The number written above is called power or index, and the number written below is called base. What are negative powers? A negative power or index shows how many times one must be divided by the number. For example, 7 _1 = 1/7, 7~5 = 1/75. What is the rule of multiplication of powers? The numbers written in their index forms can be multiplied by adding their powers. For example, 5 2 x 5 6 = 5 8 . What is the meaning of the root of a number? If a large number can be written as the power of a smaller number, the smaller number is called a root of the large number. For example, 32 = 2 5 , or 2 is, therefore, the fifth root of 32 and can be. written as 32 1/5 . What is a complimentary number? In number theory, the number which is obtained by subtracting a number from its base . For example, the compliment of 7 in numbers to the base 10 is 7. What are rational numbers? Rational numbers are those numbers that can be expressed as the ratio of two integers. They include all positive and negative integers as well as any number that can be expressed as a fraction. What is a square root and a cube root? If a number is raised to the power 1/2, it is called the square root of the 25

number. Similarly, if a number is raised to the power 1/3, it is called cube root of the number. For example, square root of 64 is 8, while cube root is 4. The sign of root is (V~~ ). What are prime numbers? A prime number is any positive integer (excluding 1) having no integral factors other than itself and unity. It is a natural number which cannot be expressed as the product of other natural numbers. For example, 2, 3, 5, 7,11,13, etc. are prime numbers. The lowest prime number is thus 2. The highest known prime number is 2 2 1 6 0 9 1 - 1 discovered in September 1985. This prime number contains 65050 digits. The lowest non-prime (excluding 1) number is 4. What are irrational numbers? The numbers which are not rational are called irrational numbers. They cannot be expressed as the ratio of two integers. How would you judge whether a number is divisible by 2? A number is divisible by 2 if its last digit is divisible by 2, i.e. its last digit is even. What is the criterion that a number is divisible by 3? A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 531 is divisible by 3 because the sum of its digits (5+3+1=9) is divisible by 3. How would you ascertain that a number is divisible by 4? If the number formed by the last two digits of a number is divisible by 4, the original number is divisible by 4. For example, 732 is divisible by 4 because its last two digits form 32, which is divisible by 4. What is the criterion that a number is divisible by 5? If the last digit of a number is 0 or 5, it is divisible by 5. How can we decide that a number is divisible by 6? If a number is divisible by both 2 and 3 it is divisible by 6. How would you test that a number is divisible by 7? To check the divisibility by 7, start at the right and separate the digits into 26

groups of three. Beginning with +, write + and - alternately in front of each group. Do the sum and if the answer is a multiple of 7, then the original number is divisible by 7. For example, 14294863492 is divisible by 7 because -14+294 - 863+492= -91=7x-13. What is the criterion of divisibility by 8? If the number formed by the last three digits of a number is divisible by 8, the number is divisible by 8. What is the criterion that a number is divisible by 9? If the sum of the digits of the number is a multiple of 9, the number is divisible by 9. How would you check that a number is divisible by 10? A number is exactly divisible by 10 if its last digit is 0. What are triangular numbers? If the dots representing a number can be arranged into a triangle, it is called the triangular number. See Fig. 3.2

Fig. 3.2 Triangular numbers What are polygonal numbers? When a number is represented by a group of dots such that dots can be arranged in a geometric shape, it is called a polygonal number. What are square numbers? If the dots representing a number can be arranged into a square, the number is called a square number (Fig. 3.3). 27

Fig. 3.3 Square numbers What are cubic numbers? The numbers which can be represented by three dimensional cubes (Fig. 3.4).

Fig. 3.4 Cubic numbers What is lowest common multiple? The smallest number among the common multiples of two numbers is called their lowest common multiple (iXM). For example, LCM of 4 and 6 is 12. What are the laws of multiplication? There are three laws of multiplication: (i) The Commutative Law, (ii) The Associative Law, and (iii) The Distributive Law. What are tetrahedral numbers? The numbers that can be represented by the layers of triangles forming a 28

tetrahedron are called tetrahedral numbers (Fig. 3.5).

Fig. 3.5 Tetrahedral numbers What are pyramidal numbers? The numbers that can be represented as layers of squares forming a pyramid are called pyramidal numbers (Fig. 3.6).

Fig. 3.6 Pyramidal numbers What are the basic operations of numbers? Additions, subtraction, multiplication and division are the four basic operations of numbers. What is addition? Calculating the total of two or more numbers is called addition. By adding one number to another, we get their sum. What are the laws of addition? There are two laws of addition: (i) Commutative law of addition, and (ii) Associative law of addition. 29

What is the commutative law of addition? The sum of two numbers does hot change even if the order of addition is changed (Fig. 3.7).

4+3

=

3+4

Fig. 3.7 Commutative law of addition What is the associative law of addition? For three numbers (A+B)+C=A+(B+C). This is called associative law (Fig. 3.8).

(4+3)+ 5 = 1 2

4 + (3+ 5) = 1 2

Fig. 3.8 Associative law of addition What is a common multiple? A number which is a multiple of two different numbers at the same time is called their common multiple. For example, 2 is a common multiple of 4 and 6. Similarly, 5 is a common multiple of 15 and 20. What is subtraction? The difference of the two numbers is called subtraction. What is multiplication? Multiplication is the process of repeated addition as many times as a number is being multiplied by another;number. It is a way of combining 30

two numbers to obtain a third symbolized by "x". f o r example, 4 x 5=20, it means 4 is repeatedly added 5 times (Fig. 3.9). 4 column

5 children

4 x 5 = 20

Fig. 3.9 Multiplication What is the commutative law of multiplication? The product of two numbers does not change, if we change the order of multiplication (Fig. 3.10).

5x3

3x5

Fig. 3.10 Commulative law of multiplication What is the associative law of multiplication? The multiplication (AxB)xC is the same as A x (BxC). This is called the associative law of multiplication. What is the distributive law of multiplication? According to this law, the value of A x (B+C) = AxB+AxC.

.31

What is division? The process of repeated subtraction is called division. On doing repeated subtraction we may or may not get some remainder. Explain the terms divisor, dividend quotient and remainder? See the example below: dividend divisor

3

7

2 - quotient

6 1

-

Remainder

What is a common factor? When a number is a factor of two numbers, this is called the common factor of the two numbers. For example, 3 is a factor of 6, and 3 is also a factor of 9. So, 3 is a common factor of 6 and 9. What is the highest common factor? The product of the set of common factors of two numbers is called the highest common factor (HCF). For example, common factors of 12 and 18 are 2 and 3. Their product is 6 so, their highest common factor is 6. What are the positive numbers? The numbers which are greater than zero are called positive numbers.

ooo

32

4. Binary Numerals What is a binary numeral system? It is a system of numerals in which all numbers can be expressed with the digits 0 and 1. It is called binary because we use only two symbols to write any number. It is also called 'base two system'. What is the use of binary system? Binary system is used in computers. Codes based on binary numbers are used to represent instructions and data in all modern digital computers. The only digits used in the binary system are 0 and 1 (Fig. 4.1).

Fig. 4.1 Zero and one of Binary Code What are the different binary place values? The value of any position in a binary number increases by the power of 2 with each move from right to left. Thus, the place values from right to left in a binary system are ones, twos, fours, eights and so on. For example, 11001 in the binary system means (1x16) + (1x8) + ( l x l ) . What are the other characteristics of binary number system? The different characteristics of base two system are as follows: 1. In base two, odd numbers always end in 1; while even numbers end in 0.

33

2.

If we want to double any binary number, a 0 is added to its right, e.g. 10011 x 2=100110. 3. To have an even binary number, remove the final 0, e.g. 101010 + 2 = 10101.

How a binary number is expressed? A binary number is expressed by writing 'two' at the bottom of the number. For example, number 1 expressed in binary system would be written as l t w o . What is a bicimal? In the binary system, a point separates numbers of 1 and above from numbers less than 1. In base 2, numbers less than 1 are called 'bicimals'. The place values to the right of the bicimal point are expressed in negative powers of 2, i.e. 2~>, 2"2, 2~3, 2~4. Can binary numeral system be used to transmit photographs from space? Yes. The photographs from space are transmitted as radio signals, representing the zeros and ones of the binary system. The digits are then converted by computer into an image consisting of a series of dots. What are the other popular names of the Hindu-Arabic system? The other popular names of the Hindu-Arabic system are decimal system and base ten system. Why it is called a base ten system? Because the system needs only ten symbols to represent any number. These symbols are called digits. The digits in a decimal system are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. What do these symbols stand for? These ten symbols stand for the numbers one, two, three, four, five, six, seven, eight, nine and zero respectively. What is a decimal system? This is the most commonly used number system, to the base ten. The use of decimals greatly simplifies addition, and multiplication of fraction. 0. Its full name is 'decimal positional notational system'. 34

Which numeration system is used throughout the world? The Hindu-Arabic system is the numeration system used throughout the civilized world today. It is called so because it originated with the Hindus and was carried to the western world by the Arabs. What does the process of 'double-babble' involve? Double-babble involves converting a binary number into its decimal equivalent. What is a quibinary code? Quibinary code is a binary coded decimal code. It is used to represent decimal numbers, in which each decimal digit is represented by seven binary digits. Can we perform addition, subtraction, multiplication and division operations in binary system? Yes, all mathematical operations can be performed in binary system. Can any number be used as the base of a number system? Yes, any number can be used. What is the octal number system? Octal system has 8 as its base or radix. The basic digits of this system are 0, 1, 2, 3,4, 5, 6 and 7. Octal number system to the base of eight is used in computing. The highest digit that can appear in octal system is seven. Where is octal system used? Octal system is used in microprocessors which are the hearts of calculators, computers, microwave ovens, etc. What is the hexadecimal number system? The hexadecimal number system, popularly known as Hex system, contains sixteen symbols and therefore, has 16 as its base. What is the advantage of binary system over decimal system? In binary system, one has to learn only four addition facts and four multiplication facts, while in decimal system one has to learn 100 addition facts and 100 multiplication facts. Therefore, binary system is the simplest system. 35

What are the different 16 symbols used in Hex system? The sixteen symbols used in this system are decimal digits 0 to 9, and alphabet A to F. Computers can be programmed in Hex system also. What do the letters A to F represent in Hex system? Letter A represents 10, B represents 11 and so on. How does the base of a number system is identified? The base in which a number is written can be identified by a 'subscript', i.e. by writing the number of the base in words or figures below and to the below of final digit. The numbers written without a subscript are normally assumed to be in base 10. What is the name of the base 60 number system? Base 60 number system is known as the sexagesimal system. Where is sexagesimal system used? It is used for expressing time (60 seconds = 1 minute, 60 minutes = 1 hour), and for measuring angles. Angles are also measured in minutes and seconds. One degree = 60 minutes and one minute = 60 seconds. How is the value of a digit determined in positional notational system? The exact position of a digit determines its value, e.g. in the number 33 the left hand digit has 10 times the value of the right hand digit. Base 2 system is known as binary system, base 10 is known as decimal system. What names will you give to the systems with base 3,4, 5, 6, 7, and 9 respectively? They are given the names Ternary, Quaternary, Quinary, Senary, Septenary and Nonary number systems. Why are zeros used? Zeros are used to keep the positions and values of other digits correct. For example, without zeros the numbers 30300 and 33 would appear to have the same value. How does the place value vary in a numeral? The far right place in any numeral is the unit or one position. The place 36

value of next position to the left is ten. In general, the place values from right to left, in any decimal numeral, are — ones, tens, hundreds, thousands and so on. For example, in decimal numbers 2,465 the 2 represents two thousand, the 4 represents four hundreds, 6 represents six tens and the 5 represents ones. Find out the place values of 4,081? The place values represented by 4,081 are: 4,081 = (4 x 1000) + (0 x 100) + (8 x 10) + (1 x 1). How can a number be changed to base 10? To change a number from any other base to base 10, multiply each digit in the number by its place value and add the products together. Change 5426 eight to its base 10 equivalent? 5426 • ht = (5x8 3 )+(4x8 2 )+(2x8)+(6x8°) = 2560 t e n + 256 t e n + 16 t e n + 6 t e n = 2838 ten . How can a number be changed to another base from base 10? To change a number from base 10 to another base, divide the base 10 number repeatedly by the new base and record the remainders in reverse order. How does the place value of a numeral vary after the decimal point? The decimal point separates the number 1 and above from numbers less than 1. Each column to the right of the decimal point has one tenth of the column before it. These place values are expressed in negative powers of 10. How can we write the place values of 0.0291? The place values represented by 0.0291 are: 0.0291 = (OxlO-1) + (2xl0- 2 ) + (9 x 10"3) +

37

(\xl0~4).

Find the equivalent of 564 ( ] 0 ) in base 6? 6

564 ten

6

94 remainder 0

6

15 remainder 4

6

2 remainder 3 0 remainder 2

So, 564 ten = 2340 s How is a decimal numeral changed into its equivalent binary form? To convert any decimal numeral into its binary form, divide it successively by 2 and record the remainders in reverse order. For example, if we divide 8 ten successively by 2, we obtain: 2

8

2

4-0

2

2-0

2

1-0 0-1

So, 8 t e n = 1000 two How are binary numbers added? When binary numbers are added we expect the sum to represent the same natural number as if the numbers were expressed in base ten. In binary addition 1+0=1, 0+0=0 and 1 + 1 = 10. 1 of 10 is carried over to the left. What will be the result of addition of 11 (2) and 10(2)? 10+11=101 Discussion: 0 + 1 = 1, so we write 1 in one's place in the sum. Then 1 + 1 = 10. Write 0 in two's place in the sum and 1 in four's place. 38

Add 11010 and 10111? 11010+ 10111 = 110001 Discussion: 0+1 = 1, write in one's place in the sum. 1+1=10, write 0 in two's place and remember 1 four. 1 + 0 + 1 = 10; write 0 in four's place and remember 1 eight. 1+1=10; write 0 in eight's place and remember 1 sixteen. 1+1+1=11; write 1 in sixteen's place and 1 in thirty two's place. Is subtraction in the binary system different from decimal system? No, it is almost the same, what is needed is some additional flexibility with binary notation. Subtract 111 from 10110? 10110 - Ill 1111 Discussion: 1 is greater than 0. Change 1 two into '10' ones. 1 from 10 is 1. Change 1 fourto '10' twos. 1 from lOis 1. Change 1 siyceeninto '10' eights, and change it to '10' fours. Now we complete the subtraction. 1 from 10 is 1.0 from 1 is 1. The difference is 1111. How is binary multiplication done? Binary multiplication is the same as decimal multiplication ; 1x0=0 and 1x1=1. Subtract 101 from 110? 110 -101

1 Discussion: We cannot subtract 1 from 0 one's. So, we take 1 unit from the next position (two's) and think of it as '10' ones. 1 one from '10' ones leaves 1 one; write 1 in one's place in the difference. 0 twos' subtracted from 0 two's leaves 0 two's; 1 four subtracted from 1 four leaves 0 fours.

39

How to multiply 10 by 11? 10 xll 10 lOx 110 Discussion: 1 x0=0; write 0 in the product. 1x1=1; write 1 to the left of 0 in the first partial product. 1x0=0; write 0 under 1 in the second partial product. 1x1=1; write 1 to the left of the 0 just written. Now add. The product of 10 and 11 is 110. Multiply 1101 by 101? 1101 xlOl 1101 0000 x 1101 xx 1000001 Discussion: For the first partial product 1x1=1, 1x0=0, 1x1=1 and 1x1=1. Every entry in the second partial product is zero. The entries in the third partial product are the same as in the first. The addition is involved but not overly difficult. Divide 11011 (2) by 11(2) 1001 11 I 11011 11 0011 It 0 40

Discussion: 11 is contained in 11 just 1 time, l x l 1=11. Subtract and get 0 for the difference. Bring down 0. 11 is contained 0 time in 0; write 0 in the quotient. Bring down 1. 11 is contained 0 time in 1; write 0 in the quotient. Bring down 1.11 is contained 1 time in 11. The remainder is 0. The answer of division is 1001. How are binary fractions converted into decimal equivalent? The position to the right of the 'binary point' has a value of 1/2, the next position has a value of 1/4; the next has a value of 1/8; and so on. Change 1.1011(2) to its decimal equivalent? 1.1011 = l x l + 1 x V2 + 0 x V 4 + 1 x V 8 + 1 x V16 = =

l + V2 + o + V8 + V16 l + l»/16orl.6875(10)

How would a decimal fraction be converted into its binary equivalent? First the fraction is written down. Then, it is multiplied by 2 using base ten multiplication fact. Now remove the integer to the left of the decimal point and record it. The process is repeated until the desired number of places in the binary fraction is obtained. The binary equivalent is obtained by reading the recorded digits from top to bottom. Convert (10110111) 2 to the equivalent base eight numeral. (10) (110) (111) = (267)g 2 eight 6 eight 7 eight How would you convert 0.721 (10) into its binary equivalent? 0.721x2 1.442 1 0.442x2 Repeating we have 0.884 0 0.884x2 1.768 1 0.768x2 1.536 1 0.536x2 1.072 1 0.072 Thus, 0 . 7 2 1 ( 1 0 ) = 0.10111 ( 2 )

41

How binary numerals are converted to octal equivalent? A number expressed as a binary numeral is readily expressed as an equivalent in base eight. For a natural number the technique is simple. Since 2 3 =8, we begin with the smallest unit, and group the 'digits' in clusters of three 'digits' each. Convert (.1101101) 2 to base eight? ,(110)(110) (100) 2 = 0.664 eight How to convert (8D) Hex into its decimal equivalent. (8D>Hex

= = =

SX^+DX^ 128 + 1 3 x 1 1 2 8 + 1 3 = 141.

0

How can we write the equivalents of 1980 ten to other bases? 1980 10 111 10111ioo 2 2201100 3 1323304 30410 5 131006 5526 7 3674 8 2640 9 7BCHex. What is EBCDIC code? EBCDIC stands for Extended Binary Coded Decimal Interchange Code. This code is used in a large computer system and consists of eight bits. It is related to punched cards, codes, which make hardware implementation very simple. It uses BCD as the basis of assignment. Hexadecimal system makes use of the digits — 0 through 9 and A through F. Based on these digits which table is used to carry out addition and subtraction in Hex? The given table in the next page is used for carrying out addition and subtraction in Hexadecimal system.

42

Hex Addition and Subtraction Table 0 0 1 2 3 4 5 6 7 8 9 A B C D E F

1 2 3 3 1 2 2 3 4 5 3 4 5 6 4 5 6 7 8 6 7 7 8 9 8 9 A 9 A B A B C B C D C D E D E F E F 10 F 10 11 10 11 12"

4 4 5 6 7 8 9 A B C D E F 10 11 12 13

5 6 5 6 7 6 7 8 8 9 9 A A B B C C D D E E F F 10 10 11 11 12 12 13 13 14 14 15

7 7 8 9 A B C D E F 10 11 12 13 14 15 16

8 8 9 A B C D E F 10 11 12 13 14 15 16 17

9 9 A B C D E F 10 11 12 13 14 15 16 17 18

A A B C D E F 10 11 12 13 14 15 16 17 18 19

B C D B C D C D E D E F E F 10 F 10 11 10 11 12 11 12 13 12 13 14 13 14 15 14 15 16 15 16 17 16 17 18 17 18 19 18 19 1A 19 1A IB 1A IB 1C

E E F 10 11 12 13 14 15 16 17 18 19 1A IB 1C ID

F F 10 11 12 13 14 15 16 17 18 19 1A IB 1C ID IE

What is BCD? BCD is the short form of Binary Coded Decimal. In BCD six bits (binary digits) are used to represent any character of the character set. What is ASCII code? ASCII code (American Standard Code for Information Interchange) uses 8 bits to represent any character of the character set. This code allows manufacturing to standardize keyboards, printers, video displays and so on. What is chunking? Chunking is a procedure of shortening the strings of binary digits or replacing longer strings of data with shorter one. How would you find the sum of 6 and A from the Hex table? First find the row that begins with 6. Then run your finger to the right along this row until you come to the column that has A at the top. Your finger will be at 10 (meaning 1 sixteen plus no ones) the sum of 6 plus A43

How would you find the sum of 2C and 81 in Hexadecimal notation? Following the Hex table sum will be calculated as follows: Hexadecimal

Meaning

2C + 81

2 sixteen + C ones 8 sixteens + 1 One

AD

A sixteens + D ones

Decimal addition 44 + 129 173

First use the fact C+1=D to add the ones. Then use the fact that 2 + 8 =A (2 sixteens + 8 sixteens = A sixteens). How would you add 6E and 34 in Hexadecimal notation? 6E and 34 can be added as follows: Hexadecimal Meaning Decimal notation addition 6E 6 sixteens + E ones +34 3 sixteens + 4 ones 110 A2

9 sixteens + 1 2 ones +52 on regrouping 162 (9+1) sixteens+2 ones A sixteens +2 ones In the ones place E+4=12 as shown in the table. Regroup 12 ones into 1 sixteen +2 ones. Write 2 in the ones place and a small 1 in the sixteens place over the 6. Then add the sixteens 6+3=9 sixteen; 9 sixteens +1 sixteen, we get A sixteen; by regrouping. Therefore sum will be A2. How would you use the table for finding the value of 12-D? First find the volume that begins with D. Next run your finger down this column until you come to 12. Then run your finger to the left along row to the first digit to find the answer 5. How would you find E8 — B6 with the help of the table? Hexadecimal subtraction E8 -B6 32

Meaning E sixteens+8 ones B sixteens+6 ones 3 sixteens +2 ones

Decimal subtraction 232 -182 50

Subtract the ones using the fact 8-6=2. In the sixteens place, use E-B=3. Answer is 32. 44

How would you find the product of 5x9 from the Hex multiplication table? First find the row that begins with 5. Then move your finger to the right and stop at the column that has 9 at the top. Your finger should be on 2D (2 sixteens +D ones). How would you find the value of 56x8 with the help of multiplication table? 56x8 can be found as follows: Hexa decimal Meaning Decimal multiplication 56 x8

5 sixteens + 6 ones 8 ones

86 x8

2B0

28 sixteens + 30 ones 688 after regrouping 2B sixteens + 0 ones In the ones place, use the fact 6x8 = 30. Regroup 30 ones as 3 sixteens and 0 ones. Write 0 in the ones place and a small 3 in the sixteens place above the 5. Then multiply 5x8 to get 28 and add the 3 to get 2B. Since 5 was in the sixteens place, the final result has B in the sixteens place and 2 in the sixteens times sixteens place. When the multiplier has more than one digit, repeat the operation for each digit and add the products. How would you divide B into 58? First find the column that begins with B. Run your finger down this column until you come to 58. All the way to the left of this row you will find the answer 8. How would you divide D8 by 8? Division will be carried out as follows: Hexadecimal Meaning division IB

1 sixteen + 8 ones

58 58

27 8_y216"

/

8

Decimal division

8 ones / D sixteen + 8 ones 8 sixteens 5 sixteens + 8 ones 5 sixteens + 8 ones

x 45

16

56 56 x

D sixteens divided by 8 is 1 sixteen. Write 1 above the D. Multiply 1 x 8 and write the product 8 below D'. Subtract 8 from D, using the hexadecimal addition table. Complete the problem as you would do in decimal division but use hexadecimal arithmetic facts. Which table is used to find multiplication and division in hexadecimal system? Following table is used to calculate multiplication and division in hexadecimal notation. Hex Multiplication and Division Table X

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

1

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

2 2

4

6

8

A

C

E

10

12

14

16

18 1A 1C IE

3 3

6

9

C

F

12

15

18 IB

IE

21 24 27 2A 2D

4

4

8

C

10

14

5 5

A

F

14

19 IE

6

6

C

12

7 7

E

15 1C 23 2A

8 8

10

18 20 28

30 38 40 48

9 9

12

18 24 2D

36 3F 48 51 5A 63 6C 75 7E

A A

14 IE

B B

16 21 2C

C C

18 24

30 3C 48 54

60 6C 78 84 90 9C A8 B4

D D 1A 27

34 41 4E 5B

68

E E 1C 2A

38 46 54 62

70 7E 8C 9A A8 B6 C4 D2

18 IE

18 1C 20 24 28 2C 30 23. 28 2D

24 2A 31

28 32 3C 46 37 42 4D

F F IE 2D 3C 4B 5A 69

30

34 38 3C

32 37 3C 41 46 4B

36 3C 42 48 4E

54 5A

38 3F 46 4D 54 58 62

50 5A

50 58 60 68 70 78

64 6E 78

58 63 6E 79

78

69

82 8C

87 96

84 8F 9A A5

75 82 8F 9C A9 B6 C3

87 96 A5 B4 C4 D2 El

ooo 46

5. Set Theory What is a set? Set in mathematics is stated as any collection of difined things (elements) provided the elements are distinct and that there is a rule to decide whether an element is a member of a set. In other words, "A set is simply a collection of ideas". The concept of sets was developed in 1874 by a German mathematician, Georg Cantor. Give some examples of sets? A herd of cows, a flock of words, a shoal of fish, a bunch of bananas, a bunch of grapes, three pineapples, etc. are the examples of sets (Fig. 5.1).

Fig. 5.1 Set of bananas, grapes and pineapples 47

What is the symbol for 'is a member o f ? The symbol for 'is a member of is o For example, the set of human beings c the set of animals. Similarly, the set of odd natural numbers { 1, 3, 5, 7, ...} c the set of natural numbers {1, 2, 3, 4, 5, 6,...}. What is the complement of a set? A subset together with its complement makes the whole set. For example, the complement of the set of odd natural numbers is the set of even natural numbers {2,4,6,...}. If a set is denoted by A, its complement is denoted by A'. What is the meaning of 'union of sets'? The union of given sets is the combination of the sets. In other words, the set composed of two sets lumped together is called the union of two sets. For example, the union of the set of two dogs and the set of three dogs is the set of five dogs (Fig. 5.2). Symbol of union is U.

Fig. 5.2 Union of sets How do we explain the intersection of two sets? If some elements of one set are also elements of another then those elements are called the intersection of the two sets. For example, Ram and Meera are brother and sister. They have certain things in common. These things make intersection of the set of things (See fig. 5.3 next page).

48

Fig. 5.3 Intersection of two sets What is a null or empty set? A set with no element is called an empty set. This is the union of two sets, which have no common part. For example, in Fig. 5.4, the boys are wrestling and the girls are skipping. Here the set of girls and the set of boy s have no common part. So, their union represents an empty set. It is denoted by 0 and should not be confused with {O}, which is a set with one number Zero.

Fig. 5.4 Null set What is a universal set? A universal set is a fixed set, such that all the sets under consideration at a particular time are subsets of this universal set. It is denoted by U. The domain may contain elements in addition to all those under consideration. 49

What do we call each member of a set? Each member of a set is called an 'element' of the set. For example, in a herd of cows, each cow is an element. What are the notations used to represent sets? Two kinds of notations are used for sets: (a) enumeration notation, and (b) set builder notation. How do we represent sets in these two notations? In enumeration notation we write a set of the numbers 1 and 3 as (1,3) and first two prime ministers as (Nehru, Shastri). In set builder notation we write the set of whole numbers as {x/x is a whole number, etc.}. Nowadays, set builder notation is most frequently used. How do we represent a subset? Suppose {a, b, c} is a set then {a, b} is the subset of {a, b, c}. We represent it as {a, b} c {a, b, c}. When do we say that two sets are equal? When we say that two sets, A and B, are equal (written as A=B), we mean that every member of A is a member of B and vice versa. For example, (5,7) = (7, 5). Order of listing the members is not significant. What is a subset? A set whose members are all members of another set, termed as subset is called a subset of the set. For example, animals in a wildlife park make a set and all the birds or lions are the subset of the set of animals. What are the rules for the union of sets? There are eight rules related to the union of sets. 1. A u A = A 2. A u B = B u A 3. A u (B u C) = (A u B) u C 4. A u (B n C) = (A u B) n (A u C) 5. A u 0 = A 6. A u A' = U 7. U' = 0 8. A u U = U

What are the rules for the intersection of sets? Following are the rules for intersection of sets: 1. A n A = A 2. A n B = B n A 3. A n ( B n C ) = ( A n B ) n C 4. A n (B u C) = (A n B) u (A n C) 5. A n U = A 6. A n A' = 0 7. 0 = U 8. A n 0 = 0 9. (A1)' = A What is a finite set? A finite set is a set that has no proper subset equivalent to it. It has a limited number of members such as the letters of the alphabet. What is a single-element set? It has only one number such as days of the week beginning with M written as (Monday). What is an infinite set? An infinite set is a set that has at least one proper subset equivalent to it. It has an unlimited number of members such as all whole numbers. How do we make a universal set in a Venn diagram? A region enclosed by a rectangle represents the universal set (Fig. 5.5). How do we represent a subset on a Venn diagram? Regions inside the rectangle that are enclosed by circles represent subsets of the universal set. Figure 5.6 represents a subset.

u

Fig. 5.5 Universal set

Fig. 5.6 Sub set in a Venn diagram 51 i

How do we represent equal sets on Venn diagrams? A circle in the rectangle is labelled with two or more letters (Fig. 5.7). Each letter stands for a set. The single circle shows that each set has exactly the same numbers.

u

Fig. 5.7 Equal sets What are Venn diagrams? Venn diagrams in mathematics are the diagrams representing the set or sets and the logical relationship between them. The method is named after the British logician John Venn (1834-1923). How do we represent two overlaping sets with the help of Venn diagram? By drawing overlaping circles as shown in fig. 5.8. This figure shows that some of the members of A also belong to B, i.e., an area of overlap between two circles (sets) containing elements that are common to both sets.

U

Fig. 5.8 Overlaping set

How do we represent disjoint sets with the help of Venn diagram? By drawing separate circles as shown in fig. 5.9. In other words, circles that do not overlap represent sets with no elements in common.

m

Fig. 5.9 Disjoint set On Venn diagram, how complement of a set is shown? It is shown by the shaded portion on the Venn diagram as shown in fig. 5.10. The shaded portion represents f he set of numbers that belong to U 52

X' Fig 5.11

but not to X. This set is called the complement of X. The symbol X' stands for the complement of X. Where do we use set theory? Set theory is used in solving several problems of arithmetic, algebra, geometry and logic. How do we represent the union of set S and set T by Venn diagram? Figure 5.11 shows the union of set S. >. and set T.

U

SuT Fig. 5.11

How complement of the union of set S and set T is represented by Venn diagram? See Fig. 5.12.

SuT' Fig. 5.12 How intersection of set S and set T represented with the help of Venn diagram? (See Fig. 5.13). SnT Fig. 5.13 How complement of the intersection of set S and set T is represented on Venn diagram. (See Fig. 5.14).

ooo

SnT Fig. 5.14

53

6

Arithmetic What is arithmetic? Arithmetic is a branch of mathematics in which computations are carried out with the numbers of the decimal system. It is originated from Greek word Arithmetika, which means 'the number science'. The origin of arithmetic goes back to 6th century B.C. Until 16th century arithmetic was viewed as the study of all properties and relation of all numbers. What are the fundamental operations in arithmetic? In modern times, it usually denotes the study of Real numbers and Zero under six fundamental operations: Addition, subtraction, multiplication, division, involution (obtaining powers of numbers) and evolution (obtaining roots of the numbers). How do you define addition operation? Addition represents the grouping or bunching together of numbers (Fig. 6.1).

rrrrr ^ rrr - rrrrrrrr Fig. 6.1 Addition method

54

What is subtraction? In subtraction we take out one or more objects from another group of objects (Fig. 6.2).

:

rrrrrrrr" rrr rrrrr

I c Q j ^ b ^ ^ l I. I I I I I Fig. 6.2 Subtraction method What is multiplication? Multiplication is actually a form of addition in which the answer to a problem is obtained by simple addition. It is also a way of combining two numbers to obtain a third by 'x' or merely the juxtaposition of the numbers where suitable. For example, 5x3 means that five is to be added three times (Fig. 6.3).

5x3 5x3 = 15 Fig. 6.3 Multiplication method

55

Who introduced the sign of multiplication? William Oughtret. What is division? Division is a kind of subtraction. If we divide 24 by 8, we want to know how many times 8 goes into 24. Division is also the inverse operation of multiplication, the determination of the number of times one number must be multiplied to equal another number. What is involution? Involution means raising a number to any desired power. The number that is to be raised to the power in question is called base and the power is called exponent. For example, 2 to the power three or 2x2x2 is written as 2 3 . Here, 2 is the base and 3 is the exponent. What would be the result if any number is raised to the power zero? The result is always one. What is evolution? Evolution is the process of finding the roots of numbers. /

What is the square root of 16? The square root of 16 is 4. Square root is always obtained by finding out a number which is multiplied by itself to give that number whose square root is being determined. What are imaginary numbers? The numbers which are multiples of V^T are called imaginary numbers. What do we write for V - l It is written as i (iota)

?

How do we define percentage? Percentage is a rate or proportion per hundred. It comes from the Latin phrase 'per centum' meaning 'by the hundred'. What is an integer? A whole number is called an integer. 56

What are negative integers? A whole number having the negative sign before it, is called negative integer. Where do we use arithmetic? Arithmetic is used in science, business, and in our everyday life. What is the other name of irrational numbers? They are also called transcendental numbers. Can you give one example of an irrational number? a/~2~ is an irrational number. 37 V3T ^PT etc., are also irrational numbers. What is usually taught in arithmetic? Every one who goes to school learns arithmetic. In arithmetic, the child is taught addition, subtraction, multiplication, division, involution, evolution, fractions, percentage, profit and loss, interest, LCM, HCF, etc. What were the first printed books on arithmetic? The first printed arithmetic book was published in Italy in 1478. The first one printed in England was De arte Supputandi (1522) written by Bishop Guthbert. The first book in English was the ground of Artesticals published in 1540. The first modern book on the theory of numbers was disquistional. Arith Medicae by Karl Friedrich Gauss was published in 1801. Were do we find the earliest written evidence of arithmetic? It is found in Babylonian tables and in Egyptian handbooks of the 12th dynasty (1991-1786 B.C.). What is a divisor? A divisor is the number by which another number, the divident, is divided. For example, in a fraction such as a/b, the divisor is b. The divisor "b" is also called as denominator.

ooo

57

7. Algebra What is algebra? The branch of mathematics dealing with the relationships and properties of number systems by use of general symbols (such as a, b, x, y) to represent mathematical quantities. How did the word algebra originate? The word algebra originated in the 9th century A.D. from the title of the work on algebra by a Persian, Mohammed-ibn-Musa Al-Kwarizmi. He wrote a book in Arabic called Kitab-Al-Jebr w'al muqabala, which means restoration and reduction. In the course of time, the word al-jebr was changed to algebra. In its modern form, algebra was invented by the French mathematician, Francois Viete in the 16th century. Who was the first to work on the problems of equations? A Greek mathematician, Diophantus, in the third century A.D. attempted some problems of equations. In fact, the basis of algebra were familiar in Babylon during 2000 B.C. and were practised by the Arabs in the middle ages. Who used the vowels and the consonants for the first time in algebra? Francois Vieta, a French mathematician of 16th century used the vowels a,e,i,o,u and the consonants b,c,d,f,g to represent unknown numbers. What are the three fundamental laws in algebra, which govern the addition, subtraction, multiplication and division of all numbers? The three fundamental laws are : a. Commutative laws of addition and multiplication, b. Associative laws of addition and multiplication, c. Distributive laws of multiplication. 58

Who proposed the system of algebraic symbols? The great 17th century French philosopher, Rene Descartes proposed the system of algebraic symbols now in use. What was the system of Descartes? In his system, a, b, c, etc.,represent the fixed numbers and x, y, z stand for the unknown numbers in a problem. What are the commutative laws of addition and multiplication? According to the commutative law of addition, a+b=b+a and according to the commutative law of multiplication axb=bxa. What are the associative laws of addition and multiplication? According to the associative laws of addition and multiplication, a+(b+c)= (a+b)+c and ax(bxc)=(axb)xc. What is the distributive law of multiplication? According to the distributive law of multiplication, ax(b+c)=axb+axc. Can we find out HCF and LCM of algebraic expressions? Yes, HCF and LCM are found out by following the methods of arithmetic. What is the meaning of a 2 and a 3 ? The meaning of a 2 is that 'a' is being multiplied with 'a' two times, i.e. a 2 =axa. a 3 means that 'a' is being multiplied with 'a' three times, i.e. a 3 =axaxa. What is the value of any number raised to the power zero? Any number raised to the power zero is equal to one, e.g. a°=l. What is the value of a ni +n? am+n =

a

m

x

an

What is the meaning of (a m ) n ? (a m ) n is equal to a m n , e.g. (a 2 ) 3 = a 6 . What is the value of a-™?

59

What is the value of (ab) m ? (ab) m = a m b m What is the value of a m /a n ? _ ^m ^

n _ ^m-n

a" When + is multiplied by +, what is the result? When + is multiplied by + the result is +. When - is multiplied by what is the result? When - is multiplied by - the result is +. When + is multiplied by what is the result? When + is multiplied by the result is - . What are complex numbers? The numbers which contain real part and imaginary part are called complex numbers. What is a conjugate complex number? If a+ib is a complex number then a - ib is called its conjugate complex. We obtain conjugate complex by changing the sign of imaginary part. When the two complex numbers are said to be equal? When the real and the imaginary parts of a complex number are equal to the real and the imaginary parts of the other complex number, they are said to be equal. Who devised the method of algebric reasoning during 19th century? The British mathematician, George Boole used the method first, in working out construction of computers. What is the inverse Operation? If two operations negate each other, they are termed as inverse operation. Addition and subtraction are inverse operations. What is an additive inverse of a complex number? It is that complex number which when added to a given complex number the result is zero. 60

What is multiplicative inverse? It is a complex number which when multiplied with the given complex number the result is zero. What are the cube roots of unity? 1, w, w 2 are the cube roots of unity, where, -

1 +

a n d

2

w

2

=

2

The sum of 1+w+w 2 = 0 and their product is equal to one. What is an equation? An equation may be looked upon as a balance with equal numerical values on each side of the 'equal' sign (=). (Fig. 7.1).

Fig. 7.1 An equation How many equations are required to solve 'n' unknowns? We need 'n' equations to solve 'n' unknowns. In other words, the number of equations should be equal to the number of unknowns. What are the cubic equations? The cubic equations are those in a single variable which appears to the power 3, but not higher. What is the degree of an equation? Generally, the degree of an equation is defined as the sum of the of the variables in the highest p o w e r term of the equation. 61

exponents

How operations are carried out on equations? All operations of addition, subtraction, multiplication or division are carried out on all members of an equation, meaning that we should add or subtract or multiply or divide the whole equation by the same quantity. What is a linear equation? The linear equation is that in which no variable term is raised to a power higher than 1 (one). What is a quadratic equation? If the highest power of the unknown quantity is two, it is called a quadratic equation. A quadratic equation always has two roots; What is an algebraic identity? When an equation is true for all the replacement values of the variables concerned, it is called an identity. Which is the most familiar identity? The most familiar identity is (a+b) 2 = a 2 + 2ab+b 2 . What is the value of (a-b) 2 ? (a-b) 2 = a 2 - 2ab + b 2 What are the factors of a 2 -b 2 ? a2-b2=(a+b)(a-b) What is the value of (a+b) 3 ? (a+b) 3 = a 3 + 3 a 2 b+3ab 2 +b 3 . What is the value of (a-b) 3 ? (a-b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3 . What are the factors of a 3 -b 3 ? a3—b3 = (a-b) (a 2 +b 2 +ab). What are the factors of a 3 +b 3 ? a 3+ b 3 = (a+b) (a2 - ab + b 2 ). What is harmomic series? It is an inverse of arithmetic series. 62

What are sequences and series? A sequence is a succession of numbers and a series is a sum of numbers in a sequence. What is an arithmetic series? In a series, if the difference of any two consecutive terms is the same, it is called an arithmetic series, e.g. 2, 4, 6, 8, 10, ... . Most series are "infinite" — containing an infinite number of terms. Imagine a cement foundation that is 16cm above the level of the ground. On this foundation, you build up 6 layers of stone blocks. Each layer is 8cm thick. As you add each layer of blocks the height of the pile becomes larger. What would be the form of equation if x represents the number of layers and y represents the height of the pile? It will be a linear equation y=8x+16 (Fig. 7.2).

16 cm Cement foundation

Fig. 7.2 f'tone blocks

63

What is a geometric series? In a series, if the ratio of any two consecutive terms is constant, it is called a geometrical series, e.g. 2, 4, 8, 16, 32,... What are the uses of algebra? Algebra has proved very useful in solving the problems of physics, chemistry, engineering, finance, probability, equations, etc. How would you solve the two equations 2y=x+4 and y=5-x graphically? Take the equation 2y=x+4 and give different values to x and correspondingly find the values of y x

0

2

4

y

2

3

4

Do the same for the equation y=5-x x

0

3

5

y

5

2

0

Draw the graph (Fig. 7.3). The point of intersection represents the values of x and y. Value of x= 2 and y =3

Fig. 7.3 64

Some terms used in Alebgra Absolute value

It is the size of a number when the number is positive or negative.

Coefficient

It is the multiple of a variable or number usually written next to the variable.

Constant

It is a number or variable whose domain is a set of one number.

Equation

In equation, function put equal to a constant, in particular zero.

Expression

Expression in algebra is a set of numbers or variables combined by operations such as addition, subtraction, multiplication or division.

Function

It is the combination of symbols standing for quantities.

Identity

The identity is an equation that holds for all values of the variable.

Monomial

It is an expression in algebra consisting of a product of numbers and variables.

Variable

It is a symbol in algebra usually a letter that can be replaced in one or more numerical values. o o o

65

6

Plane Geometry What is plane geometry? Plane geometry is the branch of mathematics which deals with points, lines, angles, triangles, quadrilaterals, circles, etc. The word 'geometry' has originated from the Greek word geometria which means earth measurement. Who was Thales? Thales (640 - 546 B.C.) was a Greek mathematician and scientist. He made advances in geometry and predicted sun's eclipse in 585 B.C. He discovered that no matter what diameter one draws in a circle, it always cuts the circle into two halves. Who is called the father of geometry? The learned Greek mathematician, Euclid, who taught Geometry at the Museum of Alexandria in Egypt about 300 B.C. (Fig. 8.1). His major work the Elements is still the basis of much of geometry. Fig. 8.1 What is a theorem? A theorem is a statement that gives certain facts about a figure and one concludes f r o m these facts that certain other fact must be true. Advanced mathematics consists almost entirely 66

Fig. 8.1 Euclid

of theorems and proofs, but even at a simple level theorems are very important. What was the contribution of Euclid to geometry? He systematised the total knowledge of geometry and presented it in 13 volumes called Elements. These volumes contain information about points, lines, circles, triangles, methods of making geometrical figures, theories of ratio and proportion and various theorems of geometry. The last three volumes deal with solid geometry. The solid and plane geometries are together called Euclidean geometry which is based on self evident postulates. His main work lay in the systematic arrangements ofprevious discoveries and the geometrical book remained a standard textbook for over 2000 years and are still in regular use today. What is a postulate in geometry? In geometry, one cannot prove certain statements of relations between figures. Such statements are called postulates. One example of a postulate is,"only one straight line can be drawn between two points". How do we define a point in geometry? A point is the simplest element in geometry. It has neither length nor width or thickness. A geometric point is impossible to make because it has no dimensions. What is a line? A collection of points is called a line and the shortest distance between the two points is called a straight line. The line has only length. It does not have any width or thickness (Fig. 8.2). A»

B «

C •

Fig. 8.2 A line

67

D •

What is a ray? The name ray is given to the part of a line that starts at a given point. What are parallel lines? The lines which meet at infinity are called parallel lines (Fig. 8.3). Railway tracks are parallel lines. What is a plane? A plane is a surface having two dimensions. For example, the surface of a table top is a plane. Fig. 8.3 Parallel lines How angles are named according to their sizes? Angles are named as follows (Fig. 8.4): Acute angle — It measures more than 0° but less than 90°. (b) Right angle - It measures 90°. (a)

(c)

Obtuse angle - It measures more than 90° but less than 180°.

(d) Straight angle - It measures 180°. (e)

Reflex angle - It measures more than 180° but less than 360°.

What is an angle? An angle in mathematics, is the amount of turn or rotation; it may be defined by a pair of rays (half-line) Fig. 8.4 Different angles

that share a common end point but do not lie in the same line. In Fig. 8.5, AB & BC are two rays with the same standing point. The angle fnrmed bv these two ravs is ABC.

Fig. 8.5 Angle formed by two rays, AB and BC What are the complementary angles? The complementary angles are two angles whose sum is 90° (Fig. 8.6).

Fig. 8.6 Complementary angles What are the equal triangles? The triangles having the same size or area are called the equal triangles. What are the supplementary angles? The supplementary angles are two angles whose sum is 180° (Fig. 8.7).

What are the conjugate angles? The conjugate angles are two angles whose sum is 360° (Fig. 8.8). What is a triangle? A plane figure bounded by three line Fig- 8.8 conjugate angles segments (Fig. 8.9) is called a triangle. The sum of the internal angles of a triangle is always 180°. Vertex

Fig. 8.9 A triangle How are triangles classified? Triangles are classified as: (a) Acute angled triangle: A triangle with three acute angles, (i.e. less than 90°) (b) Right angled triangle: A triangle that contains one right angle. The hypotenuse is the side opposite the right angle. (c) Obtuse angled triangle: A triangle that contains one obtuse angle, (i.e. more than 90°) All the three triangles are shown in Fig. 8.10.

Fig. 8.10 Acute, right and obtuse angled triangles 70

What is an equilateral triangle? In an equilateral triangle, all the sides are of the same length and all the angles are equal to 60° (Fig. 8.11) What is an ellipse? An ellipse is acurve, joining all points around two fixed points so that the sum of the distances from those points is always constant. Those two points are called/oci of the ellipse.

Fig. 8.11 Equilateral triangle

What is an isosceles triangle? A triangle whose two sides are equal and two angles are the same or equal. (Fig. 8.12). What is scalene triangle? A triangle whose all sides and angles are of different sizes (Fig. 8.13).

Fig. 8.13 Scalene triangle Fig. 8.12 Isosceles triangle

I

Which is the most famous theorem of triangles? In geometry, the most famous theorem is the Pythagoras theorem which states that in a rightangled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of lengths of the other two sides. (Fig. 8.14) Fig. 8.14 Pythagoras theorem 71

What are similar triangles? The triangles having the same shape but different size are called similar triangles (Fig. 8.15).

Fig. 8.15 Similar triangles What is a median? A line which joins the mid point of one side of a triangle to the opposite vertex is called median (Fig. 8.16).

Fig. 8.16 Medians of a triangle

Fig. 8.17 A quadrilateral

What is a quadrilateral? A quadrilateral is a plane figure with four angles and four sides. The sum of the four interior angles of a quadrilateral is 360° (Fig. 8.17). Two intersecting lines are called its diagonals.

How quadrilaterals are classified according to their shape? Quadrilaterals are classified as follows: (a) Square (b) Rectangle

All sides are equal and all angles are of 90°. Opposite sides are equal and all angles are of 90°.

(c) Rhombus

All sides are equal but none of the angles is 90°. Opposite sides are parallel and equal.

(d) Parallelogram 72

(e) Trapezium

:

(f) Kite

:

(g) Arrowhead shape :

One pair of opposite sides is parallel. Adjacent sides are of the same length and the diagonals intersect at right angles. A special shape which is in the form of an arrow.

All these are shown in Fig. 8.18.

Fig. 8.18 Different types of quadrilaterals What is the area of a triangle? The area of a triangle is half the base multiplied by the corresponding altitude.

73

What is the area of a rectangle and a square? The area of the rectangle is equal to the length multiplied by height. Area of the square is equal to the side multiplied by side. What is a circle? A circle is the path of a point which keeps a constant distance from a fixed point. This fixed point is called centre and the path is called circumference. The fixed distance is called radius (Fig. 8.19).

Circumference

Fig. 8.19 A circle

What is the area of a circle? The area of a circle is m 2 where r is the radius of the circle. What is the area of a rhombus? The area of a rhombus is half the product of the lengths of the two diagonals. What is a field book? A book in which dimensions of fields are recorded to calculate areas is called a field book. For calculating the areas of fields we usually come, across triangles and trapeziums. What do you understand by the terms radius, diameter, chord, 1 secant, tangent and arc of a circle? (a) Radius : Distance between centre and circumference. (b) Diameter : The longest distance from one side of a circle to the' other is called the diameter. It is thus twice the radius. (c) Chord

A straight line joining any two points on the circum-i ference.

(d) Secant

A straight line that cuts across the circumference of' a circle at any two points.

(e) Tangent

A straight line that touches the circumference at onej 74

(f) Arc

point only and has the same slope as the curve at the point of contact, A section of the circumference of a circle.

All these are shown in Fig. 8.20.

Fig. 8.20 (a) Radius (b) Diameter (c) Chord (d) Secant (e) Tangent ( f ) Arc of a circle What is a semi-circle? The space between a diameter and the circumference is called a semi-circle (Fig. 8.21 A). How do we define a sector? The space between any two radii is called a sector. Semi-circle, sector and segment are shown in Fig. 8.2l|B. What is a segment? The space between a chord and the circumference. Fig. 8.21C.

Fig. 8.21 (A) Semicircle (B) Sector (C) Segment

When we put a dot with a lead pencil, is it not a geometric point? No, it is not a geometric point but a physical point. However, we make use of such points in geometry.

75

What is a parabola? The parabola is the path of a point, which moves so that its distance from a fixed line is always equal to its distance from a fixed point. In mathematics, parabola is a curve formed by cutting a right circular cone with a plane parallel to the slopping side of the cone. The path of a bullet is a parabola. What is a hyperbola? The hyperbola in geometry is a curve formed by cutting a right circular cone with a plane so that the angle between the plane and the base is greater than the angle between the base and the side of the cone. Ellipse, parabola and hyperbola are called conic sections, shown in Fig. 8.22.

Fig. 8.22 Conic sections — Ellipse, parabola and hyperbola What is a catenary? The shape or curve taken by a chain or rope hanging freely between two points at the same height is known as a catenary. A suspension bridge makes a catenary. How is an involute generated? An involute is traced by the end of a piece of a sticky tape as it is unrolled of the spool. What is a cycloid? A cycloid is the name given to the curve traced by a point on the outside

76

edge of a wheel rolling along in a straight line (Fig. 8.23).

Fig. 8.23 Cycloid What is a polygon? A plane figure enclosed by several straight lines is called a polygon (Fig. 8.24).

Fig. 8.24 Polygons What is a tetrahedron? In geometry, it is a solid figure with four triangular faces; that is, a pyramid on a triangular base. A regular tetrahedron has equilateral triangles at its faces; it can be constructed by joining four points that are equidistant from each other on the surface of a sphere. What is a centroid? The point of concurrence of medians in a triangle is called centroid (Fig. 8.25). What are the applications of geometry? Geometry is used by designers, engineers, architectures, surveyors and scientists. Fig. 8.25 Centroid 11

What is an orthocentre? The point of concurrence of altitudes in a triangle is called an orthocentre (Fig..8.26). What is a circumcentre? The point of concurrence of line bisectors in a triangle is called a circumcentre (Fig. 8.27). What is incentre? The point of concurrence of angle bisectors in a triangle is called incentre (Fig. 8.28).

Fig. 8.27 Circumcentre

Fig. 8.26 Orthocentre

Fig. 8.28 Incentre. OOO

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Solid Geometry What is solid geometry? It is the study of three dimensional world, i.e. the study of the shapes which include length, breadth and thickness. Why do we study solid geometry? The two dimensional world of plane geometry is not sufficient to explain the world in which we live. It is a world of three dimensions and to understand it, we study solid geometry. What is a dihedral angle? It is the simplest angle in solid geometry and is formed by the intersection of two planes described by a point on one of the planes — the line of intersection (edge) and a point on the other plane (Fig. 9.1).

Fig. 9.1 A dihedral angle

.79

What is a polyhedral angle? If more than three planes meet at a point, the angle subtended is called a polyhedral angle (many faced angle). What is a polyhedron? Polyhedron is a three-dimensional figure bounded by four or more plane sides. What are the five common solids which are studied in solid geometry? The five common solids are: The prism, the cylinder, the pyramid, the cone and the sphere. What is a prism? It is a solid figure whose cross section is constant in planes drawn perpendicular to its axis. What is a trihedral angle? When three planes meet at a point, they form a trihedral angle (Fig. 9.2). Each of the angles making up a trihedral angle is called a face angle. In Fig. 9.2 ABC, CDB and ADB are all face angles.

Fig. 9.2 Trihedral angle

How many common shapes of a prism are known to us? The common prisms are: Triangular prism, quadrangular prism, hexagonal prism and octagonal prism (Fig. 9.3).

Fig. 9.3 Triangular, quadrangular, hexagonal and octagonal prisms What is the use of a triangular glass prism? It is used to study refraction of light. When white light passes through a prism, it is spLtted into seven colours. 80

What is the lateral and total area of a prism? The lateral area of a prism is equal to the perimeter of the base multiplied by height. On adding area of the two bases to lateral area, we get the total area of the prism. What is the volume of the prism? The volume of the prism is found out by multiplying the area of the base by the altitude. What is a cylinder? A cylinder is a tubular solid figure with a circulars base, which is obtained by rotating a rectangle completely about one of its sides. (Fig. 9.4) How do we find the lateral area and total surface area of a right circular cylinder? The lateral area of a cylinder is 2rcrh, where it is about 3.14, r is the radius of the base and h is the height. The total surface area of a cylinder is obtained by adding areas of the two circles of the bases to the lateral area (Fig. 9.5).

Fig. 9.4 A cylinder

2K r x h

S = 2 (7t r 2 + 7t r x h) Fig. 9.5 Lateral and total surface areas of a right circular cylinder.

81

What is a pyramid in geometry? The pyramid in geometry is a solid figure with triangular side-faces meeting at a common vertex (point) and with a polygon as its base. What is the volume of a pyramid? The volume of a pyramid, no matter how many faces it has, is equal to the area of the base multiplied by one-third of the perpendicular height. What are the common types of pyramids? The common types of pyramids are: Triangular pyramid, quadrangular pyramid, hexagonal pyramid and octagonal pyramid (Fig. 9.6).

Fig. 9.6 Different types of pyramids What is the frustum type of a pyramid? A pyramid with its pointed top cut off by a plane parallel to the base is a type of frustum (Fig. 9.7). What is a cone? A cone is a solid figure having a plane curve as its base and tapering to a point (the vertex) (Fig. 9.8).

A

Fig. 9.7 Frustum of a square pyramid

What is the volume of a cylinder?

The volume of a cylinder is obtained by multiplying the area of the base by height. The volume of a cylinder is given by V = nr 2 h where V is volume, r is the radius Fig. 9.8 A cone and h is the height. 82

What is the importance of volume of cylinders in our daily life? By finding out the volume of the cylinders we can know the content of tin cans, gas storage tanks, reservoirs, etc. How the volume and area of a frustum are calculated? They are calculated by subtracting the volume or area of the "missing" piece from those of the whole figure. What is the meaning of frustum in geometry? Frustum in geometry means that a 'slice' taken out of a solid figure by a pair of parallel planes. What is an axis of a cone? The line joining the vertex to the centre of the base is called the axis of a cone. What is a circular cone and a right cone? A circular cone has a circle as its base, while a cone that has its axis at right angles to the base is called a right cone. What is the frustum of a cone? If we cut a cone in two parts by passing through a plane parallel to the base, the lower part is called the frustum of a cone (Fig. 9.9). What is a sphere and what is its surface area? If a semi-circle is rotated about a diameter, the resulting figure is a sphere (Fig. 9.10). It is perfectly round with all the points on its surface, the same distance from the centre. The surface area of a sphere is 4nr 2 , where r is the radius of the sphere.

83

Fig. 9.9 Frustum of a cone

Fig. 9.10 A sphere

What is the importance of solid geometry? The solid geometry has enabled astronomers to give useful interpretation of the heavenly bodies. Measurement of areas and volumes of solid figures in our day to day life is a gift of solid geometry. What is the volume of a sphere? The volume of a sphere is -y- Ttr3. What is the volume of earth? The volume of earth = - | - x it x 6,380 3 = 1,090,000,000,000 cu. km. What is a cube? In geometry, a cube is a regular solid figure whose faces are all squares. What is the surface area and volume of a cube? It is 6 times the square of its side. It is equal to the cube of one of its sides (Fig. 9.11). For example, if the side of a cube is 4 cm, its volume will be 64 cubic centimetre. Fig. 9.11 A cube What is the volume of a cuboid? The volume of a cuboid is length x width x height (Fig. 9.12).

Fig. 9.12 A cuboid 84

What are the longitudes and the latitudes? The longitude "Lines" are circles passing through the N and S poles whose centres are the centre of the earth. All the planes that cut the earth at right angles to the axis of small circles are called latitudes. (Fig. 9.13).

Longitudes

•Latitudes

Fig. 9.13 Longitudes and latitudes What is descriptive geometry? Descriptive geometry is the presentation of methods to represent a figure in space on a plane by means of projection. How do we define differential geometry? Differential geometry is the branch of geometry dealing with the basic properties of curves and surfaces, using the techniques of calculus and analytic geometry. What is algebraic geometry? Algebraic geometry is concerned with geometric loci that can be determined by algebraic relations between coordinates of some suitable coordinate system. What is hyperbolic geometry? It is one basic type of non-Euclidean geometry. It is based on the axiom that through a point not on a given line more than one line may be drawn parallel to the given line.

85

What is elliptic geometry? It is also a basic type of non Ecuclideam geometry. It is based on the axiom that through a point not on a given line, there are no lines that do not intersect the given line. What are the main methods of projection in the descriptive geometry? There are two main systems of projection: i) The perspective view in which the object is projected from one point onto a plain. ii) The orthographic projection where the object is projected from one plane into another. o o o

86

10 Analytical and Non-Euclidean Geometry What is analytical or coordinate geometry? Analytical or coordinate geometry explains geometric figures in terms of algebraic formulae. Who was the first to combine algebra and geometry? The 12th century Persian poet, Omar Khayyam was the first who tried to combine algebra and geometry. What are the two branches of analytical geometry? Plane coordinate geometry and solid analytical geometry. Who worked out the fundamental idea of analytical geometry? The fundamental idea of analytical geometry was worked out by the great 17th century French philosopher and mathematician Rene Descartes. He is regarded as the founder of calculus. The field of co-ordinate geometry was established by him. In which book, Descartes presented the idea of analytical geometry? Descartes presented the idea of analytical geometry in his book 'Discourse on Method' in 1637. What do we call the x-axis and the y-axis? X-axis is called abscissa and the y-axis ordinate.

87

What do we call the point of intersection of the X-axis and the Y-axis? It is called origin and assigned (0, 0) coordinates. Origin is taken as reference for any measurement. What is the Cartesian Coordinate system? It is a system used to denote the position of a point on a plane or in space with reference to a set or more axis. When a cross is made by drawing two lines at right angles to each other, it sets off four areas called quadrants. The horizontal line is called x-axis and the vertical line y-axis. Each axis is subdivided into units of equal size. Any point can be represented in the four quadrants (Fig. 10.1).

Fig. 10.1 Cartesian coordinate system Who developed the Cartesian Coordinate system? R e n e Descartes.

What do we study in plane coordinate geometry? In plane coordinate geometry, we study lines, circles, ellipses, parabolas and hyperbolas.

88

How do we study all these shapes in analytical geometry? All these shapes are studied by means of algebraic equations and geometrical figures represented on the Cartesian coordinate system. How do we write coordinates of a point in the coordinate system? A point in a plane is located by giving particular values for x and y. Thus, a point (4,3) is marked by going 4 units to the right from the origin along x-axis and then 3 units upward parallel to the y-axis (Fig. 10.2).

Fig. 10.2 Representation of a point in a coordinate system How quadrants are designated? Quadrants are designated as I, II, III and IV as shown in Fig. 10.3. What are the polar coordinates? Polar coordinates define a point in a plane by its distance from a fixed point and direction from a fixed line. How do we mark positive and negative points on coordinate plane? All positive values of x are taken on the right of the origin and negative values on the left of the origin. All positive values of y are taken upward to the origin and negative values of y are taken downward to the origin.

89

What do these four quadrants show? The I quadrant shows +x and +y, the II quadrant shows - x and +y, the III quadrant shows - x and - y and the IV quadrant shows +x and - y (Fig. 10.3).

Fig. 10.3 Values ofx & y in four quadrants How do we represent a line on coordinate system? First, we take an equation of the line and then assign positive and negative values to x in the equation and find out the corresponding values of y. These points are marked on the coordinate axis and on joining the different points we get the desired line (Fig. 10.4).

Fig. 10.4 Representation of a line 90

How can we derive the equation of a circle? The equation of a circle can be derived by applying Pythagoras theorem to the system of Cartesian coordinates. How can we represent an ellipse on the coordinate system? By taking the equation of the ellipse and giving proper values to x and finding out corresponding values of y, the ellipse can be traced on coordinate plane. What are the open curves? The circle and ellipse are the closed curves, while parabola and hyperbola are the open curves. How do we represent the open curves on graphs? The open curves are traced on graphs by taking their equations and giving the appropriate values of x and finding out corresponding values of y. The values of x and y are marked on the graph and by joining the different points the desired curve is obtained. Can we solve simultaneous equations with the help of graphs? Yes, graphs are traced for the simultaneous equations and the values of x and y are obtained from the point of intersection of the lines representing the equations. What is solid analytical geometry? It is related to the study of lines, planes, spheres and other solids in three dimensions. How do we locate a point in three dimensions? A point in three dimensions is represented by taking three dimensional space x, y and z, mutually perpendicular to each other (Fig. 10.5). Fig. 10.5 Three dimensional representation 91

Is there any study which deals with more than three dimensions? Yes, this is called higher geometry.

What is a hypersphere? A sphere with more than three dimensions. Such spheres have been applied in the manufacture of television cathode ray tubes.

In the geometry of LobachevskyBolyai what is die sum of the three angles of a triangle? It is a l w a y s less than (Fig. 10.6)

180°

Fig. 10.6 Sum of three angles of a triangle in LobachevskyBolyai geometry

What is the sum of the three angles of a triangle In Riemann geometry? It is always greater than 180° (Fig. 10.7).

What was the basis of Riemannian geometry? Riemannian geometry was developed by German Mathematician, 1 G.F. Riemann on the postulate that no lines are ever parallel drawn through a Fig. 10.7 Sum of three fixed point mutually at right angles to angles of a triangle in each other. His system of Non-EuclidRiemann geometry ean geometry thought at the time to be mere mathematic curiosity, was used by Einstein to develop his General Theory of Relativity.

What was the basis of LObachevsky-Bolyai geometry? The geometry developed by Russian, Nikolai Ivanovich Lobachevsky and Hungarian, Janos Bolyai was based on the assumption that at least two lines can be drawn through a given point parallel to a given line.

What is the basis of non-Euclidean geometry? It is based on the shortest distance between two points on any kind of surface. This distance is called a geodesic. 92

Did all mathematicians of modern times concede to that the postulates are self evident? No, all mathematicians have not conceded to the idea of self evident postulates. This gave birth to a new geometry called non-Euclidean geometry.

What are the uses of the analytical geometry? Analytical geometry has wide applications in physics, engineering and space science.

What is the main use of Riemann geometry? The elliptic non-Euclidean geometry of Riemann has given better explanation to Einstein's general theory of relativity.

What is non-Euclidean geometry? The greatest upset in the history of geometry came in the 19th century. The geometries of J.K.F. Gauss, Riemann, N.I. Lobachevski and Bolyai Janos began to question the Euclidean parallel lines axiom and discovered hyperbolic geometry—the first non-Euclidean geometry. o o o

93

11 Vector Analysis What is vector analysis? Vector analysis is the branch of mathematics which deals with the study of vector and scalar quantities. Who is called the founder of vector analysis? J. Willard Gibbs (1839-1903) is called the founder of vector analysis. In the course of his research on electromagnetic theory of light, he made fundamental contributions to the art of vector analysis. What are scalar quantities? The quantities which have magnitude but no direction are called scalar quantities. Scalar quantities can be described fully in terms of its magnitude as constrasted with a vector, which has both magnitude and direction. What are vector quantities? The quantities which have both magnitude and direction are called vector quantities. They may be represented geometrically by lines with specified directions. Some examples are velocity, acceleration, force, momentum, etc. What is a vector space? It is a set of vectors together with a field of scalars such that: the sum of any two members of the set is a vector in the set, and multiplication of a member of the set by a member of the field produces a vector also in the set.

94

How do we represent a vector? A vector is denoted by a line with an arrow at one end (Fig. 11.1). The point 'O' is known as origin. The length indicates the magnitude and the

What are like vectors? The vectors are said to be like when they have the same sense of direction. What are unlike vectors? The vectors are said to be unlike when they have opposite directions. What are parallel vectors? Two or more vectors are said to be parallel when they have same direction. What are equal vectors? Two or more vectors are said to be equal if they have the same magnitude and direction (Fig. 11.2). A

>

B

C

•

D

Fig. 11.2 Two equal vectors What are coplaner vectors? The vectors are said to be coplaner when they lie in the same plane whatever their magnitude may be. How do we define a zero vector? If the origin and terminal points of a vector coincide, it is said to be a zero vector. 95

What is the magnitude of a zero vector? The magnitude of a zero vector is zero. How do we define a unit vector? A unit vector is the one whose magnitude is of unit length. What is a position vector? If O be a fixed origin and P any point, then the vector O P is called the position vector of the point P with respect to origin O. What is a scalar multiple of a vector? If the magnitude of any vector is made K times then K is called the scalar multiple of the vector. What is negative of a vector? If A is a given vector, then - A is a vector whose magnitude is the same but direction is opposite to A. What are co-initial vectors? The vectors having the same initial point are called co-initial vectors. What are localized vectors? A vector drawn parallel to a given vector through a specified point in space is called a localized vector. What are free vectors? If the origin of vectors is not specified, the vectors are called free vectors. What are the two laws applied to the addition of vectors? Triangle law of addition and parallelogram law of addition. What is triangle law of addition? If two vectors are represented both in magnitude and direction by the adjacent sides of a triangle, then their sum is represented in magnitude and direction by the third side of the triangle. What is parallelogram law of addition? If two vectors are represented both in magnitude and direction by the adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram. 96

How do we add two vectors a and b? Choose a point O as origin and draw the vector a and b such that the terminus of a coincides with the origin of b. Join O B. It represents the sum of a and b (Fig. 11.3).

Fig. 11.3 Sum of vectors a & b

Does vector addition obey commutative and associative laws of addition? Yes. How can we subtract two vectors a andb? Choose a point O as origin and draw the vector a. Now, suppose we want to subtract b from a, then we draw vector b in the negative direction of b, as shown in figure 11.4. Join O B, then O B represents a - b.

Fig. 11.4 Subtraction of vectors a&b

How can we resolve a vector along x and y axes? The resolved components of vector A making an angle 0 with the positive direction of x axis are A cos0 and A Sin 8. (Fig. 11.5).

Fig- 11.5 Resolving a vector along x a ndy axes

What are unit vectors i, j, k? These are the vectors of unit length along x, y and z axes of coordinate system. How do we resolve a vector A along x, y and z axes? A = i Ax + jAy + KAz. Can the sum of two vectors be a scalar? No, the sum of two vectors is always a vector. When can the sum of two vectors be zero? The sum of two vectors is zero when they are opposite in direction and both have the same magnitude. What is the scalar product of two vectors a and b making an angle 6 with each other? Scalar product of a and b is written as a b and is equal to a b cos 8.

Fig. 11.6 Scalar product What is another name of scalar product? It is also called a dot product. Can scalar product of two vectors give vector result? No, scalar product of two vectors is always a scalar quantity. What is another name of vector product? Cross product.

98

What is the vector product of two vectors a and b? The vector product of a and b is equal to ab Sin 0, where 0 is the angle

Fig. 11.7 Scalar product is always scalar What do we get as a result of vector product? We always get a vector quantity. What are the applications of vector analysis? Vector analysis is mainly used to solve many problems of physics, space and aeronautics.

ooo

99

12. Trigonometry What is trigonometry ? It is the branch of mathematics, that solves problems relating to plane and spherical triangles. Its principles are based on the fixed proportions of sides for a particular angle in a right-angled triangle. Who is considered as the founder of trigonometry ? The founder of trigonometry was the Greek astronomer, Hipparchus of Nicaea. He lived in the 2nd century B.C. He also discovered the precession of Equinoxes. WHy did he found trigonometry? Hipparchus wanted to measure the sizes and distances of the sun and the moon from the earth. For this purpose, he evolved trigonometry. What is the base of trigonometry? Trigonometry is based on the use of right angled triangle. It can be applied to any triangle by drawing a perpendicular from vertex to the base and

What are trigonometrical ratios ? The trigonometrical ratios show the relationship between the sides of a right angled triangle and angles. What ratios are used in trigonometry? Trigonometrical ratios are sine, cosihe, tangent, cotangent, secant and cosecant. How do we define the sine of an angle? The sine of either of the acute angles in a right triangle is the ratio of the opposite side to the hypotenuse. The sine of angle 1 is a/c and the sine of angle 2 is b'~

Fig. 12.2 Trigonometrical ratios How do we define cosine of an angle ? The cosine of either of the acute angles in a right triangle is the ratio of the adjacent side to the hypotenuse. The cosine of angle 1 is b/c and the cosine of angle 2 is a/c (Fig. 12.2) How do we define tangent of an angle ? The tangent of either of the acute angles is the ratio of the opposite side to the adjacent side. The tangent of angle 1 is a/b and the tangent of angle 2 is b/a (Fig. 12.2). How do we define cotangent of an angle? The cotangent of either of the acute angles is the ratio of the adjacent side to the opposite side. The cotangent of angle 1 is b/a and the tangent of angle 2 is a/b (Fig 12.2). 101

What is another definition of cotangent? It is defined as the reciprocal of tangent, i.e. cotangent = How do we define secant of an angle ? It is defined as the reciprocal of cosine, i.e. secant = How do we define cosecant of ant angle ? It is defined as the reciprocal of sine , i.e. cosecant =

1

tangent

* Cosine ^ Sine

What is the other definition of tangent ? Tangent is also defined as the ratio of sine and cosine. What are the maximum and minimum values of sine of an angle? The maximum value is +1 and the minimum is - 1 . What are the maximum and minimum values of cosine ? The maximum and minimum values of cosine are+1 and - 1 respectively. What is the basic formula of trigonometry? The basic formula is sin 2 0 + Cos 2 8=1. How do we get other relations from the basic formula ? By dividing the basic formula by sin 2 8 we get l+Cos 2 8 = Cosec 2 8 and by dividing it by Cos 2 6 we get l+tan 2 8 = Sec 2 8. What are the values of sin8, when 8 = 0°, 30°, 45°, 60° and 90°? The values of Sin 0°, 30°, 45°, 60°, and 90° are 0, _L , _L , and 1 respectively. 2 y 2

VH, 2

What are the values of Cos8, when 8 = 0°, 30°, 45°, 60° and 90° ? The values of Cos 0, 30, 45, 60 and 90 are 1, V T , and 8 respectively.

1

J_, 2

What are the uses of trigonometry? Trigonometry is used in surveying, engineering, navigation, mapping and astronomy. What are periodic phenomena? Any phenomenon that repeats itself on regular intervals of time is called periodic, such as tides, motion of a pendulum, etc. 102

What type of curve is obtained for sine or cosine when angle changes from 0" to 360° ? We get a periodic curve for sine and cosine as shown in Fig 12.3 and 12.4.

Fig. 12.3 Sine curve

Fig. 12.4 Cosine curve What is law of sines? In a triangle (that is not a right triangle) ABC (Fig. 12.5) with sides a, b and c the relation between sides and angle is a Sin A

b Sin B

c SinC Fig. 12.5 103

This is known as law of sines. If we know two angles and one side of a triangle we can calculate the third angle and other two sides by using this law.

c

What is law of cosines? For a triangle (that is not a right triangle) b/ ABC with sides a, b and c (Fig. t2.6) / C 2 = a 2 + b 2 - 2ab cos C A —^B Fi b 2 = c 2 + a 2 - 2ac cos B S-12.6 and a 2 = b 2 + c 2 - 2bc cos A are known as law of cosines or cosine formulae. If we know two sides of a triangle and the angle between them the remaining part of the triangle can be found by using the law of cosines. Can we study periodic phenomena by trigonometry ? Yes, by measuring angles we can study periodic phenomena. What is law of tangents? In a triangle (that is not a right triangle) ABC with sides as b and c (Fig. 12.7) we have a-b tan a+b

-(V-) • (B-C \ b-c tan { — — ) = b + c

a- c a+c a n d

tan

/A-C

\ )

Each formula represents law of tangents. Fig. 12.7 104

In the given figure 12.8, the coordinates of point P are (x y), the point is at a distance r from origin and the line drawn from origin to this point makes an angle 0 with the positive direction of x axis. How would you represent the point trigonometrically ? x = r Cos 0 and y = r Sin 0

Fig. 12.8

What is radian measure? A radian is the measure of an angle whose vertex is at the centre of a circle and which intercepts an arc whose length is equal to the radius of the circle. In Fig. 12.9 angle AOE is one radian. The relation between degree and radian is given by 2 n radian = 360°.

Fig. 12.9

What is an inverse trigonometric function? If Sin 0 = x then 0 = Sin ~'x. It is called inverse sine function. Other trigonometric ratios can also be expressed like this, e.g. 0 = cos~'x 0 = tan"'x, 0 = Cos~'x, 0 = Sec _ 1 x and 0 = Cosec _ 1 x.

What is the sine rule in spherical trigonometry? In spherical trigonometry the sine rule is : Sin a

Sin a

Sin a

Sin A

Sin B

Sin C

with angles A, B, C and sides a, b, c. 105

Fig. 12.10 Finding the height of a tree How can we find out the height of a tree by means of trigonometry ? By measuring the angle A and the distance AB we can find out the height of a tree (Fig. 12.10). o o o

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13 Calculus What is the meaning of the word 'calculus'? The word 'calculus' comes from the Latin word for pebble. This name originated because pebbles were used thousand of years ago for counting. Calculus has two branches; differential and integral. Who invented calculus ? Sir Issac Newton of England and Baron Gottfried Wilhelm Von Leibniz of Germany are credited for the invention of calculus in 17th century. For what purpose Newton used calculus? He applied calculus to his theories of motion and gravitation. What has been the major use of calculus? Calculus has proved very useful in the study of changing or varying quantities, such as Velocity, acceleration, etc. On the basis of the use, how do we define calculus? Calculus may be defined as the mathematics of change. In calculus, we study mainly instantaneous changes of velocity, acceleration, etc. It can be seen as an extension of analytic geometry, much of whose terminology it shares. What is calculus of variations? It is an extension of calculus concerned with the examination of definite integrals and the calculation of their maximum or minimum values. What are dependent and independent variables? In varying quantities, very often one variable depends on another vari107

able. For example, the distance travelled by a car depends on time. We call distance the dependent variable and time as independent variable. What are other varying quantities to which calculus is applied? Calculus is used in the study of changing population, radioactive decay, changes in the cost of living, rocketry, etc. Figure 13.1 shows the launching of a rocket. By using calculus we can determine the position and velocity of rocket.

Fig. 13.1 Launching of a rocket What is a function? If one variable depends on another, we say that first variable is a function of the second variable. For example, the distance travelled by a car is a function of the time of travel. What does a function indicate? A function indicates that there is a relationship between the dependent and independent variables. What are different types of functions? There are various types of functions such as algebraic functions, trigonometric functions, logarithmic functions, exponential functions, inverse functions, etc. 108

If a ball falls freely from the Leaning Tower of Pisa by obeying the equation S = 4.9t 2 , how can we show ; ts velocity at different times by using calculus? ds V = — = 9.8t. Velocity of free fall at different points is shown in Fig. 13.2.

Fig. 13.2 Determination of velocity offree fall at different points How do we express a function? Suppose y is a function of x, then we write y = f (x). The letter f is used as the abbreviation of function. What is an implicit function? When one variable is not given explicitly in terms of other variable we call it an implicit function. For example, x 2 + y 2 = 2xy is an implicit function, because here y is not defined explicitly. Suppose y is a function of x, then what will be its derivative? If y=f(x), t h e n i s called its first derivative, dx What is second derivative ? If V = f(x), then - ^ - i s its first derivative and is its second derivative, dx dx" 109

What is the derivative of x n . If y = x n , then its first derivative calculus.

= nx n_1 . This is called differential fix

How can a manufacturer minimise the surface area of one litre containers which he wants to manufacture? By using calculus he can minimise the surface areas of one litre containers as shown in Fig. 13.3 .

Fig. 13.3 Minimal surface areas of one litre containers Who invented the integral calculus? Francesco Bonaventura Cavalieri invented the integral calculus. What is integral calculus? Integral calculus is based upon the reverse process of differentiation. For example, differential of x 2 is 2x, while integration of 2x is x 2 . What is the concept of integration. Integration is the process of adding little by little and it is symbolised

What is the integration of x n ?

What is the meaning of integration of functions of more than one variable? We consider a surface of two variables represented by Z=25 - x 2 - y2. t h e n / / z dx dy = / / ( 2 5 - x 2 - y 2 ) dx dy 110

The two integral signs show that we are dealing with a function of two variables. What is the use of integral calculus? It is used for finding out the areas bounded by closed curves, or volumes bounded by closed surfaces (Fig. 13.4). It is also used to solve differential equations, etc.

Fig. 13.4 Area calculation by applying integral calculus

Where do we use differential equations? The basic physical principles of light, sound, motion, electricity, heat, flow of liquids and many other branches of physics and chemistry are formulated in the form of differential equations. Why is Augustin Louis Cauchy famous? Cauchy Integral (Fig. 13.5). Fig. 13.5 Augustin Louis Couchy

What is a differential equation? An equation written in the form of derivatives is called a differential equation. For example dy dx

+ x 2 = 5 is a differential equation.

Which Swiss scientist made significant contribution in calculus? Jacques Bernoulli (Fig. 13.6).

Fig. 13.6 Jacques Bernoulli

What is partial differentiation? Partial differentiation is concerned with the derivatives of the functions when the variables are more than one. For example, the area of the parallelogram given in figure is given by A = xy Sin 0. 111

We can consider the rate of change of A with respect to x, y and 0 where other two variables are held constant as each derivative is calculated. So the partial derivative. dA dA 3A = y Sin 0, -=—- = x Sin 0 and -=r— = xy cos 0. dY a 0 dX

X

ooo

112

14. Statistics What is statistics ? Statistics is the branch of mathematics which enables us to deal with the collection, sorting and interpreting numerical data on any subject of enquiry. What is numerical data? This is a collection of figures or numbers about any subject. What are the t w o types of data? Primary data and secondary data. What is a primary data? The data is said to be primary, if the investigator himself is responsible for its collection. What is a secondary data? The data is said to be secondary, if the investigator collects it from other investigators or agencies. W h a t is d a t a p r e s e n t a t i o n ? If the number of observations in the data is large, it is presented in the form of tables and charts in order to bring out their mean values. This is known as data presentation. How d a t a is p r e s e n t e d in a t a b l e ? Numerical data are arranged in row and columns in a table categorised "ito classes based on the range of raw data. 113

How do we define range of raw data? The range of raw data is the difference between the maximum and minimum number occuring in the data. How do we take size of class interval? Range divided by desired number of classes is called class interval. What is frequency of a class? The number of observations in a particular class is called the frequency of that class. What is a frequency of occurrence? Frequency in statistics is the number of times an event occurs. For example, when two dice are thrown repeatedly and the two scores added together, each of the numbers 2 to 12 may have the frequency of occurrence. What is a bar chart? A bar chart is a graphical representation of data using bars (rectangles) of the same width (Fig. 14.1). numbers

cars

lorries

buses

II

bicycles

Fig. 14.1 Bar chart

114

motor cycles

What is a frequency distribution? The set of data including the frequencies is called a frequency distribution. It is usually, presented in a frequency table or shown diagramatically by a frequency polygon. What is cumulative frequency? The cumulative frequency corresponding to a class is the sum of the frequencies of that class and all classes prior to that class. What are the different graphical methods used for presenting data? The different graphical methods are : Bar charts, histogram, pictograph and pie chart. How would you represent the following expenditure of a country on a bar chart? Departments

Expenditure (Crores)

Education Health Industry Electricity Agriculture

600 400 300 350 250

Expenditure (crores)

See Fig 14.2

Department

Fig. 14.2 Bar chart of given question 115

What is a histogram? A histogram is a graphical representation of a frequency distribution in the form of rectangles one after the other with heights proportional to frequencies. Blocks are drawn such that their areas are proportional to the frequencies within a class or across several class boundaries. There are no spaces between blocks. (Fig. 14.3). 60 I

50 |

|

40|

2 CC

30|

lu U-

20 I

10 I

0

!' I I

4

5

6

1 7

'1 I IM I M l 8

9

10

11 12 13

I • I' I t 1 I I I

14 15 16 SCORE

17 18 19

20 21

22

I I 1 'I

23

24 25

26

Fig. 14.3 Histogram How do we draw a cumulative frequency curve or ogive ? The cumulative frequency curve is obtained by first plotting the points and then joining them smoothly by free hand. What is the range, class interval, frequency and cumulative frequency of each class in the following table? Marks

Frequency

0-5 5-10 10-15 15-20

2 5 10 3

Comulative frequency (2+5) (7+10) (17+3)

2 7 17 20

The range is 20 - 0 = 20, class interval is 5. frequency of each class is 2.5,10,3 and cumulative frequencies are 2.7.17. 20.

116

What is a pictograph? In a pictograph we use a symbol (a car, a man, etc.) to represent each item with a suitable scale. A typical pictograph is shown in Fig. 14.4.

Fig. 14.4 Pictograph What is a pie chart? A pie chart is drawn by first drawing a circle of suitable radius and then dividing the angle of 360 degree sign 0 at the centre in proportion to the figures given under various heads (Fig 14.5). How do we find arithmetic mean? First, we find the total of a given data and then, divide it by the total frequency. The result fcgives the arithmetic mean.

. Fig. 14.5 Pie Chart

What are the uses of arithmetic mean? It is used for finding out average temperature, income, marks, expenditure, cost, profit, etc. What is mode? The most popular score is called mode. In other words, mode is that value which occurs most frequently.

117

How would you draw a pie chart for the following data? Out of 100 people 70 see TV, 20 see cinema and 10 enjoy sports. See Fig 14.6 What is geometric mean? The geometric mean of 'n' observations xl, x 2 , x n is defined as (x,xx2x xn)1/n What is the geometric mean of 4 and9?

Fig. 14.6 Pie chart of given question

G.M. = \ 4 X 9 = 6.

What is the geometric mean of 2,4 and 8? G.M.= (2 x 4 x 8) 1/3 = (64) 1/3 = 4 What is harmonic mean ? It is the reciprocal of arithmetic mean. What are quartiles? Quartiles are the values of the data, which divide the total observations into four equal parts. Quartiles are lower and upper. What is lower or first quartile? The term having midway between lowest extreme and median is called first or lower quartile. What is upper or third quartile? Item between median and upper most extreme is called third or upper quartile. What is median? The median is the middle score in a set of scores or the central value of the set of observations. What is quartile deviation? This is half the difference between upper and lower quartile. 118

What is normal distribution curve? It'continuously variable quantities such as heights of persons or scores of IQ tests, are plotted on a graph, a normal distribution curve is obtained (Fig. 14.7).

Below the mean.

Above the mean

Fig. 14.7 Normal distribution curve What is standard deviation? It is the square root of the mean of the squared deviations f r o m the arithmetic mean.

Who developed standard deviation? Karl Pearson.

What is degree of correlation? A degree of correlation tells about the changes in two variables such as cost of production versus cost of raw materials, size of circumference of a circle versus radius, etc.

What is scatter diagram? It is a m e t h o d of getting some idea about correlation.

119

Who was the first to use statistics for biological investigations? Francis Galton. What are the applications of statistics? Stastistics is used throughout science, wherever there is an element of probability involved, and also in industry, politics, market analysis and traffic control. What are the two main branches of statistics? Descriptive statistics — dealing with the classification and presentation of data, and inferential or analytical statistics, which studies ways of collecting data, its analysis and interpretation. o o o

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15 Probability How does a mathematician describe probability? Probability is a mathematician's way of describing the likelihood that a certain event will take place. It studies the laws of chance. What information we get from probability? The probability theory enables us to determine probable characteristics of a sample drawn from a population, whose characteristics are known. Who discovered the theory of probability? Blaise Pascal. What is the probability of getting a head or tail in a single toss of a coin? The probability of getting a head or tail in a single toss is _J_ (Fig. 15.1). 2 What is binomial distribution? It is a probability distribution expressing the probability of successes or failures.

T O S S I N G A COIN There are only two possible ways a single coin can land : head or tail.

Tail

Head

The probability of its coming up heads (or tails) is: Ui

Fig. 15.1 Tossing a coin What is the total value of probability of all events? The total probability of all events always adds upto one.

121

What is the definition of probability as given by Laplace? Probability is the ratio of numbers of favourable cases to the total of equally likely cases. I f ' p ' is the probability, 'm' the number of favourable cases and 'n' the total number of cases, then rp = — . n What is the definition of probability as given by Aristotle? Aristotle defines a probability as being "what men know to happen or not to happen to be or not to be, for the most part thus and thus". A dice is thrown once. Find out the probability of having an odd number? Here, number of favourable cases =3 Number of total cases = 6 Probability = 3/6 = — See Fig. 15.2.

Fig. 15.2 Probability of an odd number What is a random variable? A random variable is a real valued function defined on a sample space of an experiment. What is sample space and sample points? The set of all possible outcomes is called sample space and the possible outcomes are called sample points. 122

What is the probability of getting heads when we toss two coins? 1/4 (Fig. 15.3). TOSSING T W O COINS

There are 4 possible ways two coins can land. Each is equally likely. The probability that both will come up heads (or tails) is 1/4, whereas the probability of their coming up heads and tails is 1/2.

Fig. 15.3 Tossing two coins What is the probability of getting a total of 7 on throwing two dice? 1/6 (Fig. 15.4). There are 6 ways in which the sum of dice can total 7. These 6 ways represent more variations than tor any other possible total

two

TOSSING T W O DICE

Fig. 15.4 Throwing two dice

123

What is the probability of an impossible event ? Zero. If A and B are two mutually exclusive events, what is the probability of A and B ? Probability P (A and B) = 0 Give an example of two mutually exclusive events? In a single toss of a coin, getting a head and a tail simultaneously are two mutually exclusive events. What is the probability distribution model? It is a prototype of the basic dispertion of random components of position that results in the binomial distribution. In this model, balls drop one after the other from the top and migrate at random through a maze of pegs to one of the lower channels. What are mutually exclusive events? The events are said to be mutually exclusive if the occurrence of one rules out the simultaneous occurrence of the other (Fig. 15.5)

Fig. 15.5 Mutually exclusive events If A and B are two mutually exclusive events, what is the probability of A or B? Probability P (A or B) = P (A) + P ( B ) If the probability of occurrence is p 1} what is the probability of nonoccurrence? Probability of non-occurrence = 1 - Pj

124

If A and B are two no-mutually exclusive events then what is the probability of A or B? P(A or B) = P(A) + P(B) - P(A and B) What are independent events? Two events are said to be independent of the occurrence or nonoccurrence, if one does not affect the probability of the occurrence of the other. If two events A and B are independent what is the probability of occurrence of A and B? Probability of occurrence of A and B, i.e. P(A and B) = P(A) x P(B) What is the use of probability theory for insurance experts? With the help of probability theory, insurance experts find out the life expectancies so that they may set appropriate life insurance rates. Who published the first complete account of probability theory? The French mathematician, Pierre Simon Laplace published the complete account of probability theory in 1812. What is probability distribution? Probability distribution is a distribution of a random variable in which probabilities are distributed to the different values of random variable such that their sum is one. What is the main application of binomial distribution? It is used in the industries for finding out the probability of good and defective items in production. What is Poisson distribution? Poisson distribution is a discrete probability distribution and is widely used in statistics. It was discovered by French mathematician, Simeon Denis Poisson in 1837. What is the difference between binomial and Poisson's distribution? Binomial distribution is applied when the number of items under consideration is small, while Poisson's distribution is used when the number of items is indefinitely large. 125

What is the application of probability in electron physics? It has been used to calculate the positions and velocities of electrons orbiting around the nuclei of atoms. What is the application of probability in gas studies? The fluctuations in density of a given volume of gas have been analysed by applying the probability theory. What are the applications of probability theory in genetics? With the help of probability theory, it is possible to calculate the percentage of individuals with like and unlike traits in successive generations. o o o

126

16. Logic and Game Theory What is game theory? It is an application of mathematical logic to decision-making in games and by extension, in commerce, politics and warfare. How did this branch get its name? It got its name from the fact that there are so many conflicting situations in the real world of business, finance and military, which are similar in basic structure to the games of bridge, poker, chess, etc. What are the objectives of game theory? In game theory, they are assumed to employ strategies that should give the greatest gain and the smallest loss. Can you give one example of a zero sum game? Two boys playing chess is a two person zero sum game (Fig. 16.1).

Fig. 16.1 A zero sum game — Chess 127

How many persons are required for mathematical games? Mathematical games can be played by one person or by two persons or by two teams. It is called a 'two person zero-sum game.' What is the meaning of zero-sum? Zero-sum means that what one side gains, the other side loses and vice versa. The total value of the game does not increase or decrease during the course of playing the game. What is the meaning of finite games? Finite games mean that there are only a certain number of strategies or alternatives that each player can follow and the game is solved in a certain number of moves. How is the strategy determined by a player in singular and dual games? In singular games (i.e. solitaire) the player's strategy is determined solely by rules. In dual games (i.e. football, chess) one side's strategy must take into account the possible strategies of the other. What is a pay off matrix? It indicates what happens when the players select their different strategies. What do we call the figures outside the matrix? These are called rim figures. Who invented the chess-playing machine? L. Torresy Quevedo. When the game theory was developed? The theory was greatly developed during World War II by Oscar Morgenstern and John Von Neumann. Which is the famous book of John Von Neumann on game theory? 'Theory of Games and Economics Behaviour' written in 1929 and published in 1944. What are common finite games? Checkers, chess and tick-tack-toe are some common finite games. 128

What are non-zero games? A non-zero sum game is one in which the losses of one player are not necessarily the gains of the other. A two person non-zero sum game is 'the prisoner's dilemma'. What is the prisoner's dilemma game? Two criminals are arrested and put in different jails. If one of them confesses and produces evidence, he will be made free. But the other will be sentenced for 20 years jail. If both confess, they will receive 5 years sentence. If neither confesses, they will be sentenced for one year. If you are one of the prisoners, what would you like to do? What are n-person games? The games involving more than two persons are called n-person games. Who invented Icosian game? William R. Hamilton. Which is the famous book of John D. Williams? The Compleat Strategyst. What is the future of game theory? It will have increasing importance in the future. What is logic? Logic is a branch of philosophy and mathematics deals with the rules that govern correct and incorrect reasoning inferences. It is called an argument. This formalization of arguments is the fundamental of all logic. It was created by Aristotle. What is the meaning of argument? An argument consists of a set of statements called premises followed by another statement called the conclusion. If the premises support the conclusion the argument is correct. If the premises do not support the conclusion, the argument is incorrect. What are the two branches of logic? The major branches of logic are: deductive logic and inductive logic. Deductive logic deals with the theory of demonstrative arguments, that 129

is arguments whose premises necessitate their conclusions. Inductive logic deals with the theory of what is often called confirmation, this is, with arguments whose premises do not necessitate their conclusions. Who published the complete system of first order logic? German mathematician Gottlob Frega in 1879 published the complete system of first order logic. What is symbolic logic? Symbolic logic is a mathematical form of logic which is based on some special symbols. It avoids much of the ambiguity of ordinary language. What is Boolean logic? It was developed by English mathematician George Boole. In Boolean logic all problems can be solved by reducing them to a string of yes or no, true or false choices. Boolean logic is essential to the digital computer which can only deal with two choices. How venn diagrams can be applied to the following two premises: "All men are mammals" "No mammals are cold-blooded" The arguments mentioned above Spider can be represented by three overlaping circles representing men, mammals and cold-blooded animals respectively (Fig. 16.2). Lion The areas where the men and mammals circles overlap the coldMan blooded animals circle are shaded to indicate emptiness because no members of either these groups Fig. 16.2 are cold-blooded. Similarly, since no men and not mammals are coldblooded, the area of the men circle which does not fall within the mammals circle is also empty. The diagram thus shows that all men are mammals, some mammals are men but no men or mammals are coldblooded.

130

Who is said to be the father of logic? Greek philosopher Artistotle is said to be the father of logic. What do you conclude from the two premises given below on the basis of logic 'All men are mammals' and 'No mammals are cold-blooded'? Using these two premises, the conclusion would be derived by getting rid of the middle term 'mammals' which occurs in both premises to give the conclusion, 'No men are cold-blooded'. What was the concept of 'Logical Positivism'? This was the doctrine of the "Vienna Circle" — a group of philosophers founded by M. Schlick. At the heart of logical positivism was the assertion that apparently factual statements that were not sanctioned by logical or mathematical convention were meaningful only if they could conceivably be empirically verified. Thus only mathematics, logic and science were deemed meaningful; ethics, metaphysics and religion were considered worthless. The influence of logical positivism, however, evaporated after the World War II. Who have been the notable modern logicians? Notable modern logicians include the British mathematician George Boole, Alfred North Whitehead and the British philosopher Bertrand Russel. These logicians have used mathematical methods as well as techniques that involve symbols. What is the minimax theorem of Von Neumann? Von Neumann showed in his minimax theorem (first stated in 1928) that, since statistically minimax and maximin strategies negate each other, most dual games are not worth playing, in that their outcome is determined solely by the rules. What are the two major strategies available to players of dual game? The minimax — in which a player evaluates his probable maximum loss and attempts to minimize. The maximin in which a player evaluates his probable minimum gain and attempts to maximize it. o o o

131

17. Mathematical Tools and Instruments Which is the oldest tool used for mathematical calculations? Abacus is probably the simplest instrument still in use to assist computation. Decimal numbers are represented by the position of beads on wire in a frame in which beads move up and down. It was discovered in China some 5000 years ago. (Fig. 17.1)

Fig. 17.1 Abacus

132

Who invented the slide rule? The slide rule one of the most important mathematical innovations of the modern era is 400 years old. Jost Burgi, a Swiss astronomer was instrumental in devising it. What is a slide rule? A slide rule is an instrument, which is used for rapid, through approximate calculations. Two scales are calibrated identically so that on each, the distance from 1 point to any point on the scale is proportional to the logarithm of the number represented by that point. (Fig. 17.2). It works on the principle of logarithm. It is a convenient tool for carrying out multiplication and division.

Fig. 17.2 A slide rule What is a mechanical calculating machine? A mechanical calculating machine has a system of toothed-wheels inside it, which make the calculations possible. Addition and multiplication are done by turning the handle forward, while subtraction and division are achieved by turning it backward (Fig. 17.3).

Fig. 17.3 Mechanical calculating machine

133

What are the different tools used to measure length? The different tools used for measuring length are: Ruler, vernier callipers, micrometer, tape, curvimeter, etc. What is an electronic calculator?

pocket

It is a small calculating device, which makes use of an integrated circuit inside. It is used f o r carrying out addition, subtraction, multiplication, division and other operations (Fig. 17.4).

Fig. 17.4 Calculator

What is a ruler? It is a kind of strip made of wood or plastic or metal, marked in inches on one side and in centimeters on the other side (Fig. 17.5).

Fig. 17.5 A ruler What is a folding ruler? A folding ruler has more length as compared to the conventional rulers (Fig. 17.6).

Fig. 17.6 A folding ruler 134

What is a vernier callipers? A vernier callipers is a metallic tool, which has a vernier scale and main scale marked in centimeters and inches (Fig. 17.7).

^

Fig. 17.7 A vernier callipers

Upto how much accuracy can we measure the length with a verniei callipers? It can measure lengths with an accuracy of 0.01 of a millimeter. What is a measuring tape? It is a long flexible plastic tape marked with a scale on it (Fig. 17.8).

Fig. 17.8 A measuring tape What is a micrometer? It is an instrument for measuring accurately-dimensions or separations. It is generally used to measure the diameters of wires or spherical balls with an accuracy of 0.01mm (Fig. 17.9).

135

Fig. 17.9 A micrometer What is a curvimeter? The curvimeter is an instrument used to measure the distance along a curve or on a map (Fig. 17.10).

Fig. 17.10 A curvimeter What is a protractor? A device used to measure angles in degrees. It is usually semicircular and marked off in degrees along the semicircular edge. (Fig. 17.11)

Fig. 17.11 A protractor 136

What are set-squares? A pair of triangular shapes usually made of plastic. One triangle has the shape of isosceles right angled triangle and the other has the shape of a right angled triangle with other two angles of 60° and 30° (Fig. 17.12).

Fig. 17.12 A pair of set squares What is the use of set-squares? Set-squares are used for making angles and drawing parallel lines (Fig. 17.13 and 17.14).

Fig. 17.13 Fig. 17.14 Making angles and drawing parallel lines with set squares What is a compass? The compass is a mathematical tool used for drawing circles (Fig. 17.15).

Fig. 17.15 A compass 137

What are the volume measuring devices? Volumes of liquids are measured with pipettes, measuring glasses, measuring cylinders, etc. (Fig. 17.16).

Fig. 17.16 Pipettes, cylinder, glass Who invented the integrator? James Thomson. Which instrument is used to measure the value of definite integrals? Integrator. What is the use of beam compass? It is used for drawing circles of large diameter. What is a spherometer? It is an instrument used to measure curvature of the surfaces. With which machine chess was played first? Thinking machine.

138

Which machine is used to solve differential equations? Wheel and disc integrator' is used to solve differential equations. What is a pantograph? A pantograph is an instrument used for duplicating geometrical shapes to a reduced or enlarged scale on a sheet of paper. (Fig. 17.17)

^J^Pivot

T

Pivof

end of bolt filed to a point Pencil holders

T

M l rest

wooden strips

wooden strips

Pencil draws map half original size wooden disk Tracer mover over outline of map

tracer lead pipe AC CD DG DF FA AB FE FJ JH

= = = = = = = = =

21 cm 42 cm 42 cm 21 cm 42 cm 7cm 7 cm 21cm 7 cm

control reel pencil weight of lead | presses pencil onto paper

end of bbolt oltT filed to point

ho)der

Fig. 17.17 A pantograph What is a planimeter? It is a simple integrating instrument for measuring the area of a regular or irregular plane surface. It consists of two hinged arms: one is kept free and the other is traced around the boundary of the area. What is a theodolite? A theodolite is a surveying instrument used for measuring horizontal and vertical angles usually used in surveying. It consists of a small telescope mounted so as to make on two graduated circles, one horizontal and other 139

vertical, while its axis pass through the centre of the circles. (Fig. 17.18).

Fig. 17.18 A theodolite What is a sextant? It is an instrument used to measure the angle of elevation of the sun above the horizon (Fig. 17.19).

Fig. 17.19 A sextant 140

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18 Computer and Mathematics What is a computer? A computer is an automatic electronic machine which can solve complex mathematical problems at great speed without committing any mistake. It can add, subtract, multiply and divide large numbers rapidly and accurately. It can solve most complex equations. A large amount of data and information can be stored in computer's memory and can be kept for very long durations. A computer is shown in Fig. 18.1

Fig. 18.1 A computer Which was the most successful computing device in ancient times? Abacus. This device was developed in China and has been in use for over 2500 years. It is still used in advanced countries like Japan and Russia for primary education.

141

What is Napier's Bones? It is a calculating device invented by the Scottish mathematician John Napier (1550-1617). This device consists of a set of eleven rods, which Napier called 'bones'. Numbers were carved on these bones. 'Napier's Bones' was very useful in calculations involving multiplication and division of large numbers. What are the major parts of a computer? The major parts of a computer are: 1. Input unit, 2. Central processing unit, and 3. Output unit. How were calculations performed in Pascal's calculator? Pascal's calculator (Fig. 18.2) was named after its inventor Blaise Pascal, who invented it in 1642. This calculating machine consisted of gears and levers. The numbers were displayed by rotation of gears by 1 or 0 steps and had a ratchet to rotate the next wheel by one step when the previous wheel completed 10 rotations. Odometers (used to measure speed) are based on Pascal's machine.

Fig. 18.2 Pascal's calculator Who invented the 'punched tape' programme? Joseph-Marie Jacquard invented the 'punched tape' programme. These were developed for weaving textile designs automatically, not for performing calculations. What are the different subunits of Central Processing Unit (CPU)? The different subunits in CPU are: 1. Arithmetic and Logic Unit (ALU), 2. Memory Unit, and 3. Control Unit. 142

What are the functions of CPU? The functions of CPU are to: (a) Store data and instructions. (b) Control the sequence of operations as per stored instructions. (c) Issue commands to all part of the computer system and (d) Carry out data processing and to send results to output. Who is known as the father of modern computer science? Charles Babbage, a 19th century English mathematician is known as the father of modern computer sciFig 18 3 Charles Babbage ence (Fig. 18.3). He took the first step towards the automatic computation in 1823. Babbage gave the models of two mechanical computers, Difference Engine and Analytical Engine. Who was Lady Ada Lovelace? Lady Ada Lovelace was the daughter of the famous English Poet, Lord Byron. She developed binary number system and explained how this system can be used to run Babbage's machines. What were the basic concepts presented in John-von Neumann report? John-von Neumann was a very well known Hungarian mathematician. In 1946, he explained how programmes can be stored. He laid stress on using binary numbers (0 and 1) to represent data. What is the function of ALU? ALU performs all the arithmetical calculations and takes logical decisions. What are the three types of computers? They are: Digital computer, Analog computer, and Hybrid computer.

143

What is a digital computer? In a digital computer, problems are Solved by using numbers. In this system, all expressions are coded in binary digits (0 and 1). What is an analog computer? The analog computer measures one quantity in terms of another quantity. It can record temperature, speed, or other measurements which change constantly. Analog computers operate by measuring rather than counting. What is a hybrid computer? Hybrid computer is a combination computer using all the good qualities of both the analog and digital computers. In such computers, some calculations are done in the analog portion and others done on the digital portion of the same computer. What is the input unit of a computer? The input unit of a computer accepts instructions and data from the user and communicates them to the computer. The input to a computer in binary language can be given by light pen, Joy Sticks, punched card, Magnetic Drum, etc. Input output unit of a computer are shown in fig. 18.4 ^fc^n \

Magnetic disk , x £

G r a p h i c input V i d e o display unit

Electronic keyboard (musical)

Sound Synthesizer Punch card

Central Processing Unit

Line Printer

Visual display unit

Floppy disk

Fig. 18.4 Input md output unit of a computer / 144

G r a p h i c Plotter

Daisy printer

What is the output unit of a computer? The output unit provides the results of different operations to the operator. It is a communication link between the computer and the user. The different types of output devices are card puncher, optical printer, magnetic tape, visual display unit, laser printer, etc. What is logic? Logic is a method of thinking or at least arguing. It proceeds in a step-bystep manner to a conclusion. Logic was developed and formularized by the Greek philosophers. After its development, it also became the basis of certain kinds of mathematics. What is a super computer? Supercomputers (Fig. 18.5) are the most powerful computers. These computers can execute 10,000 million instructions per second. The memory size of these computers varies from 8 megabytes to 256 megabytes. Seymour Cray is the inventor of supercomputer. What is PASCAL? PASCAL is a high level computer programming language. It is used for scientific and numeric work.

Fig. 18.5 A super computer

Who invented the first personal computer? The personal computers are designed for personal use of individuals or small business establishments. It was invented by David Ahl, Edward Roberts and Steve Wozniak. What is Boolean logic? Boolean logic is a form of a mathematical logic developed by a 1'^th century English mathematician. George Boole. Boolean logic is essential 145

to the digital computer, which can only deal with two choices- Yes or No, True or False. When and by whom was PASCAL developed? PASCAL was developed by Nicolaus Wirth for teaching computer science at the Zurich Engineering University in early seventies. It is named after the great French physicist and mathematician, Blaise Pascal, who contributed a lot to the evolution of computers by developing calculating machines in the 17th century. What is a 'byte' A cluster of bits, usually eight, is called a byte (Fig. 18.6). In computers, a group of eight bits is normally needed to form a single letter or number.

Fig. 18.6 A byte Which was the first computer employed for performing statistical calculations? Dr. Hollerith's tabulator was the first computer employed for performing statistical calculations. It was employed by the US Census to conduct its studies regarding American population. How are programmes given to a computer? Programmes are represented using binary codes. What does 'binary' mean? 'Binary' is a Latin word, which means 'two'. In digital computers, it refers to the two conditions—the presence or absence of a pulse. 146

Who was the first to employ binary mathematics in the working of his calculating machine? Konrad Zuse. What is 'bit'? A single '0' or ' 1' is called a bit. It is a digit in the binary system, which uses only combination of 0 or 1 to denote all numbers. Why computers are referred to as binary? It is because all information processed by digital computers is ultimately reduced to a series of electronic pulses or signals. The presence of a pulse is represented as 1 and the absence is represented as 0. What is a 'number cruncher' ? Number cruncher is a computer that can handle a large volume of numbers. Define NIBBLE? A group of four binary digits is called a NIBBLE. What was the name of the mathematician who developed first compiler while working on UNIVAC? The mathematician to develop first compiler was Grace Murray Hopper. What is coding? Coding is the process of representation of numeric or non-numeric information in terms of binary digits. Which was the first electro-mechanical computer? ASCC (Automatic Sequence Controlled Calculator), designed by Dr. Howard Aiken of Harvard University, was the first electro-mechanical computer. Its other name is Harvard Mark I. What is IBM? IBM stands for International Business Machines Corporation. It is a multinational computer manufacturing company, established by Dr. Hollerith in 1896, and was sold out in 1924.

147

ENIAC (Electronic Numerical Integreter and Calculator) is known as the grand daddy of the present computers. Who invented it? John W. Mauchly and J. Presper Eckert, the two scientists serving in Pennsylvania State University were responsible for the development of this machine in 1946. (Fig. 18.7).

Fig. 18.7 John W. Mauchly and J. Presper Eckert Complex dynamics is one mathematical area opened up by the invention of computers. What is its use? Complex dynamics or chaotic dynamics is used in making science fiction films. In which base calculations are performed in computers? Computer uses base 2 to perform calculations. Why is Zuse-3 famous? Zuse-3 was the first operational general purpose programme controlled calculator. What is the meaning of 'K' in computer science? 'K' usually stands for kilo or 1,000. However, in computers 'K' stands for 2 1 0 or 1024. Why a fault in the computer system or programme is called a 'bug'? It was in 1940, when mathematician and programmer, Grace Hopper while working on Harvard Mark I computer found a real bug trapped in 148

one of the electromagnetic relays. Since then, the term 'bug' derived from the incident is used for any mistake related to computer system. The bug is shown in Fig. 18.8. Which was the first special purpose digital computer? Colussus was the first special purpose digital computer. What is data and how it is stored in a computer? Data is the name given to the facts about activities. It is simply an input to a computer. Data is stored in computer using binary code What are flow charts? Flow charts are graphic means of describing a sequence of operations, which have to be done on data. For drawing flow charts, a set of conventional symbols are used. Mathematical symbols used in a flow chart are shown in Fig. 18.9.

Fig. 18.9 A flow chart

Fig. 18.8 A bug

Star or stop program

Decision

)

Flowline

O

Connector

Processing

Input or outpur

Offpage connector

Symbol

Meaning




Greater than or equal to

=

Equal to

< >

Not equal to

149

s What types of data are processed by computers ? A computer processes data of military interest, business, education, engineering, government and other fields. What is a data file? A group of records is called a data file. There are files on almost every subject: birth, death, crime, scientific research, etc. What is a computer data bank? A computer data bank is a large storage facility. How data files are processed? Data files are processed by batch processing and on line processing. What does 'Baud rate' means? Baud rate is defined as the number of data elements sent per second e.g. one baud is equal to one bit per second. What is data aggregate? Any collection of data items within a record that is given a name is called data aggregate. Where was 'C' developed? 'C' is a programming language, and was developed by Dennis M. Ritchie at Bell Laboratories, USA. It is used to create graphics. What is a mathematical software package? A mathematical software package is a building bridge between numerical analysts who devise numerical algorithms and computer users who need reliable numerical software. What is PROLOG? PROLOG stands for Programming in Logic. Originally intended for theorem solving, but now used more generally, in artificial intelligence. What is COGO? COGO is a problem oriented programming language used for solving geometric problems. COGO stands for Coordinate Geometry.

150

What is ALGOL? ALGOL stands for Algorithmic Language. It is an Algebraic language. It was developed in Europe in the early 1960s. What is FORTRAN? Fortran is the short name of Formula Translation. It was first developed by IBM in 1957. Fortran finds its maximum use in scientific calculations. Fortran-90 is the latest version of this language. What is MIN PACK? MIN. PACK is a computer package and it consists of FORTRAN Programs for numerical solution of non-linear equations and non-linear least square problems. What are fractals? Fractal is a branch of mathematics and it deals with covers and surfaces with non-fractional dimension. This can be used in computer graphics for obtaining a degree of complexity analogues to that in nature from a handful of data points. What does GIGO mean? GIGO stands for Garbsge In, Garbage Out. It means that, if you put faulty, incomplete or meaningless data into a computer, you are going to get faulty, incomplete or silly answers from the computer. Are those funny looking numbers on the bottom of cheques meant to be read by computers? Yes, they are printed in a form that is easily recognised by the computer we too can read them with little practice. In fig. 18.10 numbers used on the bottom of cheques are shown.

ABCDEFGH I JKLP1N OPCLRSTUVWXYZ Fig. 18.10 Alphabets and numbers written on cheques 151

Can pictures be drawn on a computer screen? Yes, they can be drawn by using 'light pen'. It can be used to write or sketch on a video screen. The sketch is not only displayed visually, it is also communicated to the computer. Milestones of Computer Development 1614

John Napier invented logarithms.

1615

William Oughtred invented the slide rule.

1623

Wilhelm Schickard (1592-1635) invented the mechanical calculating machine.

1645

Blaise Pascal produced a calculator.

1672-74

Gottfried Leibniz built his first calculator, the Stepped Reckoner.

1801

Joseph-Marie Jacquard developed an automatic loom controlled by punch cards.

1820

The first mass-produced calculator, the Arithometer, was developed by Charles Thomas de Colmar (1785-1870).

1822

Charles Babbage completed his first model for the difference engine.

1830s

Babbage created the first design for the analytical engine.

1890

Herman Hollerith developed the punched-card ruler for the US census.

1936

Alan Turing published the mathematical theory of computing.

1938

Konrad Zuse constructed the first binary calculator, using Boolean algebra.

1939

US mathematician and physicist J V Atanasoff (1903- ) became the first to use electronic means for mechanizing arithmetical operations.

1943

The Colossus electronic code-breaker was developed at Bletchley Park, England. The Harvard University Mark I or Automatic Sequence Controlled Calculator (partly financed by IBM) became the first programcontrolled calculator.

1946

ENIAC (acronym for electronic numerator, integrator, analyser, and computer), the first general purpose, fully electronic digital computer, was completed at the University of Pennsylvania, USA.

1948

Manchester University (England) Mark I, the first stored-program computer was completed. William Shockley of Bell laboratories invented the transistor.

1951

Launch of Ferranti Mark I, the first commercially produced computer. Whirlwind, the first real-time computer, was built for the US air-defence system. Grace Murray Hopper of Remington Rand invented the compiler computer program.

1952

EDVAC (acronym for electronic discrete variable computer) was completed at the Institute for Advanced Study, Princeton, USA (by John Von Neumann and others).

152

1953

Magnetic core memory was developed.

1958

The first integrated circuit was constructed.

1963

The first minicomputer was built by Digital Equipment (DEC). The first electronic calculator was built by Bell Punch Company.

1964

Launch of IBM System/360, the first compatible family of computers, John Kemeny and Thomas Kurtz of Dartmouth College invented BASIC (Beginner's All-purpose Symbolic Instruction Code), a computer language similar to FORTRAN.

1965

The first supercomputer, the Control Data CD6600, was developed.

1971

The first microprocessor, the Intel 4004, was announced.

1974

CLIP-4, the first computer with a parallel architecture, was developed by John Backus at IBM.

1975

Altair 8800, the first personal computer (PC), or microcomputer, was launched.

1981

The Xerox Star system, the first WIMP system (acronym for windows, icons, menus, and pointing devices), was developed. IBM launched the IBM PC.

1984

Apple launched the Macintosh computer.

1985

The Inmos T414 transputer, the first 'off-the-shelf microprocessor for building parallel computers, was announced.

1988

The first optical microprocessor, which uses light instead of electricity, was developed.

1989

Wafer-scale silicon memory chips, able to store 200 million characters, were launched.

1990

Microsoft released Windows 3, a popular windowing environment for PCs.

1992

Philips launched the CD-I (Compact-Disc Interactive) player, based on CD audio technology, to provide interactive multimedia programs for the home user.

1993

Intel launched the Pentium chip containing 3.1 million transistors and capable of 100 MIPs (millions of instructions, per second). The Personal Digital Assistant (PDA), which recognizes user's handwriting, went on sale.

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153

19. Famous People in Mathematics Who was Thales? Thales (640 - 546 B.C.) was a philosopher, who studied mathematics, astronomy, physics and other sciences. He was born in Greece but went to Egypt to study. What was his contribution in mathematics? He measured the height of a pyramid using the idea of similarity and predicted the date of an eclipse of the sun. He is sometimes called the father of mathematics and astronomy. Thales is shown in Fig. 19.1. Fig. 19.1 Thales Who was Pythagoras? Pythagoras (580 — 500 B.C.) was a philosopher, who studied mathematics, music and other subjects. Pythagoras is shown in Fig. 19.2. He is famous because of the theorem named after him. Fig. 19.2 Pythagoras 154

What was the main contribution of William Jones? William Jones gave the symbol of n to denote the ratio of the circumference to the diameter of a circle. Why is Euclid famous? Euclid, the Greek mathematician who lived in Alexandria is famous for geometry. He wrote 13 volumes on geometry, which became the most important works in the study of geometry and have been used throughout the world. He was born 300 BC in Greece (Fig. 19.3).

Fig. 19.3 Euclid

W h o was Archimedes? Archimedes (287-212 B.C.) was a famous Greek physicist and mathematician (Fig. 19.4). He made important discoveries in geometry, hydrostatics and mechanics. He also invented the principle of lever and gave the concept of density.

Fig. 19.4 Archimedes Who is called 'ten men in one'? Leonardo da Vinci (1452-1519) was called 'ten men in one', because he was a painter, inventor, lute player, sculptor, military engineer, scientific observer, anatomist, architect, town planner and designer (Fig. 19.5). Who was Leonard Euler? Leonard Euler (1707-83), was a Swiss mathematician, whose prolific output was 155

Fig. 19.5 Leonardo da Vinci

such that his papers were still being published for the first time more than fifty years after his death. His collected work have been printed bit by bit since 1910 and will eventually occupy more than 75 large quarto volumes. What did Archimedes say about a lever? Archimedes said, "If you give me a long enough lever and a point to put it, then I can even move the earth." What was the contribution of Leonardo da Vinci to mathematics? He used perspective to paint solid figures on a plane canvas. Perspective is a way of showing three dimensional objects on paper. His enquiring scientific mind led him to investigate every aspect of natural world from anatomy to aerodynamics. Who was Nicolaus Copernicus? Copernicus (1473-1543) was a famous astronomer, mathematician and physicist of Poland (Fig. 19.6). His famous idea was that the sun is the centre of the universe and the earth revolves round the sun and became prime founder of modern astronomy. In which book this famous and revolu- „. , Fig. 19.6 Nicolaus Copernicus tionary idea was put forward? This idea was put forward in his famous book The Revolution of Celestial Bodies. Who was Galileo? Galileo Galilei (1564-1642) was the famous mathematician, physicist and astronomer of Italy. He revolutionized scientific thought in his day and prepared the ground for Newton (Fig. 19.7). What is the law of falling bodies given by Galileo? Fig. 19.7 Galileo

Before Galileo, people believed that the 156

speed of a falling body depends on its weight. Galileo showed for the first time by dropping two metal balls of different weights from the top of the Leaning Tower of Pisa that the speed of falling objects does not depend on their weight. He argued convincingly that free-falling bodies, heavy or light, had the same constant acceleration and that a body moving on a perfectly smooth horizontal surface would neither speed up nor slow down. (Fig. 19.8).

Fig. 19.8 Showing the law of a falling body Mention some other contribution of Galileo? Galileo also contributed to the discovery of pendulum clocks. He made a telescope and studied several heavenly bodies. He supported the Copernicus' idea that the earth revolves round the sun. Who was Descartes? Rene Descartes was a French mathematician who lived from 1596 to 1650. He is regarded as the discoverer of analytical geometry and founder of the science Of optics (Fig. 19.9). 157

Figw I 9 . 9 R e n e D e s c a r t e s

W h a t w e r e the c o n t r i b u t i o n s of Descartes in mathematics? He developed the coordinate system for representing points in a plane and space. He was the first person to have used letters of the alphabet to represent numbers. Who was Blaise Pascal? Blaise Pascal was a French mathematician and physicist, who lived from 1623 to 1662. (Fig. 19.10).

Fig. 19.10 Blaise Pascal

What did Pascal contribute to geometry? At the age of 12, he discovered that the sum of interior angles of a triangle is always 180°. He contributed to the development of calculus and probability theory. What was the contribution of Blaise Pascal in the science of computing? At the age of 19, Blaise Pascal invented the first calculating machine, which used gear wheels. Who was Seki Takakazu? Seki Takakazu was a Japanese mathematician who lived from 1642 to 1708 (Fig. 19.11). Fig. 19.11 Seki Takakazu Who was Newton? Newton was the most famous mathematician and physicist of England, who lived from 1642 to 1727 (Fig. 19.12).

Fig. 19.12 Newton 158

What is the most famous discovery of Sir Isaac Newton? The most famous discovery of Newton is the law of gravity, which he discovered when he was observing an apple falling from a tree. He also developed theories about other natural forces motion and light. He invented a type of reflecting telescope, which came to be named after him. (Fig. 19.13.)

Fig. 19.13 Law of gravity Who was Leibniz? Leibniz was a German mathematician, who lived from 1646to 1716 (Fig. 19.14). What are the contributions of Leibniz in mathematics? Leibniz and Newton formulated the basic ideas of differential calculus. Leibniz also invented a type of calculating machine and a symbolic mathematical logic. What did Seki contribute to mathFig. 19.14 Leibniz ematics? He invented a method of measuring area of figures bounded by curves and the volume of irregular solid figures. 159

Who was Ino Tadataka? He was a Japanese mathematician; who lived from 1745 to 1818 (Fig. 19.15). His main contribution was to the making of the map of his whole country. Who was John Dalton? Dalton (1766-1844) was an English mathematician and chemist. He proposed atomic theory and prepared table of atomic weights. Fig. 19.15 Ino Tadataka

Fig. 19.16 John Gauss

Who was Johann Gauss? Jahann Gauss was a German mathematician, who lived from 1777 to 1855 (Fig. 19.16). Gauss as one of the world's greatest mathematician contributed a lot in the fields of astronomy and surveying. He also worked on the theory of numbers, non-Euclidian geometry and on the mathematical development of electric and magnetic theory.

What was the contribution of Simon Steven? He advocated the use of decimal numbers in mathematics. Who is considered as the founder of the theory of numbers 7 Pierre de Fermat (Fig. 19.17). Fermat's principle of least path was his another contribution.

Fig. 19.17 Pierre de Fermat 160

What was the main contribution of Ptolemy? Claudius Ptolemy (100-178 A.D.) calculated the value of n for the first time and told that its value was 22/7. His greatest work was known as the Almagest in which he developed the theory of Aristotle. Which Scientific Academy offered a prize of 1,00,000 Marks (about Rs. 1,850,000) for a proof that the theorem of Fermat was correct? In 1908, the German Academy of Sciences offerred the above price, which still stands. As a result of inflation, the price money has now been reduced to 7,500 Deutsche Marks (about Rs. 1,4000). What is known as "Fermat's Last Theorem"? Fermat's last theorem states that there are no solutions to such an equation when n is a whole number greater than 2. He died without offering the proof. Who was the founder of matrics algebra? Arthur Cayley is considered as the founder of matrics algebra. Why is Willard Gibbs famous? Josiah Willard Gibbs founded the theory of vector analysis. Who discovered parabola, ellipse and hyperbola? Menaechmus discovered conic sections, i.e. parabola, ellipse and hyperbola. Who is known as the discoverer of science of infinity? George Cantor is considered as the discoverer of science of infinity. Which discovery John Napier is famous for? John Napier is famous for the discovery of logarithm. Fig. 19.18. Fig. 19.18 John Napier

161

Why is Emmy Noether famous? For his discovery of abstract mathematical rings. What did Rose Mary Stemmler contribute to mathematics? She developed early computer languages. Ada Lovelace is very famous in computer science. What for? She was the first computer programmer. Who was Hipparchus? He was a Greek astronomer, who founded trigonometry. Who invented Venn diagrams Venn diagrams were invented by the British mathematician John Venn in the 19th Century. Who gave the theory of games? John von Neumann. Who devised abstract calculus for the use in probability theory? Academician Andrei Kolmogorov of Russia. Who wrote the famous book, "The Great Art"? The Great Art was written by Jerome Cardon in 1545. He made significant contributions in the field of complex numbers. What for french mathematician Francois Vieta is famous? For making major contributions in Algebra. Who made major contribution in fluid dynamics? Daniel Bernoulli What was the contribution of Leonhard Euler in Mathmatics? He showed that the operations of differentiation and integration were opposite. o o o

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20 Mathematicians of India Bose-Einstein statistics is the famous branch of modern physics. Who discovered it ? S.N. Bose developed Bose-Einstein statistics (Fig.20.1). What are Bosons? The particles which obey Bose-Einstein statistics are called Bosons. When was S.N. Bose declared NaFig. 20.1 S.N. Bose tional Professor? S.N. Bose was declared National Professor in 1958. In the same year, he was elected Fellow of Royal Society of London and was awarded Padma Vibhushan. He was a Fellow of Royal Society (FRS). Who was Srinivasa Ramanujan? A famous Indian mathematician who lived from 1887 to 1920 (Fig. 20.2). The theory of numbers brought worldwide fame to Ramanujan.

Hg. 20.2 Srinivasa Ramajujan

With whom Ramanujan worked at Cambridge University? He worked with the great mathematician, G.H. Hardy at Cambridge University. His birth centenary was celebrated in 1987. 163

Who was Aryabhata? Aryabhata was a famous Indian mathematician who was born in Pataliputra, Bihar in 476 AD . Aryabhata completed his study at the University of Nalanda, which was the then great centre of learning. What are the famous books of Aryabhata? Aryabhatiya and Aryabhatasidhanta. What were the contributions of Aryabhata in mathematics? He gave the value of n. He gave astronomical data for preparing panchangs. India's first satellite was named in his honour as Aryabhata. Who was Brahmagupta? Brahmagupta was a famous mathematician of India, who was born in 598 AD in Gujarat. His writings influenced Arabs and hence medieval European scholars. What did he contribute to mathematics? He gave many operations of zero. He is considered as the founder of numerical analysis. He also framed laws to solve simple and quadratic equations. Which is the most famous book of Brahmagupta? Brahmasphutasidhanta. Who was Bhaskara? Bhaskara was a famous mathematician and astronomer of India, who was born at Bijapur in 1114 A.D. He told that if any number is divided by zero the result is infinity. What were the contributions of Bhaskara to mathematics? He is known as the founder of calculus. He gave cyclic method to solve algebraic equations. He considered Brahmagupta as his guru in Algebra. He gave many formulae and theorems of trigonometry. What are his famous books in mathematics? Siddhantasiromani, which he wrote at the age of 30. 'LEELAVATI' is a chapter in this book which deals with arithmetic and was named after his daughter. The other book is Karanakutuhala, which he wrote at the age of 69. It deals with the astronomical calculations. 164

Which Indian lady is called mathematics wizard and human computer? Shakuntala Devi.

Which Indian student became Britain's youngest graduate in mathematics? In 1992, a 13 year-old Indian boy — Ganesh Sittampalan, studied for Surrey University Maths degree (B.Sc.) for just 60 days and beat the previous record holder, Ruth Lawrenec by seven month. He secured a first class honors degree in mathematics

Who was Mahalanobis? P.C. Mahalanobis (1893-1972) was a famous statistician of India. He invented ' M a h a l a n o b i s d i s t a n c e ' and 'fractile graphical analysis'. 'Mahalanobis distance' is a widely used concept in taxonomical classifications. He founded Indian Statistical Institute (ISI) Calcutta. He was elected F R S in 1945 (Fig. 20.3) Fig. 20.3 Mahalanobis

P.C.

Who was D.R. Kaprekar? A famous mathematician of India, born on Jan. 17, 1905 at Dahanu near Bombay (Fig. 20.4). In 1946, he discovered 'Kaprekar Constant' in 1976. It is the number 6174. He also gave 'self numbers' and Demlo numbers.

Fig. 20.4. D.R. Kaprekar

What was the indirect contribution of Sawai Jai Singh II to mathematics? He asked the experts to translate Euclid's Elements into Sanskrit.

165

What are the contributions of C.S. Seshadri to mathematics? Prof. Seshadri is distinguished for his valuable contributions to algebraic geometry, vector bundles, moduli problems and algebraic varieties. He was elected FRS. (Fig. 20.5) What is the contribution of J.V. Narlikar to mathematics? He contributed a lot to the theory of Fig. 20.5 C.S. Seshadri relativity and cosmology. His significant contributions include generalization of Bode's law and studies of gravitional space-time matrices. What are the contributions of C.R. Rao in the field of mathematics? Prof. C.R. Rao's contributions to mathematics cover several areas of statistical theory and applications including characterization of probability distributions, matrix algebra and analysis of diversity. He was elected FRS. (Fig. 20.6)

Fig. 20.6 C.R. Rao

Which numerals have Indian origin? Brahmi numerals.

o o o

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21 Uses of Mathematics How did ancient men used mathematics? The ancient people used mathematics mainly for counting their animals and measuring the size of their fields. The Egyptians used mathematics for marking the boundaries of their fields after floods (Fig. 21.1).

Fig. 21.1 Use of mathematics by Egyptians How has mathematics affected our daily lives? We use mathematics in our daily life—for seeing the time in a watch, for calculating the cost of purchases or for keeping the score in tennis or for measuring the size, volume, weight, etc. In fact, arithmetic is rooted in every kind of human activity. Which branch of mathematics is utilised by the manufacturers. The manufacturers make use of calculus in order to be able to utilize raw materials most effectively.

167

Which branch of mathematics is used by pilots of ships or planes? Geometry is used by them for plotting their course. Which branch of mathematics is mainly used by surveyors? Trigonometry. Surveyors use the method of triangulation for measurement. What is the famous statement of noted Scottish-American mathematician, Eric Temple Bell? He called mathematics as "Queen and servant of all sciences". What is the famous mathematical statement of Thomas Malthus? He said, "The world population increases in geometrical ratio and food only in an arithmetical ratio" Which techniques are used for quality control in industries? Statistical techniques of quality control. What is the use of mathematics in music? The system of musical scales and the theories of harmony are basically mathematical. Which musical instruments were designed with the help of mathematics? Mathematics has helped greatly in the design of pianos, organs, violins and flutes. Which branch of mathematics has helped artists in perceiving three dimensional effects? Three dimensional geometry has helped artists in sketching three dimensional effects on flat surfaces. Which branch of mathematics was born out of painting? Projective geometry. *

Which branches are generally used by the physicists? The physicists mainly make use of algebra, geometry, calculus and trigonometry.

168

What is the most poetic mathematical statement made by James Clerk Maxwell on Fourier's Analytical Theory of Heat?, In 19th century, Maxwell pronunced Fourier's Analytical Theory of Heat as "a great mathematical poem." "Nature is pleased with simplicity, and affects not pomp of superfluous cases" — who wrote this? Newton wrote in his "Principia" — 1687. What is the application of graphs in science? Graphs are used to give a clear picture of the relationship between different quantities (Fig. 21.2).

SPEED AND DISTANCE Time (hours) Man

4 km/h

Bicycle

20 km/h

> Car

?

50 km/h

*

1

2

3.

4

5

4

8

12

16

20

Bicycle

20

40

60

80

100

Car

50

100

150

200

250

Man

Putting jr = time and y = distance, the graphs of the equation? - vx, where v stands for the speed, are straight lines . The faster the speed, the steeper the slope of the graph becomes.

2

3 Time (hours)

Fig. 21.2 Application of graphs 169

Which branches of mathematics are used by chemists? Mainly arithmetical and algebraic operations. What is the use of plane and solid geometry in chemistry? These two branches of mathematics are used in chemistry to study the way in which atoms or ions are combined. What is the use of algebraic formulae? The laws of physics, chemical equations and chemical laws are expressed by means of algebraic formulae. What is the use of mathematics in astronomy? The astronomers measure angles and carry out mathematical calculations regarding apparent motions of the sun, stars, moon and planets. What for the figure 420 is used? It is used for indicating forgery and disgrace. Which branches of mathematics are used in astronomy? Arithmetic, algebra, plane geometry, solid geometry, trigonometry, calculus, etc. are used in astronomy. What is the use of mathematics in biology? Mathematics is proving very useful in genetics to calculate the percentage of individuals with like and unlike traits in succeeding generations. The geneticists also make use of the theory of probability. What is the use of dimensional analysis in biology? Dimensional analysis is used to analyse the growth patterns of certain animals. It is also used to determine the ratio between the life time of a given animal and the time required by the animal to draw a single breath. Where do we see parallel lines in our daily life? Staircases, railway tracks, etc. are the examples of parallel lines. Which number is used for black money? Number two.

170

Which number is considered unlucky? 13 (thirteen). Which geometrical figure is considered auspicious in Christian religion? A triangle. Why mathematics is considered an endless source of entertainment? For many centuries, man has used mathematics as a source of entertainment in the form of games, constructions and brain twisters. Which geometrical shape is observed in a honeycomb? In a beehive, we see a network of regular hexagons. In which branch of physics, are complex numbers used? These are used in electricity. o o o

171

22. Miscellany Give an example of permutation? Suppose you have three Cards—plane, dotted and striped, these cards can be arranged in six ways as shown in Fig. 22.1

Fig. 22.1 Permutation of three cards How do we calculate the population density of a country? Dividing the total population by the area in square kilometers we get the population density of a country. Where do we use Dewey decimal system? The Dewey decimal system is used to classify and order books in a library. 172

Who gave the present form of decimal point? John Napier gave the present form of decimal point. How do we differentiate between permutations and combinations? Give one example. Suppose we have four items as shown in Fig. 22.2. They can be combined in 6 ways by taking two together. But their permutations will be 12.

Fig. 22.2 Combination What is the meaning of symmetry in mathematics? Two shapes are said to be symmetric, if they are mirror images of each other or, if they can be placed one upon the other exactly in all respects by 180 degrees rotation around a point. For example, the diagonals of a parallelogram divide it into four symmetrical figures. Symmetry preserves length, angle but not necessarily orientation, it has exact likeness in shape about a given line (axis), point or line. (Fig. 22.3).

Fig. 22.3 Symmetrical figures 173

What are congruent shapes? The shapes exactly of the same size are called congruent shapes such as two envelopes, two spoons as shown in Fig. 22.4.

Fig. 22.4 Congruent shapes What is permutation? When we choose a few things from many and arrange them in various ways, this is called permutation. In other words it is a speciafied arrangement of a group of objects. What is combination? When we choose a few things from several without bothering about their order, such way of choosing is called combination.

174

What are similar shapes? Similar shapes are those which resemble each other, not necessarily of the same size. For example, the two leaves of a plant are similar in shape (Fig. 22.5) What are magic squares? Magic squares are the squares with some digits. When these digits are Fig. 22.5 Similar shapes added horizontally, vertically or diagonally the sum is always the same (Fig. 22.6).

1 15 12 6 8 10 13 3

14 7 11 2

4 9 5 16

Fig. 22.6 Magic square

What is a Harshad number? A number which is divisible by the sum of its own digits is called a Harshad number. What is the shape of an umbrella? Its cloth is in the shape of a regular octagon (Fig. 22.7).

Fig. 22.7 Umbrella shape 175

How does a regular dodecahedron appear? It appears as shown in Fig. 22.8.

Fig. 22.8 Dodecahedron How does a regular icosahedron look like? It looks like as shown in Fig. 22.9.

Fig. 22.9 Icosahedron How does a regular octahedron look like? It looks like as shown in Fie. 22.10

Fig. 22.10 Octahedron 176

How do we trace a French curve? It is traced as shown in Fig. 22.11 How do we trace limacon? It is traced as shown in Fig. 22.12. How do we trace serpentine? It is traced as shown in Fig. 22.13.

Fig. 22.11 French curve

Fig. 22.13 Serpentine

Fig. 22.12 Limacon

Square root of - 1 is denoted by i(iota). Who gave this letter? Leonhard Euler, a Swiss mathematician and physicist. Which number is used for millennium? 1000. Which numbers are used for billion and trillion? 109 and 10 12 How do we define logarithm? If a" =m, then log a m = x, where, a is known as the base. What is De Moivres theorem? (Cos 0+ / Sin9) n =(Cos n6+ i sin6) What is the equation of ellipse? x

a

2 2

2

+ y b2

= 1, where a and b are semi-major and semi-minor axis.

What is spherics? It is the study of geometry and trigonometry of figures on the surface ot' a sphere. 177

Who developed theory of relativity? Albert Einstein. Who gave the mathematical form to electromagnetic theory? James Clerk Maxwell. Which mathematician won the Nobel Prize in literature? Bertrand Russell. What is a Rubic's cube? It is a type of cube which has become the most popular^juzzle today. Who is Narendra Karmarkar? He is a mathematician working in USA and best known for discovering Karmarkar Algorithm. Indian National Science Academy publishes a reputed journal of mathematics. What is its title? Indian Journal of Pure and Applied Mathematics. It is now a monthly journal. Indian Statistical Institute, publishes a renowned Journal of Statistics. What is the title? Sankhya. It is still one of the few professional journals published from India with a global reputation. Which medal is awarded to outstanding mathematicians by Indian National Science Academy? Ramanujan Medal. Where is the famous Indian Statistical Institute situated? Calcutta. Which institute of mathematics is situated in Hyderabad? School of Mathematics and Computer Information Sciences. In which city Ramanujan Institute of Mathematics is situated? Chennai (Madras).

178

W h a t is a Moebius strip? It is a strip of paper whose two ends are pasted together in the form of a ring, after giving one end a twist of 180\ Its one side is coloured. W e observe as if the whole ring inside and out is coloured (Fig. 22.14).

Fig. 22.14 Moebius strip W h a t are M a y a n M o n u m e n t s ? The Mayans were a powerful Indian Nation in Mexico and central America about 1500 years ago. There people used numerals which looked like human faces for recording dates. W h a t is the Rhind Papyrus? The Rhind Papyrus, written in Egypt more than 3500 years ago is the oldest known book on mathematics. It contains problems about the areas of triangles and rectangles. H o w does a cycloid look like? It looks like as shown in Fie. 22.15

Fig. 22.15 Cycloid

179

In the King James Version of the Bible, what do you read about the average span of a human life? It is given as 'three score and ten', which means 70 years. Each score is equal to 20. What are transcendental numbers? It is the other name for irrational numbers. What is a mathematical table? Arranging things or numerical data in rows and columns is called a mathematical table (Fig. 22.16).

© 20

0 6© 16

6

4

14

Fig. 22.16 Mathematical table What would we call the figure 22.17? Crescent. It is just like the shape of a moon when it appears less than half-illuminated. What is a hexahedron? A hexahedron is a solid figure which has six faces. The common name for a regular hexahedron is cube. Buddhist use a number asankhyeya. Asankhyeya is equal to 10 !4 °.

Fig. 22.17 Crescent

What does it mean?

What do we call 10 1 0 0 ? It is called Googol. This term was devised by Dr. Edward Kasner of USA. To how many decimal places, the value of K has been calculated by the French mathematicians Guilloud and Boyer? The greatest number of decimal places to which pi (it) has been calculated

180

is 1,000,000 by the French mathematicians Jean Guilloud and Mile Martine Bouyer on CDC 7600 computer. What the most accurate version of "pi"? The most decimal places to which pi (re) has been calculated is 1073740000 by Yasumasa Kanada and Yoshiaki Tamura of the University of Tokyo, Japan on 19 November, 1989 using a Hitae S-820/80E computer. What is a cuboctahedron? A figure having 14 faces with triangles and squares (Fig. 22.18)

Fig. 22.18 Cuboctahedron What is a icosadodecahedron? A figure having 32 faces with triangles and pentagons (Fig. 22.19).

Fig. 22.19 Icosadodecahedron

181

Which mathematical term means without end or limitless? Infinity. It is denoted by the digit of eight written horizontally (°°).

What is the earliest measure of weight known to man? The earliest measure of weight known to us is the bequa. It erusted in Egyption civilization some 3800 BC. The weights were cylindrical with rounded ends f r o m 188.7 to 211.7 gm.

Which people do not use any number system? The Nambiquara, an Indian people who live in Brazil's Mato Grosso province on the fringes of the A m a z o n jungle, use no system of numbers at all.

Which is the oldest magic square? The world's oldest magic square is Chinese square described in the Chinese book of Divination. It is called the One Ching. It was written in 12th century BC. It makes use of numbers from 1 to 9. Every row, every column and both diagonals add up to 15. (See Fig 22.20)

Fig. 22.20 Oldest magic square

182

Which is the longest Romman Numeral? The date requiring most Roman letters is AD 1888 with 13 viz MDCCCLXXXVIII. It was used on the entrance to the high court of New South Wales completed in that year so drawing the comment that the building would become equally famous for the length of its sentences. What is the longest measure of time? The longest measure of time is the KALPA in Hindu Chronology. It is equivalent to 4320 million years. Which of the lines b or c is aligned with oblique line a in fig 22.21? c

b

a

Fig. 22.21 Line a is aligned with c What is the meaning "Congruence" in geometry? Congruence in geometry is related to figures having the same shape and size, as applied to two-dimensional or solid figures.

ooo

183

23 Mathematical Brain Twisters A shopkeeper has four weights for weighing 1 kg to 40 kg (integral values only). What are these four weights? 1 kg, 3 kg, 9 kg and 27 kg weights. Can you imagine a number which is equal to the cube of the sum or its digits? 4913 is the cube of 17 which is the sum of its digits. Can you tell the number of triangles in the following figures? Fig. 23.1 has 20 triangles, while Fig. 23.2 has 47 triangles.

Fig.23.2

Fig. 23.1

184

What is the sum of 'n' natural numbers? n

( n + l ) is the sum of 'n' natural numbers. 2 What is the base of the binary number system? 2 (two). An old lady deposited one rupee with a shopkeeper on interest. The interest rate told to her was to make her money double every year. After fifteen years,she demanded back her money. How much should she get. Rs. 32768. Which number appears same if read in conventional manner or reverse manner? 1961. A boy celebrates his birthday only after every four years. What is his date of birth? 29 February. ^ ^ Can you tell which line is longer in Fig. 23.3. Both are of equal lengths.

»

c Fig. 23.3

How can you arrange 25 mangoes in 12 rows so that each row contains 5 mangoes? See Fig. 23.4, 23.5 and 23.6.

Fig. 23.4

Fig. 23.5 185

Fig. 23.6

One cat tells the other cat that there are two cats in front of me. The other cat also tells that I also have two cats behind me. How many cats are there? Three. How can 17 oranges be shared by three boys, so that the first gets half of it, the second gets one third of it and the third gets one ninth of it? It seems to be impossible, but if you add one more orange to these so that the number becomes 18, these can be distributed. First will get 9, second will get 6 and third will get 2. Sum of the oranges remains 17 and your orange is saved. A car has a three digit number which is the square of some number. The other car also has a three digit square number, but the first digit of the number of first car has become the last digit of the second car. What are the numbers of these two cars? 196 and 961 The small hand of a clock is at 12 and the large hand makes an angle of 90° with the small hand. What is the time in the clock? Twelve past fifteen minutes. • From the dots in Fig. 23.7, how many equilateral « « triangles can be formed? • • • Fifteen equilateral triangles. Fig. 23.7 • • • • How many squares can be drawn from the dots of Fig. 23.8? Five.

• • • • • • • • •

How many squares can be drawn from the dots of Fig. 23.9 ? 20 squares How many triangles can you make with 17 matches? (each side of the triangle should consist of whole matches) Eight triangles: 8-8-1, 8-7-2, 8-6-3,8-5-4,77.3, 7-6-4, 7-5-5, 6-6-5 186

2 1 8

•

•

•

•

•

•

•

•

•

•

23 9

What MDCCCLXXXVUI stands for ? 1888.

Which biggest number can you write with four Is? II11 Can you prove 2=1.? Suppose,

a = a2 = a 2 -b 2 = (a+b) (a-b) = a+b = 2b = 2 = (Here, assumption a=b is wrong)

b ab ab-b 2 b(a-b) b b 1

Can you tell, how many squares are there in Fig. 23.10. 14 squares You have a figure written in Roman numerals MDCLXVI. What number does it represent? M = 1000, D =500, C=100, L=50, X=10 and VI=6. So, the number is 1666. Fig. 23.10 How would you write 1789 in Roman numerals? MDCCLXXXIX. How many squares are there in the Fig. 23.11? 11 squares. What is the biggest number that can be represented by two digits? 9 9 and not 99. Fig. 23.11 187

How many triangles are there in Fig. 23.12 5 (five). From a deck of 52 cards, one card is drawn. What is the probability of getting an ace? 1/13. What is the sum of the squares of first 'n' natural numbers? n(n+l)(2n+l)

6

Fig. 23.12

What will be the sum of the squares of the first nine digits? 285. What is the sum of the first 70 odd numbers? 4900. Which is a two digit number whose square root is equal to the sum of its digits? 81.

Which two digit number is equal to the area and perimeter of a square? 16.

A shopkeeper has six weights and by combining them he can weigh from 1 kg to 364 kg. What are these six weights? 1 kg, 3 kg, 9 kg, 27 kg, 81 kg and 243 kg. What is the absolute value of - 4.5 4.5 Can you find out three digits such that the sum of the cubes of each digit gives a three digit number containing the same digits? I 3 + 5 3 + 3 3 = 153.

188

What does 1530 hr. mean? 1530 hr. is actually 30 minutes past 3 PM. How many cubes are there in the Fig. 23.13? The shaded part of the top shows 6 cubes, while the shaded part of the bottom shows 7 cubes.

Fig. 23.13 Are the letters in the Fig. 23.14 inclined or vertical? Vertical.

How can you divide 500 into two parts so that one part is a multiple of 47 and the other a multiple of 19?. (7x47) + ( 9 x 19) = 5 0 0 . A Which line is longer, A or B in the Fig. 23.15? Both are of the same length. At 6 ' 0 clock the hands of a clock are exactly opposite to each other. When are they next exactly opposite to each other? At 5 — L minutes past seven. 2 How many triangles are there in the Fig. 23.16? n

Fig. 23.15

B

a

Fig. 23.16 What is the probability of getting 5 in a single throw of a die? 1/6. If a clock in a mirror tells twenty to three, what time it is? Twenty past nine.

190

What is the area of the shaded portion, in fig. 23.17 when the diameter of each circle is 1 metre 1+

sq.m. 4 How you a r r a n g e the n u m e r a l s 1,2,3,4,5,7,8,9 in two groups, four in each group, so that the sums of the numbers placed in each group will be the same? 173 + 4 = 177 85 + 92 = 177

Fig. 23.17

How can you fill the circles with numbers from 1 to 12 in fig. 23.18 so that each row when added gives a sum of 26. Now number should be written twice. See Fig. 23.18

Fig. 23.18 How would you fill the magic star with numbers so that each of the 5 lines add up to 24. No number is written twice. See fig. 23.19 Fig. 23.19

191

Six lollipos are arranged as shown in figure 23.20. How would you write digits from 1 to 7 in such a way so that each row with three digits adds up to 12. See fig. 23.20 How would you fill the digits 1 to 14 in the seven point star such that digits in the four circles of each of the eight rows add up to 30. See fig. 23.21. How would you put the digits from 1 to 16 in the circles of an eight point star such that each line of four circles adds up to 34. See fig. 23.22

Fig. 23.21

Fig. 23.20

Fig. 23.22

The average age of 10 students in a class is 18 years. When one more student joins the class the average age becomes 17 years. What is the age of 11th student. 7 years. In a row of girls Anita is 14th from the right and 13th from the left. How many girls are there in the row? 26 girls. 192

Rakesh started a business after investing Rs. 70,000. After 8 months Sumit joined him with a capital of Rs. 1,80,000. In what ratio whould Rakesh and Sumit share the profit? The required ratio = 70000 x 12 : 1 8 0 0 0 0 x 4 = 84 : 72 = 7 : € Note: Rakesh invested Rs. 70,000 for 12 months and Sumit invested Rs. 180000 for four months) If in an NCC parade, the cadets are made to stand in columns of 45, then 18 columns are formed. If only 30 cadets are made to stand in each column, how many columns would be formed? 45 x 1 8 / 3 0 = 27. What is the oldest mathematical puzzle? This puzzle was found in the "Rhind papyrus", an Egyptian scroll bearing mathematical table and problems, copied by the scribe Ahmes (c. 1650 B.C.) This reads as follows: "As I was going to St. Ives, I met a man with seven wives. Every wife had seven sacks, and every sack had seven cats. Every cat had seven kits. Kits, cats, sacks and wives — how many were going to St. Ives?"

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193

24 Units and Measurement System What are the three systems of units used in the world? The three systems of units are: i) MKS system, ii) CGS system, and iii) FPS system. What does MKS stand for? MKS stands for meter, kilogram and second. Meter is the unit of length, kilogram is the unit of mass and second is the unit of time. What does CGS stand for ? CGS stands for centimeter, gram and second. Centimeter is the unit of length, gram is the unit of mass and second is the unit of time. What does FPS stand for? FPS stands for foot, pound, second. Foot is the unit of distance, pound is the unit of mass and second is the unit of time. What is SI system of units? It is am internationally agreed coherent system of units derived from the MKS system, now in use for all scientific purposes. In this system mass is measured in kilograms, length in meters, time in seconds, current in Amperes, luminous intensity in Candela, temperature in Kelvin and quantities of chemicals in moles. In 1960 an international conference on weights and measures recommended the universal adoption of SI units. 194

The table below shows the basic units/supplementary units, their symbols and measurements. SI UNITS — Symbol / Measurement Symbol

Basic Units Metre Kilogram Second Ampere Kelvin Mole Candela Supplementary Units Radian Steradian

Measurement

m Kg S A K mol cd

length mass time electric current thermodynamic temperature amount of substance luminous intensity

rad sr

plane angle solid angle

For multiple births we use the terms — twins, triplets, quadruplets, quintuplets and sextuplets? What are the meanings of these terms. When more than one child is born at a time we use the term twins for two, triplets for 3, quadruplets for 4, quintuplets for 5 and sextuplets for 6 (Fig 24.1).

Fig. 24.1 Multiple births (triplets) Multiple Births: 2 Twins, 3 Triplets, 4 Quadruplets (quads), 5 Quintuplets (quina), 6 Sextuplets

195

What is a score? A score is equal to 20 items. How many items are there in a gross? A gross contains 144 items or 12 dozen items. How many papersheets are there in a ream? 480 paper sheets. Nowadays we call a bunch of 500 sheets as a ream. Which prefix is used to write 10 2 ? Hecto. In USA, people use the words Pony, Century, Monkey and Grand as slangs for money. What do these terms indicate? Pony is used for 25 dollars, Century for 100 dollars, Monkey for 500 dollars and Grand for 1000 dollars. What do milli, micro, nano, pico, femto and atto stand for ? Milli is equal to 10 -3 , micro is 10 -6 , nano is 10~9, pico is 10 -12 , femto is 10~15 and atto is 10" 18 What do kilo, mega, giga and tera stand for? Kilo is equal to 103, mega is 106, giga is 109 and tera is 10 12 . What do million, billion and trillion stand for? One million is 106, one billion is 109 and one trillion is 1012. Which number is indicated by deci, centi and deca respectively? Deci indicates 1/10, centi l/100th, while deca indicates 10. What do the prefixes semi, hemi and demi stand for? All these three prefixes are used to show 1/2. Which number is shown by the prefix 'uni'? It represents 1. Bi, di, tri, ter, tetra, quadri, penta, quint are well known prefixes. What do they indicate? Bi and di are used to indicate 2; tri and ter for 3, tetra and quadri for 4, and penta and quint for 5. 196

Which numbers are represented by the prefixes sexa or hexa, hepta or septa, octa, non or nona? Sexa or hexa is used for 6, hepta or septa for 7, octa for 8 and non or nona for 9, Which prefixes are used to show 11,12,15 and 20? For 11, hendeca, undec or undeca is used. For 12, dodeca is used, for 15 quindeca is used and for 20 icos, icosa or icosi is used. Which number is indicated by googol? It indicates 10 100 or one followed by 100 zeros. What is the relation between mile and kilometer.? One mile is equal to 1.6093 km. What is the relation between a yard and a meter? 1 yard is equal to 0.9144 meter. How many centimeters are there in an inch? One inch = 2.54 cm. Which unit is used to measure large distances such as of stars? Light Year. What is a Light Year? It is the distance travelled by light in one year. It is equal to 58,80,030 million miles (Fig.24.2) \ • moon

-(SUN

alpha centauri

4.3 years

Newvork

1/50 in second London

Fig. 24.2 Light year 197

What is the relation between a kilogram and a pound? One kg = 2.2046 pounds. What is the relation between ounce and grams? 1 ounce is equal to 28.35 gms. What is the relation between a pint and litres? One pint is equal to 0.57 litres. What is the relation between a gallon and litres? One gallon is equal to 4.55 litres. What is the unit of area? Square meters or square centimeters. What is the unit of volume? Cubic meter or cubic centimeter. What is the unit of speed? Speed is expressed in meter/sec or kilometer/hr. What is the unit of force? In MKS system, unit of force is Newton, while in CGS system it is dyne. 1 Newton = 105 dynes. Which units are used to measure angles? Angles are measured in degrees and radians. One radian equals 57.2958°. What are the further sub-divisions of a degree? Degree is further sub-divided into minutes and seconds. One degree=60 minutes and one minute= 60 seconds. How is time divided into different units? Time is measured in day, hours, minutes and seconds. One day is equal to 24 hrs, one hour is equal to 60 minutes and one minute is equal to 60 seconds.

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How do we define a radian? One radian is the angle at the centre of a circle that cuts off an arc on the circumference, which is equal in length to the radius (Fig. 24.3). What is the value of 7u ? ri =3.14159 How many litres are there in one cubic meter? 1000 litres are equal to one cubic meter. How many cubic centimeters are there in a litre? 1000 cubic centimeters are equal to one litre.

Fig. 24.3 Radian

Which unit measures water depths? Fathom. One fathom = 6 ft. Fermi is used for? Fermi is used to measure the radii of nuclei. One Fermi = 10~15m. How yard came into existence? The English King Henry I, established the yard early in the 12th century as the distance from the tip of his nose to the tip of his outstretched thumb. Edward I redefined it as 3 feet in 1305 A.D. What is the relation between barrel and gallons? One barrel = 31.5 gallons. What is par sec? It is the unit of distance. One par sec is about 200,000 times the distance between the earth and the sun. What is mach number? It is a unit used to measure supersonic speeds. One mach number is equal to the speed of sound in air, i.e. 340m/sec. 199

How do we define a degree? One degree is the angle at the centre of a circle that cuts off an arc that is 1/360th of the circumference (Fig. 24.4)

Fig. 24.4 Degree What is an Angstrom unit? One Angstrom unit = 10~10m. What is astronomical unit? It represents the average distance between the sun and the earth. Its value is 149,600,00 km. One light year =60,000 A.U. Who gave the unit acre? The word acre comes from the Latin 'ager' meaning field. It was introduced initially by Edward I of England. An acre was the amount of land, a yoke of oxen could plough in a day. As the performance of oxen could vary widely, Edward fixed the acre 40 rods long by 4 rods wide (each rod being 16.5ft long). This measurement has survived unchanged. Where did foot originate? It originated in Rome. A foot originally was the length of a man's foot from heel to big toe. The Romans divided it into 12 parts now known as

Which unit is used to measure the speed of ships? Knot. One knot = 6080 ft (Fig. 24.5).

Fig. 24.5 Knot How cubit had its origin? The cubit was introduced by Noah. It was the length from the elbow to the tip of the middle finger. It is about 460 mm or 18 inch. How furlong came into existence? The furlong comes from the phrase, a furrow long, meaning the length of a furrow in a standard square field of 10 acres. Later it was defined by Romans as one- eighth of a mile i.e. 220 yards or 201m. How mile came to exist? The Romans were the first to invent the mile. For them it was equivalent to one thousand paces. Now it is taken as 1760 yds or 5280 ft or 1.6 Km. What is a cosmic year? It is the period of rotation of the sun around the centre of the milky way gallery. It is about 225 million years What is the smallest linear unit? The shortest unit of length is the atto-metre which is 1.0 x 10"' 6 cm. What is knows as megalithic yard? The unit of length used by the megalithic tomb-ouilders in north-western Europe, C. 3500 B.C. This was deducted by Prof. Alexander Thom (1894-1985)in 1966.

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25. Fun with Numbers A wonder sum that is always the same. Take any three digit number in which the first digit is larger than the last say 725. Reverse it making 527 and subtract the smaller from the larger, making 198. Now add the result to the same number reversed, that is 891. The answer is 1089 — and will be 1089 whatever number you start with. Time you would take to count one billion. If a person counted at the rate of 100 numbers a minute, and kept on counting for eight hours a day, five days a week, it would take a little over four weeks to count to 1 million and just over 80 years to reach one billion. How can you tell the age of your friend? Ask him to write down his age say 16. Then double it, making 32. Add one to it, making 33. Multiply it by 5, making 165. Now add 5 to it, making 170. Now multiply it by 10 making 1700. Then subtract 100 making 1600. Now ask him to tell the number. You delete the last two digits. The remaining number will be his age i.e. 16. Describe a number game such that by manipulations answer is always 15 Think any number say 12, multiply it by 5 making 60. Add 25 to it, making 85, divide it by 5, making the quotient 17, subtract the number you started with 17-12 = 5. Multiply the remainder by 3 making it 15. So answer will always be 15.

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To tell the sum of a square of three numbers in a calender. Ask your friend to choose three dates in a line from a calender. Ask him to choose three dates in a column starting with the lowest number. Then you can tell the sum of 9 digits of the square so formed (Fig 25.1). How? Ask your friend the least number which he has thought. Add 8 to it and multiply by 9. This will give you the sum. Mon

Tue

Wed

Thu

Fri

Sat

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

IO ro

Sun

23

24

25

26

27

2S

29

30

31

Fig. 25.1 The Age Detector Table The table overleaf will help you in guessing the age. What you have to do is the following Ask your friend to tell you in which colums his age appears. After knowing that add up the numbers at the top of the columns indicated and that sum will be the age of your friend. For example Your friends' age is 23. This number as you can see is present in first, second, third and fifth columns of the table. The top numbers in those four columns are 1,2, 4, 16 - which add up to 23 (see table next page)

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The table is 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63

2 3 6 7 10 11 14 15 18 19 22. 23 26 27 30 31 34 35 38 39 42 43 46 47 50 51 54 55 58 59 62 63

4 5 6 7 12 13 14 15 20 21 22 23 28 29 30 31 36 37 38 39 44 45 46 47 52 53 54 55 60 61 62 63

8 9 10 11 12 13 14 15 24 25 26 27 28 29 30 31 40 41 42 43 44 45 46 47 56 57 58 59 60 61 62 63

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16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

Tell number of brothers and sister of your friend If both mother and father are alive: Here is a simple trick by which you can tell the number of brothers and sisters of your friend. Step-1 Add the number of parents (obviously it will be two in each case). Step-2 Add to step 1 the number of brother to it and multiply the result by 2 and add one to it. Step-3 Now the result of step-2 is multiplied by 5 and the number of sisters is added to it. Step-4 Subtract 25 from step-3 result. Step-5 In the result of step-4 the unit's place indicates the number of sisters and the tenth place the number of brothers. Hence we can write the formula as (parents + brothers) x 2 + 1) x5 + sisters - 2 5 How to prove 3=4 By the given method you can prove 3 =4 As you all know - 12 = - 12 This can be written also as 9-21 = 16-28 or (3) 2 - 7 x3 = (4) 2 - 7x4 (3) 2 - 7 / 2 x 3 x 2 = (4) 2 - 7 / 2 x 4 x2 Adding (7) 2 /2 on both sides we get or or or

(3) 2 - 7/2 x 3 x 2 + (7) 2 /2 = (4) 2 - (1)12 x 4 x 2 + (7) 2 /2 (3) - (7) 2 /2 = ( 4 - 7 / 2 ) 2 3 - 7 / 2 = 4 - 7/2 3= 4

Juglary of three digit numbers 1. Consider any three-digit number. 2.

Repeat this three-digit number to form a six digit number.

3.

Now divide the number by 7.

4.

Divide the answer by 11.

5.

Now divide the answer by 13.

You have ended with the same number you started with. (See examples — next page) 205

Example - 1 : 1. Consider the three-digit number : 2. Now repeat this three-digit number to form a six-digit number : 3. Dividing it by 7, we get: 4. Dividing 51766 by 11, gives: 5. And now dividing 4701 by 13, gives: Example 2: 1. Consider the three-digit number 2. Now repeat this three-digit number to from a six-digit number : 3. Dividing it by 7, we get: 4. Dividing 112827 by 11, gives: 5. And Now, dividing 10257 by 13 gives: Magic number Here is another number which will give you a lot of surprises. This magic number is 142857. Look at these 142857 x 2 = 285714 142857x3=428571 142857x4 = 571428 4 142857 x 5 = 714285 142857x6 = 857142

362

362,362 51766 4701 362 789 789,789 112827 10257 789.

5

7

It will be very clear to you by now that if you multiply 142857 by 2,3,4, 5, 6 you will get same figures in the same order, starting in a different place each time, as if they were written round the edge of a circle. Get a Lots of '2' If you want to get lots of 2 just by adding numbers do the following. Write number from 1 to 9 in one line and then 9 to 1 in the other and again 1 to 9 on the third and 9 to 1 in the fourth. In the fifth just write 2 and add. Then you will get lots of 2. 1 9 1 9

2 8 2 8

3 7 3 7

4 6 4 6

5 5 5 5

6 4 6 4

7 3 7 3

8 2 8 2

9 1 9 1 2

2 2 2 2 2 2 2 2 2 2

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26. Mathematical Signs and Symbols +

Plus, the sign of addition, e.g. 5+3. It also denotes a positive quantity, e.g. +3. Minus, the sign of subtraction. It also denotes a negative quantity.

x

Sign of multiplication.

+

Sign of division. Dot at the centre of the two n u m b e r s is the sign of multiplication.

>

S ign of greater than and equal to.