Mathematics: The Children's Treasury of Knowledge

THE CHILDREN'S TREASURY OF KNOWLEDGE

M a t h e m a t i c s

TREE OF NUMBERS

April weather

licence plate

22-56 classification and ordering ; ^ ^ e l e c t r o n i c computer

pocket c a l c u l a t o r the binary system

abacus

discovery of zero

Roman numerals

1 2 3 4 5

Chinese numerals

Mayan monuments

Arabic numerals

the Rhind papyrus

the Rosetta stone

THE ROSETTA STONE The Rosetta stone is a tablet that was found at one of the mouths of the River Nile. It became a key to the meaning of Egyptian hieroglyphics (picture-writing). The tablet also recorded ancient Egyptian numerals.

MAYAN M O N U M E N T S The Maya were a powerful Indian nation in Mexico and Central America about 1 500 years ago. There, people used numerals that looked like human faces for recording dates.

THE RHIND PAPYRUS The Rhind papyrus, written in Egypt more than 3 500 years ago, is the oldest known book on mathematics. It contains problems about the areas of triangles and rectangles.

THE CHILDREN'S TREASURY OF KNOWLEDGE

Mathematics Translated from Kodansha's Children's Colour Encyclopaedia

Adapted and edited by the editors of FEP International Ltd.

Distributed by Time-Life Books

Printed in Singapore under the supervision of Time-Life Libraries (Asia) Pte., Ltd.

Text by Yoshikazu Horiba Teacher of Saginomiya High School

Layout by Mitsumasa Anno

Book design by AD 5

ACKNOWLEDGEMENTS Photographs, illustrations, and data appearing in this book have been made available through the courtesy of Agency of Industrial Science and Technology; Eiji Hamano; Fujikato; Geographical Survey Institute; Hagley and Hoyle Pte., Ltd.; Haruo Fujiwara; Hiroo Tachibana; J.O.; John Bartholomew & Son Limited; Kiyoshi Kuwana; Kokunai Jigyo Koku Co. Ltd., Kozo Kakimoto; Kyodo Tsushin; Mitsumasa Anno; National Theatre; North American Newspaper Alliance; Pan-Asia Newspaper Alliance; St. Mary's International School; Seisen International School; Tadao Tominari; Takeo Nakamura; Tsurunosuke Fujiyoshi; Yasuji Mori. The publishers wish to thank Mrs Fay Palmer for her assistance.

©

Kodansha Ltd. 1970,1975 All rights reserved.

CONTENTS Page 7

SETS Making sets; Relationship between sets NUMBERS

11

The history of numerals; Numerals of today; What numbers stand for; How to write big numbers; Addition; Subtraction; Addition and subtraction; Rules of addition; Multiplication; Multiplication table; Multiples and common multiples; Division; Multiplication and division; Rules of multiplication; Factors and common factors; Fractions; Decimals; Inequalities and equations; Tools of calculation; Positive numbers and negative numbers SHAPES

47

Interesting shapes; Simple shapes; Lines and angles; Parallel and perpendicular; Triangles; Quadrilaterals; Circles; Various curves; Solid shapes; Positions of points; Mathematical models; Similarity and congruence; Reduced copies and enlarged copies; Symmetries QUANTITIES

73

Length; Area; Volume; Weight; Time; Motion and speed; Direct proportions; Inverse proportions; Ratio and percentage; Probability STATISTICS

95

Tables; Graphs; Classification and ordering FAMOUS PEOPLE IN MATHEMATICS

103

TABLE OF UNITS

110

INDEX

111

Abbreviations used in this series:

LENGTH metre centimetre kilometre millimetre

= = = =

DENSITY kilogramme per cubic metre = kg/m ! gramme per cubic centimetre = g/cm3

m cm km mm

VELOCITY AND SPEED metre per second = m/s kilometre per hour = km/h

MASS kilogramme = kg gramme = g tonne = t

POWER watt = W kilowatt = kW horse power = h.p.

TIME second = s minute = min hour = h AREA square metre square centimetre square millimetre hectare

TEMPERATURE Temperature (common) = degree Celsius = °C Absolute temperature = K m2 cm2 mm2 ha

VOLUME m cubic metre cubic centimetre = cm3 litre I millilitre = ml

PRESSURE (FOR METEOROLOGY) millibar = mb bar = b

SETS

Groups of things that go together are called sets. Two water-melons make up a set. Three pineapples also make up a set. A bunch of green grapes and a bunch of purple grapes make up a set. Even a single melon can be called a set. We can also put tangerines and oranges together to form a single set. The idea of sets is basic to arithmetic. From now on, arithmetic will unfold in terms of the idea of sets. We shall learn how to split up a group that contains several different things. We shall learn the relations between various sets that contain different objects, and much more.

'ARIOUS KINDS OF SETS

7

M A K I N G

Q F T 9

A set of animals in a w i l d l i f e park.

*A is a group of things put together. * A part of a set is called a ^t/A^e? of the set. * We say that a subset of a set is contained in the set. In a wildlife park, for example, we can say that the giraffes are a subset contained in the set of animals. All the animals kept in the wildlife park together form a set.

Animals in a wildlife park make a set. The set of animals in the park is a part of the set of all animals. A part of a set is called a subset. The set of elephants, the set of crocodiles, the set of

lions, and the set of birds are all subsets of the set of all animals. The meat-eating animals make a subset of the set of all animals. This subset contains the set of lions. The animals that fly also make

a subset of the set of all animals. This subset contains most birds. The set of all animals is divided into the subset of flying animals and the subset of non-flying animals.

Sets of horses and cows in corrals.

The idea of sets is the most basic in modern arithmetic. A set is a collection of clearly defined things. For example, the nations of the world, the numbers, or the letters of the alphabet all make sets. However, our neighbours cannot pass as a set because it is not always clear whether a certain person is our neighbour or not. Each member of a set is called an element of the set. An element of the set of monkeys is an individual monkey. Each element of the set of monkeys belongs to the set of all mammals. When each element of a set also belongs to another set, we say that the first set is contained in the second set. We use the symbol <=. to mean 'is contained in'. For example, the set of lions c: the set of meat-eating animals. The set of human beings cz the set of animals. The set of odd natural numbers (1, 3, 5, 7, 9, ...) ci the set of natural numbers (1, 2, 3, 4, 5, ...). 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, 8, ...).

RELATIONSHIP BETWEEN SETS

" T h e set composed of two sets lumped together is called the union of the two sets. * A set consisting of all objects in both of two sets is called the intersection of the two sets. * A set with no element is called an empty set. This is the union of two sets which have no common part.

INTERSECTIONS

John and Ann are brother and sister. Let's see what they have. They share the things in the part coloured green. These things make the intersection of the set of things John has and the set of things Ann has. SETS THAT HAVE NO C O M M O N PART

The boys are wrestling, and the girls are skipping. Here, the set of girls and the set of boys have no common part.

The purple area in Fig. 1 shows the union of two sets of points inside the two circles. The red area in Fig. 2 shows the intersection of two sets of points inside the two circles. The intersection of the set of boys wrestling and the set of girls skipping shown on this page is an empty set. 10

The intersection of t w o or more sets is the set of elements that are shared by all of the sets. For example, the intersection of the set of meat-eating animals and the set of mammals contains the set of lions. The union of given sets is the combination of the sets. For example, the union of the set of an orange and the set of two other oranges is the set of three oranges. The union of the set of flying animals and the set of non-flying animals is the set of all animals, but the intersection of these two sets has nothing in it. A set that has nothing in it is called an empty set.

Numerals, the symbols for numbers, were invented first, then fractions and decimals. Machines are now used for fast and accurate calculations involving large numbers. Let's learn about the history of numerals and the addition, subtraction, multiplication, and division of numbers.

NUMBERS

1

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17 21 25 29 33

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HKFS KUSB OHS SflIF USOFKJ

...

The opening ceremony of the World Student Games.

11

THE HISTORY OF NUMERALS

"The idea of matching, or one-to-one correspondence, was used by our ancestors to count things. * People first used their fingers or other familiar things as symbols for numbers. *Arrangements of knots were used to record numbers. "The number zero was invented in India. * Mot all countries use the same numerals.

ONE-TO-ONE CORRESPONDENCE SHEEP AND MARKS ON A TREE

Our ancestors did not have numerals, but they could keep count of their sheep. How did they do it? 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 in, they matched each sheep with a mark on the tree. In this way, they could tell if there was any change in the number of sheep.

Familiar things such as fingers were used to indicate numbers. The members of a community might agree to use the picture of a lion's head to indicate one. feathers of an eagle to indicate two, the leaves of a clover to indicate three, and so on.

Some people pointed to parts of their body to indicate numbers. For example, the little finger stood for one, the third finger for two, and so on. For eight, they pointed to the elbow. A similar method is still used by a native people in New Guinea.

USING FAMILIAR THINGS TO TAKE THE PLACE OF NUMBERS

USING PARTS OF THE BODY TO TAKE THE PLACE OF NUMBERS

1

2

3

4 12

DISCOVERY OF ZERO *

H

The idea of zero was originally invented in India. It was introduced to Europe and took the form we know today.

USING KNOTS IN STRINGS TO RECORD NUMBERS

w

W

Roman numerals.

Some primitive peoples used knots for recording numbers. One such system was developed in South America by the Inca Indians. They used knots in strings as numerals. Each number had a special arrangement of knots, which was memorised. The whole arrangement of strings was called a quipu, and was also used for recording events.

Chinese numerals.

Ancient Japanese numerals (short wooden sticks were used).

Ancient Egyptian numerals.

The history of numbers goes back to the time when people began to match different things (one-to-one correspondence). For example, one sheep was matched with one finger, two sheep with two fingers, and so on. The need to record numbers became stronger when people began to barter (exchange) things. Methods of using fingers or other familiar things to count numbers were not good enough when they had to keep a record of those numbers. One of the first methods used was the 'knots' system. As a result, knots can be regarded as being the first numerals. Different kinds of numerals were adopted in different countries. The numerals 1, 2, 3, 4 . . were developed in the Arab countries. Although there are many highly developed number systems, the most important idea in counting is still the idea of one-to-one correspondence. The number zero is said to have been invented in India. There are many opinions about what zero originally stood for. Some experts say that zero stood for the sun. Others claim that it was the symbol of demons. In any case, the invention of zero was very important in the development of numbers. 13

NUMERALS

"The numerals 1, 2, 3, ... were originally developed in India. They reached Europe by way of the Arab countries. They are called Arabic numerals. *Chinese numerals, which are written vertically, are sometimes used in China and Japan. 'Roman numerals are sometimes used on the face of a clock, or for numbering things, or for dates on monuments. "The set of numbers 1, 2, 3, 4,... is divided into the set of even natural numbers 2,4, 6, 8, ... and the set of odd natural numbers 1, 3, 5, 7, ...

OF TODAY

ODD AND EVEN NATURAL NUMBERS

even

even

14

ARABIC NUMERALS

CHINESE NUMERALS

even

seven even

W H A T V V I I r t l

I M I I M R F R Q lilUIVIIJI.no

j

'Numbers are used to count and compare, to represent the positions of points on a line, and to represent the order of things. *They are used

.

to denote length, weight, and volume, to tell the time, and to time intervals. "The series of numbers 1, 2, 3, 4, ... has no end.

STAND FOR

POINTS ON A NUMBER LINE

The number of apples on a plate may be represented by marking off distances on a line w i t h the numbers 0, 1, 2, 3

NUMBERS MAY BE USED TO COUNT AND COMPARE

3>1 t * j

J * .

3=3

3<4

NUMBERS MAY STAND FOR THE ORDER OF THINGS IN A LINE

16

Numbers are used to count and compare. The flower bed on the left has 12 flowers. The flower bed on the right has 7 flowers. We use symbols to compare the size of numbers. The symbol > means is larger than. The symbol = means is equal to. The symbol < means is smaller than. For example, 3 > 1 , 3 = 3, and 3 < 4 .

Numbers may also stand for the order of things in a line. The orange coach is third from the left, second from the right.

These days, we find numbers all around us. Some people use numbers without really knowing what they stand for. Let's not regard numbers merely as numbers, but try to realise how they are being used.

Numbers may stand for volume.

jfc

m O

—M^Jjm.— At.. * * * *

+ *

v *

'i *

*

4

Numbers may stand for length or distance.

4

*

/

4

^

Numbers may stand for time.

Numbers may stand for weight.

ONE NUMBER COMES AFTER ANOTHER

Lights of a lighthouse.

The series of numbers 1, 2, 3,... is infinite, that is, it has no end. When we match the set of numbers 0, 1, 2, 3, ... with a certain set of points on a line, we usually start by choosing a point, called the origin, matched with the number 0. Then we move on from left to right. Pick the first point (1) at a certain distance from the origin, and move the same distance to the right and mark the second point (2), and so on.

Skipping.

A metronome.

A clock.

In a railway train, for instance, the number 1 can be matched with the first coach, the number 2 with the second coach, and so on. A train has only a limited number of coaches, but when we match numbers with a set of points on a line, there need be no end to those points. Just as a hand of a clock goes round and round, one number comes after another. When the set of numbers is matched with a certain set of points on a line, the numbers stand for the positions of the points with which they are matched. Numbers may also represent length, weight, volume, length of time, time of day, and so on. They are used on a metronome for marking exact time when we play the piano. The light in a lighthouse flashes at measured intervals. Hands on a clock go round in a certain time. Children skipping count one, two, three, and so on. 17

HOW TO WRITE

^

* We usually use the decimal system for writing numbers. "The number 0 plays an important part when we write big numbers. "The numbers 1, 2, 3, 4, ... go on for ever. * Besides the decimal system, there is the binary system, the quinary system, the duodecimal system, the sexagesimal system, and many others.

BIG NUMBERS - v

10

100

T h e n u m b e r t e n is w r i t t e n 10. Ten 10s make one hundred, which is w r i t t e n 100. Ten 100s make one thousand, which is w r i t t e n 1 000.

ZERO AND BIG NUMBERS

Zero or 0 fish means no fish. But 10 fish means ten fish, 100 fish means a hundred fish. Using 0 we can write big numbers easily.

1fi

How do you count a lot of cents? Sometimes it is easiest to divide them into sets of ten cents. If you have three sets of ten cents, you know that you have 30 (thirty) cents. If you have ten sets of ten cents, then the number of cents is

one hundred, which is written 100. The number 37 has three tens and seven ones; it is thirtyseven. The number 245 has two hundreds, four tens, and five ones; it is two hundred and forty-five. Ten hundreds is written 1 000; it is

one thousand. The method of writing numbers like this, where ten is used as the basis, or base, is called the decimal system. We say that 245 has 2 in the hundreds' place, 4 in the tens' place, and 5 in the ones' place.

The cash register makes use of the decimal system.

The gas meter makes use of the decimal system.

HOW TO WRITE TWO HUNDRED AMD FORTY-FIVE

In writing a number, the figure on the right is the number of ones, or units, the next figure (to the left) is the number of tens, the next figure the number of hundreds, and so on. Two hundred and forty-five is two hundreds, four tens, and five ones. It is written 245.

o

•

o

•

o

o

Numbers keep increasing without end.

The commonly used Arabic numerals follow the decimal system, which is convenient for writing big numbers. The value of a numeral depends on its place in the number. Its value is increased by a factor 10 for each place it is moved to the left. In Roman numerals, however, there is no such thing as place value. The number 337 is written CCCXXXVII. You can see that 337 is a much simpler way of writing it. There are many systems other than the decimal system that can be used for writing big numbers. For example, the binary system is used in

computers, the quinary system is used in abaci, the duodecimal system is used to count things in dozens, and the sexagesimal system (based on 60) is used to express time or angles (60 seconds in a minute). Roman numerals, Greek numerals, and Mayan numerals all follow the quinary system. Egyptian numerals and Babylonian numerals follow the decimal system. Traces of the duodecimal system are found in the number names 'eleven' and 'twelve', and in such measurements as a foot, which is divided into 12 inches.

LET'S TRY (1) Let's turn to page 11 and try to count the number of men in white trousers marching at the bottom of the picture. (2) Let's find examples of counting methods which use the decimal system. (3) Try to read the following numbers: 463 893 4652

19

THE BINARY SYSTEM The binary system uses two instead of ten as its base. Using only the numerals 0 and 1, we can write any number. Instead of 2 we write 10 (read as one-zero), and instead of 3 we write 11 (one-one). 11 has one in the 2's place and one in the 1's place; it adds up to 3. 4 is written 100 (one-zero-zero); 5 is written 101 (one-zero-one). 8 is written 1 000; 16 is written 10 000. In the diagrams below the numbers in the decimal system are printed in black, w i t h their binary equivalents in red. The diagrams also show the corresponding bead positions on an abacus.

The numbers along the top indicate place values.

16

8

4

2 The positions of the beads show the number (0 or 1) of the one's place, t w o ' s place, four's place, and so on. When a bead is on the upper side, it indicates the number 1 in that place; on the lower side it indicates 0.

2i I, •

13

5=4+0+I

13=8+4+0+1

I

20

20

2=2+0

6=4+2+0

3=2+ I

7=4+2+I

4=4+0+0

8= 8 + 0 + 0 + 0

20= I 6 + 0 + 4 + 0 + 0

29

29= 16+8 + 4 + 0 + I

31

31= 1 6 + 8 + 4 + 2 + 1

9

1001

17

10001

25

1100T

10

10

1010

18

10010

26

11010

3

11

11

1011

19

10011

27

11011

4

100

12

1100

20

10100

28

1 1100

5

101

13

1 101

21

10101

29

1 1101

6

1 10 14

1 1 10 22

10110

30

11110

7

111

15

1111

23

10111

31

11111

8

1000

16

10000

24

11000

32

100000

1

1

2

Counting w i t h the fingers is an example of the quinary system.

The numbers 1 to 32 w r i t t e n in the binary system.

The abacus makes use of the quinary system.

These pencils are sold in a box of 12, an example of the duodecimal system.

The (flock makes use of the sexagesimal system. hour is 60 minutes, one minute is 60 seconds.

One

The protractor is an example of the sexagesimal system. One degree is 60 minutes, one minute is 60 seconds.

The sexagesimal system, w i t h degrees and minutes, is used for fixing positions on the earth.

NUMBER SYSTEMS The decimal system is the most commonly used system of counting, but there are many other systems of writing numbers. The binary system, using two as its base, is used in computers. The quinary system, which has five as its base, is used on the abacus. The duodecimal system

has twelve as its base, while the sexagesimal system has sixty as its base. However, when things are counted in dozens (as in the duodecimal system), or angles or time measured in sixties, the actual numbers are written in the decimal system.

Each system of writing numbers has a long history. In the beginning, people used five digits (four fingers and a thumb) and then ten digits to count numbers. The decimal system was developed later by people who frequently had to count their crops of large numbers of animals.

FUN WITH NUMBERS Once upon a time, when Japan was ruled by a powerful general, Toyotomi Hideyoshi, there lived a clever person who was especially favoured by the general. His name was Sorori Shinzaemon. One day, the general decided to give Sorori a prize. 'What would you l i k e ? ' the general asked. Sorori answered,^Please gjye me a aram of rice, my lord! Then, tomorrow, give me two grains, and the next day, please give me four more, and the day after that eight more, and so on for thirty days.' Rice was very important in those days, but Hideyoshi of course granted t h i s apparently humble wish. Well, on the thirtieth day, Hideyoshi was surprised to see Sorori w a i t i n g with an enormous wrapping cloth, large enough to hold the contents of a whole storage house. 'What are you doing with that?' asked Hideyoshi. 'Well, my lord,' Sorori explained with a smile, 'it is for carrying the rice you promised me. If you do the sum 1 + 2 + 4 + 8 + 1 6 + 3 2 + 6 4 + . . . for thirty days, it adds up to 1 073 741 823 grains of rice. I shall need a whole storage house to hold it.' 21

ADDITION

"The union of two sets that share no common part is called the sum of those sets. ' T h e sum of the set of three apples and the set of two apples contains five apples. The number of the sum of two sets is also called the sum of the numbers of those sets. * Making the sum of two numbers is called the addition of those numbers. By adding one number to another, we get their sum. "When we add numbers, we sometimes have to carry over a number from one column to the next.

SUM AND ADDITION

Four birds on the tree.

ADDITION

The set of four boys playing football and the set of three girls playing house have no common part. The union of these sets is their sum.

The

There are four birds on a tree. Three more birds come along and join them. Three added to four makes seven.

sum has seven children.

The sum of four and three is seven.

When we have two sets with no common part their union is called the sum of those sets. The sum of the set of four boys and the set of three girls contains seven children. If we add three birds to the set of four birds, we get the set of seven birds. The addition of 3 to 4 is written 4 + 3 (read 'four plus three'). To show that the sum of four and three is seven, we write 4 + 3 = 7 (read 'four plus three equals seven').

The game shown on the next page may be used to practise the addition of numbers. Cards are made up with, for example, on the front 4 + 3 and on the back 7. Cards with 6 on the back will have 5 + 1 , 1 + 5, 4 + 2 , 2 + 4 , or 3 + 3 on the front. To start playing, place the cards with their front sides up. The dealer calls out the numbers. If he says 5, for example, you may pick up any one of the cards having 1 + 4 , 2 + 3 , 3 + 2 , or 4 + 1 . The player who gets the most cards wins the game.

CARD GAME

TWO CHILDREN+THREE PUPPIES

Three to five people can play the game. The dealer calls out numbers which are the sums of the numbers on the front sides of the cards. The player who gets the most cards wins the game.

The union of the set of a boy and a puppy and the set of a girl and two puppies is, of course, the set of two children and three puppies. But in arithmetic, the plus sign ( + ) may be put only between two numbers.

DOING ADDITION S U M S

34

23

23 34

+ ADDITIONS WITHOUT CARRYING

57

To add 23 and 34, we add the ones' column first and put down 7 in the ones' column, then we add the tens' column and put tiown 5. The sum is 57.

23 + 34 =

57

+

57

65

57 +

122

CARRYING OVER TO THE NEXT COLUMN To add 57 and 65, first add the ones' column, getting 12. Write down 2 (for two ones). But we have to carry over the 1 from the 12 to the tens' column (because it stands for one ten) and add the one to the tens' column. Then, adding 5, 6, and 1, we get 12 (standing for 12 tens). Carrying over the 1 to the hundreds' column, we finally get the sum 122.

io

I00

LET'S TRY

243 + 625

I 04 + 298

478 + 588

io

I00

I00 1. Suppose we have a lot of oranges. 2. After giving one orange to each of 98 people, we still have 27 oranges left. How many oranges did we start with?

65

io

IO

io

i

i

i

i

57 | + | 65 | = 122 23

SUBTRACTION

"When a set is contained in another set, the complement of the first set in the second set is also called the difference between the sets. The difference between 3 apples and 2 apples is 1 apple. *Subtraction is the opposite of addition. By adding 3 to 4 we get 7. By subtracting 3 from 7 we get 4. The number of the difference between the set of 3 apples and the set of 7 apples is found by subtracting 3 from 7. "In subtraction, we sometimes have to borrow a number from the higher column. * By subtracting one number from another, we get their difference. " I t is impossible to subtract a number from a smaller number.

DIFFERENCE AND SUBTRACTION

DIFFERENCE

SUBTRACTION

There are six frogs in the pond. Two of them are sitting on lily-pads. The difference between the second set of frogs and all the frogs in the pond is the set of frogs on the island.

There were six cows in the pen. Two of them were taken away by the farmer. By subtracting 2 from 6 we find the number of cows left in the pen.

6

-

2

The number of the difference between the set of 2 frogs and the set of 6 frogs or between the set of 2 cows and the set of 6 cows is found by subtracting 2 from 6. The answer in each case is 4.

Prepare two lots of cards carrying numbers from 0 to 10, laying one lot face up on the table. The dealer places the other lot in a pack face downwards, and then shows the first number, for example 7. The players try to get the card with the difference between that number and 10. In this case, it is the card showing 3. The dealer continues to show each card in turn while the other players try to get the 'difference' card. The player who gets the most cards wins the game.

DOING SUBTRACTION SUMS SUBTRACTION WITHOUT BORROWING

35

13

mmm

245 mnmmm

mmm

io 10

-

1 1

10

35

3 =

35 13 22

23

00 100

10 10 10 10

•••••

10 10

• • •

100

10

22

10

245 -123 122

245

23 =

122

SUBTRACTION BY BORROWING

8

32

343 -

154

•• •• •• •• ••

10 10

mm

10 _

10 10 10 10 10

10

32

• •• I

H

I

1

1 1 1

8 =

H

32

100 10

8 14

14

10

10

IO

10

10

10

10

10

10

10

10

100 10

343

10

10 10

10

•mmmmm •••• • • • • 1 1

343 154

1 1 1 1 1 1

189

154 = 189

LET'S TRY

1 98 -45

836 -568

I 84 - 67

980 -88 I

2. There are 438 workers in a factory. Among them, 75 are women. How many men work in the factory? 3. Somebody gave me 70 marbles, so now I have 345 marbles. How many did I have in the beginning?

The four subtraction sums illustrated above show clearly what happens when you subtract one number from another. Try to follow each subtraction and see how you arrive at the answer. In each case, the number in blue is being subtracted from the number in red. The answer — the difference — is in yellow.

When we subtract one number from another, the difference is smaller than the second number. For example, 70—30 = 40. 40 is smaller than 70. The difference 40 shows how much larger the number 70 is than the number 30. In the picture on the left, there are 3 apples and 5 children. Each of the children wants an apple. Two children will have no apples. You will later learn how to subtract 5 from 3 by using something called 'negative numbers'. 25

ADDITION AND SUBTRACTION

"Subtraction 6—1 + 2—3, from left to they must be

is the opposite of addition. "To do calculations like we deal with the first pair of numbers first, and then go on right. "However, if there are calculations inside brackets, worked out first.

ADDITION

+

i

8

rrrrr+ rrr

To add 3 to 5, we start from the number 5 on the line and jump 3 points forward. We reach the number 8 which is the sum of 5 and 3. In this way, we find that 5 + 3 = 8. Let's try the same thing for 3 + 3 and 5 + 8 .

SUBTRACTION

rrrrrrrr A

+

B

H

H

h

C

C - B

A

C

A

(-

To subtract 3 from 8, we start from the number 8 on the line and jump back 3 points. We reach the number 5, so 8—3 = 5. Let's try the same thing for 5—2 and 10—6.

Let's add a number B (for example 3) to a number A (for example 5), and call their sum C (in this case it is 8). We have A + B = C. Then, by subtracting B from the sum C, we get the number A. We have C—B = A. Subtraction is the opposite of addition. Also, C—A = B.

CALCULATIONS WITH BRACKETS

rrrrrrr

To calculate 7 — ( 3 + 2 ) , we first carry out the calculation 3 + 2 inside the brackets and find the answer 5. Then we put 5 in the place of ( 3 + 2 ) above, and carry out the calculation 7—5 = 2. In this way we have 7 — ( 3 + 2 ) = 2. Calculations inside brackets come first. Let's try to calculate 1 2 — ( 6 + 3 ) and 1 5 — ( 2 + 3 + 5 ) .

COMBINATIONS OF ADDITIONS AND SUBTRACTIONS

13-6+2-5 =4 To calculate 1 3 — 6 + 2 — 5 , we first look at the first pair of numbers and calculate 1 3 - 6 = 7. We put 7 in the place of 1 3 - 6 and get 7 + 2 — 5 . Then we work out the calculation 7 + 2 = 9, and put the number 9 in the place of 7 + 2 . We now get 9—5. Then subtracting 5 from 9, we finally get 9 — 5 = 4. In this way we have the answer 1 3 - 6 + 2 - 5 = 4.

p y

Q p '

A n n i T l f l l U M U U M I U I i l

*

' ^

"The sum of two numbers does not change if the order of the addition is changed. This is called the commutative law of addition. *For three numbers A, B, and C, we have ( A + B ) + C = A + ( B + C ) . This is called the associative law of addition.

COMMUTATIVE LAW OF ADDITION

• + 3 = 3 + 1 The union of the set of 4 oranges and the set of 3 bananas is the same as the union of the set of 3 bananas and the set of 4 oranges. They both contain the same 7 fruits.

+ B = B

+ A The sum of two numbers does not change if we change the order of the addition. Using A and B to represent the numbers, we have A + B = B + A . This rule is called the commutative law of addition.

ASSOCIATIVE LAW OF ADDITION

Jill first bought 4 apples and 3 oranges and then she also bought 5 grapefruit. She bought 12 pieces of fruit in all.

Jane bought 4 apples. She already had 3 oranges and 5 grapefruit. She finished up w i t h 12 pieces of fruit in all.

4 + ( 3 + 5) =

( 4 + 3 ) + 5: (A + B | + H = B +

I 2

|B + C ) For three numbers, represented by A, B, and C, we have ( A + B ) + C = A + ( B + C ) . This rule is called the associative law of addition. 27

MULTIPLICATION

^

^

V.

©

* To calculate a certain number times a given number, we use multiplication. For example, to get 3 times 5 we multiply 5 by 3. *When we multiply two numbers, we sometimes carry over a number to the next higher column to be added later. In multiplying a number by another, we get their product. "To be able to carry out multiplication guickly and correctly, it is important to learn the multiplication table (see page 30).

• •

m

100

Two 10s equal 2 times 10, which is 1 0 + 1 0 = 20.

100

Three 100s equal 1 0 0 + 1 0 0 + 1 0 0 = 300.

2 + 2 = 4

2+2+2+2

2 + 2 + 2 = 6

On each of the 4 dessert dishes you see 2 slices of cake. The number of slices of cake is 4 times 2, which is written 4 x 2. 4 x 2 is equal to 2 + 2 + 2 + 2 = 8.

HOW MANY LEAVES?

4 X 2 = 8

HOW MANY CHILDRE N?

• •

5 chilo

I I 4X»

• I

|

| X 3 =

4 co lumns

* * * * * * - * * r*

H

100

I

5

Three times 10 is 1 0 + 1 0 + 1 0 , which is also written 3 x 1 0 . So, 3 x 10 = 30. We also know that 1 0 x 1 0 = 100. It is not so simple to calculate 2 0 x 1 0 by adding 10 twenty times, but if we use some simple laws of multiplication which we shall learn later, we have 2 0 x 10 = ( 2 x 1 0 ) x 10 = 2 x ( 1 0 x 10) =

ili

* x * *

* * X* X* 20

2 x 1 0 0 = 100+100 = 200. On the next page some examples of multiplication are illustrated. In each case the number being multiplied is in blue, the number doing the multiplying in red, and the answer in yellow. Try to follow the examples step by step.

MULTIPLICATION MULTIPLICATION WHICH OOES NOT TAKE A FIGURE UP

x

|2

/"S

x

4

G r

(10)

(T)(j

(10 + 2 )

X 10

4

(10)

+

x

4

4 x

8

(10)

2

x

x

+ B [ x

2

2

4 8

4 0 10

I

4 8

4 0 + 8 MULTIPLICATION WHICH TAKES A FIGURE UP

x

24

3 ! x

0

(20+4) 20 +

3

3

•

•

•

•

(To) (m)

r r1\ r r \ / - r1\ f r\

(To) (To)

1 t~?\t

X 20

\ / vJ_A i \J,' lr

x

M I\ \ l\ I St/VI L'V '

+

1

1

x

3

I

'

'

2 4 I

2

6

0

L

x

2 4 7 2

7 2

60 MULTIPLICATION (A NUMBER WITH TWO FIGURES) X (A NUMBER WITH TWO FIGURES)

26

x

37

2 6

26

x

(30+7)

26

x

30 +

X

26

X

7

2 6 x 3 7 is equal to 2 6 x ( 3 0 + 7 ) . This is an application of the distributive iaw of multiplication, which w i l l be explained later.

2 6

3 0

7

X

I 8 0

4 2

6 0 0

I 4 0

7 8 0

I 8 2

7 8 0

+

2 6 X

3 7 I 8 2

8 2 9 6 2

To multiply numbers accurately and quickly, you need to know the multiplication table by heart. The table is given on page 30.

LET'S TRY

1

X X

I3 3

I 8

5

X

X

38 4

248

8

X X

28 I 2

84 67

A garage owner bought 29 car batteries, each of which cost 16 dollars. How much did he pay? The teacher is buying coloured pencils. He wants to give 3 pencils to each of the 37 children in the class. Each pencil costs 12 cents. How much must the teacher pay for all the pencils he needs?

By adding a number A to itself B times we get B x A A + A + A + . . . B times = B x A The multiplication table is a basic tool for doing calculations, and so it must be learnt well. After mastering multiplication, it is easy to learn about division. 29

MULTIPLICATION TABLE

0x0

0 x

0

I x |

0 x |

* It is most important to learn the multiplication table because multiplication is basic to arithmetic. "The multiplication table has the Os table, the 1s table, the 2s table, the 3s table, and so on up to the 9s table. " T o learn the multiplication table by heart, a card game may be helpful.

j u F ^

0

x

I x

2

x

0

I x

3

0

x

x

0

x

0

I x 4

I x 5

I x 6

I

x

x

0

7

x

I x

0

8

x

I x 9

0 X 2

2

x

|

2 x 2

2 x 3

2 x 4

2 x 5

2 x 6

2 x 7

2 x 8

2 x 9

0 x 3

3

x

|

3 x 2

3 x 3

3 x 4

3 x 5

3 x 6

3 x 7

3 x 8

3 x 9

0 x 4

4

x

|

4 x 2

4 x 3

4 x 4

4 x 5

4 x 6

4 x 7

4 x 8

4 x 9

0 x 5

5 x |

5 x 2

5 x 3

5 x 4

5 x 5

5 x 6

5 x 7

5 x 8

5 x 9

0 x 6

6 x |

6 x 2

6 x 3

6 x 4

6 x 5

6 x 6

6 x 7

6 x 8

6 x 9

0 x 7

7

x

7 x 2

7 x 3

7 x 4

7 x 5

7 x 6

7 x 7

7 x 8

7 x 9

0 x 8

8

x

8

8 x 3

8

8

8

8

8 x 8

8 x 9

0 x 9

9

x

9 x 2

9 x 3

9 x 4

9 x 8

9 x 9

x

x

x

9 x 5

x

9 x 6

\

x

9 x 7

After learning the multiplication table by heart, it will be easy to multiply or write their products (6, 24, 64, and so on). The dealer calls out a product, while divide any numbers. The product of multiplying a number by itself, such as the players try to take as many cards as possible carrying the numbers which Ox 0, 1 x 1, 2 x 2, and so on, is called the square of the number. You may use a multiply together to give this product (shown on the backs of the cards). For card game to learn the multiplication table. On the faces of the cards, write example, if the dealer calls 63, the players should look for cards showing 9 x 7 the multiplications 2 x 3 , 4 x 6 , 8 x 8 , and so on. On the backs of the cards or 7 x 9 .

6

7

8

9

6

7

8

9

I0

12

14

I 6

18

12

15

18

21

24

27

12

16

20

24

28

32

36

I0

15

20

25

30

35

40

45

6

12

18

24

30

36

42

48

54

7

14

21

28

35

42 ^ H H

49

56

63

8

8

16

24

32

40

48

56

64

72

9

9

18

27

36

45

54

63

72

81

I

2

3

4

I

I

2

3

4

2

2

4

6

8

3

3

6

9

4

4

8

5

5

6 7 H H

30

5

5

MULTIPLES AND COMMON MULTIPLES

*By multiplying any number by another number (1, 2, 3, ...), we get a multiple of the first number. * A number which is a multiple of two different numbers at the same time is called their common multiple. "The smallest number among the common multiples of two numbers is called their lowest common multiple.

rprprprpr rpr MULTIPLES OF 2

I0

I2

I 4

Sets of t w o flags, one red and one w h i t e , come one after another. The multiples of 2 are 2, 4, 6, 8 found by multiplying the number series 1 , 2 , 3, 4, . . . b y 2.

MULTIPLES OF 3

mm\

a

| |

|

| •

p| I:

|

|»pi I!

I;

Sets of three flags are lined up one after another. The multiples of 3 are 3, 6, 9, 12 .... found by multiplying the number series 1, 2, 3 , 4 , ... by 3.

COMMON MULTIPLES OF 2 AND 3

T h e n u m b e r s o n t h e r e d f l a g s a r e m u l t i p l e s of 2.

The numbers on the blue flags are multiples of 3. The numbers on the yellow flags are multiples of both 2 and 3. They are called common multiples of 2 and 3. The smallest common multiple of 2 and 3 is 6, which is called the lowest common multiple of 2 and 3.

By multiplying a number by 2, 3 we get the multiples of that number. A number which is a multiple of two or more numbers is called their common multiple. The set of common multiples of two numbers is the intersection of the sets of multiples of the two numbers. What is the lowest common multiple of 4 and 6? Multiplying the larger of

the two numbers by 1, 2, 3, 4 we get 6, 12, 18, 24 ... The smallest of these that are divisible by 4 (without a remainder) is 12. So, the lowest common multiple of 4 and 6 is 12. Zero (0) is a common multiple of all numbers, although usually it is not considered as such. 31

DIVISION

*To find out how often one number contains another, we use division. *7 contains 3 twice with 1 left over. By dividing 7 (the dividend) by 3 (the divisor), we get 2 as the quotient and 1 as the remainder. * By multiplying the quotient and the divisor, and then adding the remainder, we get the dividend. For example, we have 2 x 3 + 1 = 7. * If we get no remainder when we divide one number by another, we say that the first number is divisible by the second. For example, 7 is not divisible by 3, but 6 is divisible by 3.

There are 12 flowers. By dividing them equal lots, each of the three children 4 flowers. 12 contains 4 three times. symbol 4 - which means 'divided by', we

DIVIDING THE FLOWERS

*

• EALING THE CARDS

into three can have Using the may write

4

There are 15 cards. After dealing them to 5 children 3 times, they are all distributed. This means that by multiplying 5 by 3 we get 15. We have 3 x 5 = 1 5 . 15 contains 5 three times. So, we have 1 5 - ^ 3 = 5 .

[ Deal another card to each child.

Deal a card to each child.

DIVISION AND MULTIPLICATION

I M I

32

I

it

.

.

A

It

Deal another card to each child.

Twelve trucks are pulled by a locomotive. The train is divided into four equal parts each of which contains 3 trucks. We have 1 2 h - 4 = 3. This also means that by multiplying 3 by 4, w e get 12.

2X 3< I2

3 X 3 < I2

5 X 3 > I2

6 X 3 > |2

DIVISION WITHOUT A REMAINDER

42

4

m

o

mm mm m mm

'O'jlia

10

^ m

4 tens divided by 3 gives

There

1 in the tens'

Dividing by 3, w e get 4.

place

and

are

now

12

ones.

leaves a spare 10 to carry over to the ones.

0 10

I

42

^

I

I

I

! 4

3

I 32

,1° l 0 Y ' 0 i f <«'#.

3 3

$ # Cm m)

,0M°

^ n

i 3 2 m ^ 100

•

y

f

I 3 2 I

e

Y There are now 12 ones. Dividing by 4, w e get 3.

After carrying over 100, w e have 13 tens. Dividing by 4, w e get 3 in the tens' place, w i t h 10 carried over.

10

I 32 DIVISION WITH A REMAINDER

10

2

t

I

2

I

2 0

10

=

33

How can we divide 14 chocolate bars among 4 children? After giving 3 chocolate bars to each child, w e have 2 left over. We say that 14 is not exactly divisible by 4. By dividing 14 by 4, w e get 3 as the quotient and 2 as the remainder.

When a number C contains a number B A times, we have C - ^ B = A . In this case we also have A x B = C . Division is the opposite of multiplication.

LET'S TRY 1.

4)48 26)494

91387" 4787""

12)144 31

)463

2. There are 7 boxes of chocolate bars to be shared among 3 people. Each box contains 6 bars. How many whole boxes and how many additional chocolate bars does a person get? 33

MULTIPLICATION AND DIVISION

"Division is the opposite of multiplication. "The remainder is smaller than the divisor. " I f a calculation involves addition, subtraction, multiplication, and division, we work out the multiplication and division first. However, if there are calculations inside brackets, they must always be worked out first.

DIVISION IS THE OPPOSITE OF MULTIPLICATION

-v V

; i me.T*» -

«*

••

•m

m : IX

k

i

CALCULATIONS WITH BOTH MULTIPLICATION AND DIVISION

She needs

Each of her baskets holds 3 oranges

Ann has 2

4.

oranges

baskets

sS m

i

£

5

I 5

-'.Jf'lV

He needs

Each of his baskets -holds 5 oranges

John has

... J -

oranges

1 =

Adding her baskets and his, w e have

3

baskets

Division is the opposite of multiplication. Knowing that 2 x 3 = 6 , we can tell that 6 + 3 = 2, and 6 + 2 = 3. In general, if A and B are any numbers except 0, and if Ax B = C then we have C + A = B and C + B = A. The remainder is smaller than the divisor. For example, if we divide 7 by 2, the remainder is 1. We have 7 + 2 = 3 with the remainder 1. In this case, we have 2 x 3 + 1 = 7. In general, if A and B are non-zero numbers, and A + B = C with the remainder D, then D < B, and B x C + D = A. When we carry out calculations such as 2x4-6+3, we work out the multiplications and the divisions first. In our example, we first work out 2 x 4 = 8, and then 6 + 3 = 2. Then we work out the sum 8 - 2 =

7

34

baskets

6.

But if there are brackets in the calculations, we have to work out the calculations inside the brackets first.

RULES OF MULTIPLICATION

"The product of two numbers does not change if we change the order of the multiplication. This is called the commutative law of multiplication. "The multiplication ( A x B ) x C is the same as A x ( B x C ) . This is called the associative law of multiplication. *We have A x ( B + C ) = A x B + A x C. This is called the distributive law. • 'i

Five dishes with three strawberries in each together contain as many strawberries as three dishes with five strawberries.

THE COMMUTATIVE LAW

Hx |

f x ® In general The picture of the rabbits can explain the commutative law A x B = B x A. There are 4 rows of 3 rabbits at them differently, there are rabbits each. So 4 x 3 = 3 x 4 . the law A x B = B x A by looking tion table. The associate law is ( A x B ) x C = Ax(BxC), which we can see b,y counting rabbits. The distributive law is A x (B+C) = A x B + A x C .

B

X

X|

also be used to THE ASSOCIATIVE LAW each or, looking 3 columns of 4 You may check at the multiplica-

»

*l

\

#

*

;

%J-

*

1

»••_ » J .

x •/)

.• X • 1

the ears of the

> * v w

•

/

•

1j

' \ / j ?»

*

«!„ ;

;» • J v * /

How many rabbits' ears can you f i n d ? There are t w o ways of counting them. One way is to find the number of ears in a row by multiplying 2 (the number of ears on a rabbit) by 3 ( = 6), and then multiplying the product (6) by 4, which is the number of rows. In this way, we get 4 x ( 3 x 2 ) = 4 x 6 = 24. The other way is to multiply 2 (the number of ears per rabbit) by the total number of rabbits, which is 4 x 3 = 1 2 . In this way, we get ( 4 x 3 ) x 2 = 1 2 x 2 = 24. Comparing these t w o ways, we find that 4 x ( 3 x 2 ) = (4x3)x2. In fact, we have proved the general statement A x ( B x C) = ( A x B ) x C . for any three numbers A, B, and C.

»i x./

THE DISTRIBUTIVE LAW

m m m

M M M

m

The three laws above, together with the similar laws of addition, are very important. Later, we shall see that the idea of numbers may be expanded and find that there are many 'numbers' other than 0, 1, 2, 3, 4 The above laws are all satisfied by these other 'numbers'. In fact, even in advanced mathematics, these laws are used as basic relations among numbers.

n

n

MMM i n V \t ¥¥

How many flowers can you find in the picture? There are 4 x 3 = 12 red flowers, and 4 x 2 = 8 yellow flowers. In all, there are 4 x 3 + 4 x 2 = 20 flowers. Or, looking at it another way, each row has 3 + 2 = 5 f l o w e r s , and there are 4 rows. 4 x ( 3 + 2 ) = 20 also. Comparing the t w o calculations, we see that 4x (3+2)= 4 x 3 + 4 x 2 . In general, w e always have A x (B+C) = A x B + A x C .

35

FACTORS AND COMMON FACTORS

*The numbers 1, 2, 3, ... are called natural numbers, or positive integers. *When a number A is divisible by a number B, we say that B is a factor of A. "When a number D is a factor of both A and B, we call D a common factor of A and B. *The set of common factors of A and B has as its largest member the number which is called the highest common factor of A and B.

mziz NATURAL NUMBERS

•

m 0

E

H

m

• •• • • • • •: * "i •

a

FACTORS OF 12 AND FACTORS OF 18

12

| x

=

s

^"N

(

12 may be written as the product of three different pairs of numbers. The numbers in the red boxes and those in the blue boxes are all factors of 12.

!8

B

=

d =

x B

x

=M

X

18 may also be written as the product of three different pairs of numbers. The numbers in the red and blue boxes are the factors of 18.

18 |

9

I6

S -"s

v

\ I

18

\ I

\

I

-+4-

J K J\ 2

I

3 i

v.

J

The factors of 12

The factors of 18

I 6 I V l

18

Common factors of 12 and 18

COMMON FACTORS OF 12 AND 18 The factors of 12 are in the upper circle. The factors of 18 are in the lower circle. The set of common factors of 12 and 18 is the intersection of the set of factors of 12 and the set of factors of 18 (yellow flags). Among the set of the common factors, 6 is the largest. This, the largest member of the set of the common factors of two numbers, is called their highest common factor.

1 is a factor of every natural number 1, 2, 3 As a result, every natural number except 1 has at least two factors: 1 and itself. Natural numbers, except 1, with only two factors (one of which is 1) are called prime numbers. For example, 2, 3, 5, 7, 11 ... are prime numbers. When the highest common factor of two numbers is 1, we say that these two numbers are mutually prime. For example, 2 and 3 are mutually prime. When A and B are any natural numbers, then the highest common factor of the products A x C and B x C is equal to the product of the highest common factor of A and B times C. For example, 1 2 = 2x6, and 18 = 3 x 6 . Therefore, the highest common factor of 12 and 18 is the product of 1 (the highest common factor of 2 and 3) and 6, which is 6. 36

FRACTIONS

"Using pairs of integers, we can form fractions such as 5, |, f, and so on. * In the fraction A/B, A is the numerator and B is the denominator. * When the numerator is less than the denominator, the fraction is called a proper fraction. When the numerator is greater than the denominator, it is an improper fraction. 'Numbers such as 3;, and so on are called mixed numbers.

When you cut a jam roll into two equal parts, each part is \ of the whole. _2_

6

I

Ml 2

i

3

6

2 6

6

•

, l 1 1 1

I

3

2

i 3

6

•

1

1

^H

CONSTRUCTION OF A FRACTION

Numerator

1 5

1 5

1 5

1 5

1 5

If we use fractions, we can divide any number into equal-sized parts. For example, J is one-half of 1. By dividing 1 into three equal parts, each part is 3. | is read as three-fifths, and is equal to 3 times In the fraction I, three is the numerator and five is the denominator, j is a proper fraction because its numerator is smaller than its denominator. 1 is smaller than 1. We have to add I to I to get I, which is equal to 1 (see the picture). A proper fraction is smaller than 1. When the numerator of a fraction is greater than its denominator, it is called an improper fraction, f and 1 are examples of improper fractions. The numbers 1 j , 2 j and so on are equal to their integer parts plus their fractional parts. For example, I5 is equal to one and a half. They are called mixed numbers.

_3 5 Denominator

3

_5_ FRACTIONS ON A LINE

J_

4

3

H JL

3

h JL

4

3 37

1. Change the first two mixed numbers into improper fractions, and the last two improper fractions into mixed numbers:

MIXED NUMBERS AND IMPROPER FRACTIONS

_3 l

5

16

13

n

3

5


2. Reduce these fractions to the lowest terms:

The mixed number 2* is the same as 2 + 4

2 is equal to f , so the mixed number 2% is equal to the improper fraction t

A ± 8 U L 16 15 24 36 3. Change these fractions to the lowest common denominators: I

<1

)

("

CHANGING A FRACTION TO LOWER TERMS

CHANGING A FRACTION TO HIGHER TERMS

Both the numerator and denominator of a fraction may be multiplied by the same number w i t h o u t changing the value of the fraction.

20

I 0

Hii

By dividing both numerator and denominator of a fraction by a common divisor, w e can r e d u c e t h e f r a c t i o n to lowest terms w i t h o u t changing its value.

I 6

12

ADDITION OF FRACTIONS

i

5

SUBTRACTION OF FRACTIONS

I

f

_3 5

+ +1

c

mm

6

5

J

3

38

+

8_

+

15

6

MULTIPLICATION OF FRACTIONS

X

DIVISION OF FRACTIONS

2_

A

5

4

_3

3_

_4_ 5

2

I 2

X

2 3

X

2

2 b

<

8

4 _3_

8

4 b

_3

4

By dividing a number by \ , we get twice (2 times) the number.

By multiplying a number by we get j of the number.

• I

_2_

X

6 I 2

X

2 3

2 6

_6_

4

_3

2

To divide by a fraction, you invert the fraction and multiply.

FUN WITH NUMBERS Once upon a time, there lived an old man and his three sons, Tom, Jim, and Jack. One day, the old man fell ill. He called his three sons and said, 'I leave my 17 sheep for you to take care of. Tom, you take I , Jim you take L and Jack, you take J of the sheep. Share the sheep according to my word, but you should not kill any one of them.' The old man died after saying this. The three sons were very sad, but they were also puzzled as to how they could share the 17 sheep. A wise man passed by and heard of their trouble. He said, 'I can lend you an extra sheep to make 18 in all, and that will solve the problem.' Indeed, of the 18 sheep is 9 sheep, J is 6 sheep and 5 is 2 sheep, and this way the three sons could share the original 17 sheep, and give the wise man his sheep back. Can you discover why the wise man's advice worked? 39

DECIMALS

"0.3, read as 'zero point three', is a decimal. The dot in the expression is called a decimal point. *0.3 is equal to j | . Any decimal may be written as a fraction. 'Numbers such as 4.3, read as 'four point three', which is equal to 4 plus 0.3, are called mixed decimals. To add or subtract numbers with decimal points, it is useful to write the numbers in a column in such a way that their decimal points line up exactly under each other. * By rounding off a mixed decimal, we get an integer.

By cutting pieces of coloured paper into many equal parts, we can compare the sizes of many fractions and decimals.

0.1, read as 'zero point one', is called a decimal and is equal to The dot in the expression of a decimal is called a decimal point. 0.01 means 100 • 0.001 means moo • The value of a number depends not only on the figure but also on its position in the number. The number 2 in 0.2 stands for ^ , and the same number in 0.02 stands for js . The place next to the decimal point is the tenths' position, the one next to it is the hundredths' position, and so on. Numbers such as 2.3 are called mixed decimals. 2.3 is equal to 2 plus 0.3. Decimals are larger than 0 but smaller than 1. For example, 0 < 0 . 1 < 1 . Similarly, 2 < 2 . 3 < 3 . Decimals may be added, subtracted, multiplied, or divided. When we carry out calculations using decimals, we have to be careful to see what each figure stands for. For example, when we add or subtract decimals, we usually write the numbers in columns so that their decimal points come under each other. The decimal 0.2 is equal to which is equal to 0.2 is two times 0.1. Therefore, 2 x 0 . 1 __ ^

10

i

5 .

By dividing the numerator of a fraction by its denominator, we can write the fraction in the form of a decimal.

A CLOSE LOOK AT DECIMALS

By magnifying a part of a ruler, we can clearly see the decimal markings.

DECIMALS AND FRACTIONS

DECIMALS AND FRACTIONS

0.5

0.2 —I—

1.3

0.8 H

1-

— i —

1.5 -H

1

5

5

•I

SUBTRACTION OF DECIMALS

0.5

1 . 2

^

"

0

.

8

Look at the number line.

Look at the number line.

1.2

0.5

0.2

0.4

0.7

0.8

Look at the coloured tiles.

Look at the coloured tiles.

0.2

(—

4

10

ADDITION OF DECIMALS

0.2 +

1.8 -i

(—

+

0.7

0.5

0.4

0.8

.2

FIGURES AND THEIR PLACES

+

500

0. I 7

70 +

2 30

23

+

40

400

2.3

+ 0.2 3

60

4.0

0.4 0

440

0.5

50 44

0.6

0.0 6

4.4

0.4 4

When me add or subtract decimals, we w r i t e numbers so that their decimal points come under each other.

MULTIPLICATION OF DECIMALS

X 2 x

3

1.2

x

1.2

x

3 3.1

.2

=

3.6

x 3.6

Decimals and mixed decimals satisfy the commutative, associative, and distributive laws just as integers do. Using the distributive law, w e can w r i t e :

3.1

x

(3+0.1) = 3

+ 0.1 2

3.72 3

1.2

x

|.2

=

3.6

0.7 0.02

x 1.2

x 1.2 +

0.1 x 1.2

0.1

.2

= 0.12

3.1

x

.2

=

3.72 41

DIVISION OF DECIMALS

Actual calculation.

4.4

2.2

4.4

4.4 4.4

-

4 4

2

2.2

0

I . I

44 44 0 Divide into 44 equal parts.

I . I

4.4 4.4

ROUNDING OFF DECIMALS

SIZE OF NUMBERS

Imagine an orange w i t h 10 sections. We can use it to demonstrate the three ways to get an integer from a mixed decimal.

Greater than or equal to

=

Discarding decimals.

I

M

I

In this case, we count any decimal as 0. Counting the decimal 0.4 as 0, we discard it from 1.4 to leave 1.

^

1.4

|

The natural numbers greater than or equal to 3 are 3, 4, 5, 6

Raising to a unit.

ft

In this case, we consider any decimal as 1. Counting the decimal 0.4 as 1, w e thus raise 1.4 to 2.

I

Less than or equal to

1.4 •=> 2

»»»

Rounding off decimals to the nearest whole number.

1.3 •=> I

ftftftft »»

#

The natural numbers less than or equal to 3 are 3, 2, and 1. In this case, any decimal less than 0.5 is discarded (put equal to 0) and any decimal greater than or equal to 0.5 is raised to a unit.

Less than

The natural numbers less than 3 are 2 and 1.

1.7 42

2

4

INEQUALITIES AND EQUATIONS

An inequality or an equation tells us the relation between two quantities, generally referred to as its left-hand side and its right-hand side. The symbols ^ and ^ stand for greater than or equal to and less than or equal to.

"Inequalities are statements using symbols to show that some quantities are less than ( < ) or greater than ( > ) some other quantities. Other symbols are used to mean less than or equal to and greater than or equal to *An equation is a statement of equality between two quantities written in symbols and using the equal sign ( = ) . An equation may contain an unknown.

INEQUALITIES

The relation between the terms represented in an inequality or equation will not change if we add or subtract equal quantities to both sides of the statement. For example, if we have an equation containing'an unknown x. 3/—2 = 4 (3/ means 3 x / ) , we can add 2 to both sides and get 3* = 6. In the same way, if we have an inequality x-\-l>Z, we can subtract 2 from both sides and get x> 1. We can also multiply" or divide both sides of an inequality or equation by equal quantities without changing the relation between the sides. (In this case, however, we have to be careful not to multiply or divide by 0.) For example, if we have an inequality 3 / > 6 , we can divide both sides by 3 and g e t x > 2. Similarly, when 3 * = 6, dividing both sides by 3 gives / =

2.

Is 3 dollars (300 cents) enough to buy four 40-cent notebooks, eight 10-cent pencils, and two 15-cent pens? Since ( 4 x 4 0 ) + ( 8 x 1 0 ) + ( 2 x 1 5 ) = 270. and 2 7 0 < 3 0 0 , 300 cents is more than enough. How many 40-cent note-books can you buy if you have

3 dollars to spend? Let x be the number of note-books. Then, working in cents, we have the inequality 40^300. Dividing both sides by 40, we get jr«S7.5. Therefore, x. the number of note-books, must be 7.

EQUALITIES

LET'S TRY 1.

There are 3 cats and an unknown number of chickens. The total number of legs they have is 20."" How many chickens are there? Let x be the number of chickens. Then, since each chicken has 2 legs, the total number of legs the chickens have is equal to 2 times x. written as Ix. The 3 cats, of course, have 3 x 4 = 12 legs in all.

So we can write the equation: 2x+U= 20 Subtracting 12 from both sides of the equation, we get: lx= 8 Dividing both sides by 2, we finally get x = 4. In this way, we have found that there are 4 chickens.

2.

Jim gave 1 dollar (100 cents) for two note-books, and got some change. What can you tell about the price of one notebook? Try to express your answer by using inequalities. Mother gave Jane 3 dollars (300 cents). She spent 40 cents each day. She now has 20 cents left. For how many days has she been spending the money?

43

TOOLS OF CALCULATION

* To use an abacus we move the beads up and down. * A slide-rule is made by combining two specially made rulers side by side. *To operate a calculator we turn the handle or push the buttons. * Computers are the most recent kind of calculating machines.

Since ancient times, many tools for calculation have been invented. Among them, the abacus is one of the most simple and useful, and is still commonly used in Asia. The European and oriental abaci have different forms. Japanese abaci were originally imported from China, and were then modified for Japanese use. They have two sections divided by a beam. The upper section has beads of value 5 (1 bead on each wire). The lower section has beads of value 1 (4 beads on each wire). By pushing up a bead of value 1 to just below the marking point we can mark 1, and by pushing down the bead of value 5 to just above the mark, we can record 5. The number 5 is used as the base for expressing numbers, but as we move from right to left, the numbers expressed on the wires are multiplied by 10 (see the pictures below). Merely by moving the beads, we can add, subtract, multiply, or divide numbers.

The Chinese abacus has five beads on the lower section of each wire, two on the upper.

Various parts of a Japanese abacus. — Marking

Wire

2863

HOW TO CARRY OUT THE CALCULATION Addition

Subtraction

•

32

47

794

44

Bead of value 5

A Japanese boy using an abacus.

759

HOW TO EXPRESS NUMBERS

point

1

- I 73 32

To use a mechanical calculating machine, the operator sets the key and turns the handle.

The slide-rule is a convenient tool for carrying out multiplication and division.

To work out a multiplication ( 2 x 4), first slide the middle ruler until the 1 on the C-scale lines up w i t h the 2 marked on the D-scale. Then read number on the D-scale which is opposite the 4 marked on the C-scale. To work out a division (4-h 2), slide the middle ruler so that the 4 on the lines up w i t h the 2 on the C-scale. Then read off the number on the which is opposite the 1 on the C-scale.

How to use a slide-rule. .

marked off the O-scale D-scale

The electric calculating machine w o r k s by pushing buttons.

The most recently developed calculating machine is the computer, which uses the binary number system.

As well as abaci, there are many kinds of calculating machines. A slide-rule is made of two sets of rulers which can slide along parallel to each other. The principle of logarithms is used to make the special scales, which are numbered from 1 to 10. The spaces between the numbers get smaller as the numbers get larger. Because of the special property of logarithms, we can multiply numbers by adding spaces on the rulers, and divide numbers by subtracting spaces on the rulers. A mechanical calculating machine has a system of toothed wheels inside it which make the calculations possible. This type of machine was invented by a French mathematician Blaise Pascal about 300 years ago. Addition and multiplication are done by turning the handle forward, while subtraction and division are achieved by turning it backward. Electric calculating machines are getting smaller in size and are becoming more popular. Computers are the most advanced calculating machines. By using them, a large number of complex calculations can be carried out very quickly. 45

POSITIVE NUMBERS AND NEGATIVE NUMBERS

* Numbers such as 1, 2, 3, ... or \, | which are larger than 0, are called positive numbers. Negative numbers are smaller than 0. Zero (0) is neither positive nor negative. * If A is a positive number, — A (read as 'minus A') is a negative number. For example, —1, —2, —3, ... or — are negative numbers. * —2 is smaller than — 1 ; —3 is smaller than — 2 , . . . The series of negative numbers —1, —2, —3, ... keeps getting smaller and smaller with no end.

POSITIVE NUMBERS AND NEGATIVE NUMBERS

-4

-3

-2

-1

-1.5

-0.5

l u j U X x j e U t

0.5 .uu-LLULU™

1.5

_L 1

1

Negative numbers are smaller than 0. Positive numbers are larger than 0. Zero (0) is neither positive nor negative.

+

UPSTAIRS AND DOWNSTAIRS

Let's call the ground-level step the zero step. Steps leading upwards are positive steps, and steps going downwards are negative steps.

The centigrade thermometer measures temperatures. When the temperature is higher than 0 degrees, it is positive; temperatures lower than 0 degrees are negative.

If the sea-level is 0 metres, then heights above sea-level are positive and heights below sea-level (depths) are negative.

DEPOSITING AND WITHDRAWING MONEY

If w e c o u n t p u t t i n g money in the bank as positive, then drawing money out of the bank is negative.

GOING RIGHT AND GOING LEFT

4i

Numbers we have used on earlier pages (0, natural numbers 1, 2, 3 , . . . ; decimals 0.1, 0.2, ...; fractions J, ...) are all greater than or equal to 0. Numbers greater than 0 are usually marked to the right of the origin 0 on the line of numbers, and are called positive numbers. The line also goes to the left of the origin, with no end. To mark 1, 2, 3, ... on the line we moved one, two, three, ... steps to the right of the origin 0. But now we can mark —1, —2, —3, ... (minus one, minus two, minus three, ...) by moving one, two, three, ... steps to the left of the origin. If the position of the tree is set to be 0, and if positions to the right of the tree are positive, then positions to the left of the tree are negative.

Negative numbers are like mirror images of positive numbers. Positive numbers 1, 2, 3, ... get larger and larger with no end, but negative numbers — 1, —2, —3, ... get smaller and smaller with no end. Using the symbol < (less than), we have . . . — 2 < — 1 < — } < 0 < i < 1 < 2 ...

We are surrounded by all kinds of shapes: triangles, squares, circles, and so on. The study of shapes, called geometric figures, is also an important subject in mathematics. Let's study basic shapes and their properties:

SHAPES

parallel

lines,

perpendicular

lines,

rotation

of figures,

symmetry,

similarity,

congruence, and so on, by looking at many examples. We shall also see how we can indicate the position of a point on a line, on a plane, or in space.

Many different kinds of shapes can be found in this picture of the pavilions at Expo '70.

The Swiss pavilion.

The Italian pavilion.

47

INTERESTING

* We can discover many interesting shapes around us. * They may be found in nature and among man-made objects. The golden mean is considered by artists to be the most well-balanced ratio of length to width. * Ths balance of a figure is the basis of its beauty. "Interesting shapes can be made by arranging lines and curves.

SHAPES

A seashell has a coiled shape

A flower is based on curved shapes.

We can discover many beautiful shapes in nature all around us, such as flower petals, snowflakes, the network of a honeycomb, and coils formed by seashells. We can also find many beautiful shapes in man-made things. Among them there are rectangles whose long and short sides are in a special ratio called the golden mean, which is a proportion of about 1 : 0.62. The designs of many classical

THE GOLDEN A

MEAN

buildings are influenced by the idea that the golden mean is the ideal ratio. Even today, many rectangular shapes such as those of books and playing cards are formed using the golden mean. The Parthenon, built in Athens nearly 2 500 years ago, has a shape based on the golden mean. There are also other beautiful proportions. We can generally say a shape is beautiful if it is well-balanced.

This mathematical calculation was developed by Greek artists 2500 years ago. — D

E

HOPE

In the above diagram, G is the mid-point of the side BC of the square ABCD. We draw a circle with the centre at G and with radius GD. The circle meets the extension of the line BC at F. The ratio BF : AB is then the golden mean (about 1 : 0.62).

48

The design of the Parthenon, built 2 500 years ago, is based on the mathematical calculation of the golden mean.

These two packets are examples of modern shapes based on the golden mean.

Each surface of this building, the Tokyo cathedral, is formed from many sweeping lines.

A honeycomb in a bee-hive is a network of regular hexagons.

A diamond can be cut so that each facet reflects light

A pattern of floor-tiles in a room gives a well-balanced

totally.

effect.

SIMPLE SHAPES

^

0k

TRIANGLES

* A triangle is a basic shape. "The shape of a handkerchief, a window, or a picture-frame is a quadrilateral or rectangle. * A circle is the most well-balanced curved shape. * A sphere (or ball) is the most well-balanced solid shape. * A cone is a solid, with a circular (or other curved) base, which gradually tapers to a point. * A cylinder is a solid such as a roller or a column. * A prism is a solid such as a cube or a box.

QUADRILATERALS

jHHHk

CIRCLES

JHHHBHHHHBHHHHm.

V

V

*

*

Triangles have three sides and are among the most basic shapes. Their uses include sails, drawing instruments, and the musical instrument called a triangle (below).

Quadrilaterals are shapes w i t h four sides. They include rectangles, squares, and diamonds. Tables, pictureframes, and flags are all quadrilaterals (below).

Circles are the most well-balanced shapes made from a curve. Disks, coins, round tables, car tyres, and some watch-faces are circular in shape (below).

A triangular road sign stands on the right side of the road.

Rectangular windows.

A circular clock-face.

ieaJ*

We find many different shapes around us. They are divided into plane shapes and solid shapes. Plane shapes may be drawn on a plane, have length and width, and include triangles, rectangles, and circles. Solid shapes have length, width, and depth. Spheres, cones, pyramids, and prisms are common solid shapes.

CONES AND PYRAMIDS

CYLINDERS AND PRISMS

SPHERES

Spheres, or balls, are the most well-balanced solid shapes. Footballs, some melons, globes, and containers for storing gas in refineries are all spheres (below).

Cones and pyramids have broad bases and pointed tops. Examples of these are the Egyptian pyramids, tripods, and tall party-hats (below).

Cylinders are roller-shaped objects. Prisms are solids with both ends of the same shape. Most cans, boxes, and buildings (below) are cylindrical or prismatic.

Spherical sweets.

An Egyptian pyramid.

Cylindrical and prismatic buildings.

51

LINES AND ANGLES

•Rectilinear shapes are made of a series of lines. * A straight line does not have an end, but a half-line does. * An angle is formed where half-lines meet. * Angles are measured in units called degrees, minutes, and seconds. "The pairs of plastic triangles found in a set of drawing instruments are called set-squares. Each has a standard shape. * Using a pair of set-squares we can make many different angles. *To describe a point in terms of another point we use directions. '

TONY'S ERRAND

Tony went on an errand to the post-office. Tracing the way he walked (pink line on the street plan) we get a rectilinear shape.

RECTILINEAR SHAPES Exterior

angle

A rectilinear shape is made op of lines, which are called the sides of the figore. Two sides meet at a vertex, where they make an angle.

HALF-LINE AND STRAIGHT LINE A half-line is the term osed to describe a line stretching from a point (the end) in only one direction.

A straight line stretches in both directions and has no ends.

52

The part of a straight line between two points on it is called a line segment. A half-line is an unlimited part of a straight line with only one end-point. The sides of a rectilinear shape are line segments. The angle between two half-lines indicates how wide they open at the point they meet. To measure angles we use units called degrees, minutes, and seconds. The sexagesimal system is used to measure angles. In this system, 1 degree = 60 minutes and 1 minute = 60 seconds. We use the symbols 0 ,', and " to stand for degrees, minutes, and seconds. A straight angle is made by half-lines meeting at a point and forming a continuous straight line. It is equal to 180°. A half of a straight angle, 90°, is called a right angle. Set-squares used in mechanical drawing are generally flat pieces of plastic in the shape of right-angled triangles, which come in pairs. In addition to the right angle, one has two angles of 45°, while the other has angles of 60° and 30°.

MEASURING ANGLES

To survey land, engineers use an instrument called a transit to measure horizontal angles.

Angles of small things may be compared directly.

A protractor is used to measure angles accurately.

MAKING ANGLES USING SET-SQUARES

SET-SQUARES

45"

45

i k

Set-squares come in pairs. One of them has the shape of an isosceles (two sides equal) right-angled triangle. The other has the shape of a right-angled triangle with other angles of 60° and 30°.

Using a pair of set-squares we can make many different angles. Try to make angles different from the ones in the picture.

DIRECTIONS

POINTS OF A COMPASS

ENE

145° W

130° E

To explain where a point is in terms of another point, we use directions. By measuring angles we can tell the directions. For example, in the picture on the left, the factory is located 130° east of the house, while the school is located 30° west of the house. Worth, south, east, and west are the four points of the compass — the four basic directions. By splitting these still further, 16 different directions may be defined, as above. 53

PARALLEL AND PERPENDICULAR

"Straight lines in the same plane are parallel if they never meet. "Two straight lines, a straight line and a plane, or two planes which meet at right angles are said to be perpendicular to each other. "There are many examples of parallel lines as well as perpendicular planes and lines around us. " T w o straight lines not in the same plane are said to be in a twisted (or skewed) position if they never meet.

In ancient Egypt, the markings of field boundaries were washed away every year when the Nile River flooded. After the flood had subsided, men called 'rope stretchers' used ropes to re-survey the land and mark out the fields. The ropes were knotted at regular intervals, and so could be used like a flexible ruler. To mark out a right angle, they used the ropes to make a triangle whose sides were in the ratio of 3:4:5. All triangles with sides in these proportions are right-angled triangles.

PYTHAGORAS' THEOREM

Pythagoras' theorem states that, in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. The hypotenuse of a right-angled triangle is its longest side (always opposite the right angle). If its length is c, and the lengths of the other two sides are a and b, the theorem can be expressed as the equation a2 + b 2 = c2 where a2 = a x a, and so on.

PARALLEL LINES An escalator.

II

Stripes on ties.

Straight railway tracks.

54

How to draw parallel lines.

PERPENDICULAR LINES AND PLANES

PARALLEL AND PERPENDICULAR LINES

How to make right angles.

These wall tiles form a pattern of parallel and perpendicular lines.

Fronts of buildings are perpendicular to the road.

A plumb-line.

Using set-squares, it is easy to make right angles.

The road and the railway tracks never meet. on the same plane.

But these two straight lines are not parallel

Two straight lines are said to be in a twisted,

and if they are not parallel.

Two straight lines on the same plane are called parallel lines if they never meet. Two straight lines are said to be perpendicular to each other if they meet at right angles. The ancient Egyptians used knotted ropes to make right angles, and the people who used the ropes to measure the land after the Nile floods were called rope stretchers. Nowadays, people use plumb-lines to get a line perpendicular to the ground. There are many examples of parallel and perpendicular lines around us. The lines on graph paper can be considered as two sets: horizontal and vertical. The lines belonging to each set are parallel, but the lines belonging to the different sets are perpendicular to each other. Try to find more examples of parallel lines and perpendicular lines and planes. As you can see in the picture [right), some lines never meet and yet they are not parallel. In such a case, we say that the lines are in a twisted (or skewed) position.

A Japanese window-frame.

or skewed,

because they are not

position if they do not meet

TRIANGLES

* A triangle is a plane figure bounded by three lines which are called its sides. * A triangle has three sides, three angles (the 'openings' between the sides), and three vertices (the 'corners' of the triangle). * Triangles are classified according to their shapes. "The sum of the three interior angles of any triangle is 180°.

The railway bridge in the picture is built of steel girders bolted together to make a series of triangles.

TYPES OF TRIANGLES

Equilateral triangle.

Scalene triangle.

Isosceles triangle.

Right-angled triangle.

56

Isosceles right-angled triangle.

Triangles can be grouped into several kinds according to the relations of their three sides and angles. A scalene triangle has unequal sides and angles. An isosceles triangle has t w o equal sides, while an equilateral triangle has all its sides equal.

PARTS OF A TRIANGLE

USING PENCILS TO MAKE TRIANGLES

Vertex

Vertex

Side

Vertex

Scalene triangle. Isosceles triangle.

Isosceles right-angled triangle.

y

Right-angled triangle.

PROPERTIES OF TRIANGLES The sum of the three interior angles of any triangle is 180°.

Base angles of an isosceles triangle are equal.

The sum of any two sides of a triangle is always greater than the length of the remaining side. The difference between any two sides of a triangle is smaller than the remaining side.

The three angles of an equilateral triangle are all equal. Triangles are made up of three straight lines. These lines are called the sides of the triangle. The points where the sides meet are called vertices (or corners). It follows that triangles have three sides, three vertices, and three angles. There are triangles with many different shapes. These shapes can be classified in terms of the lengths of the sides and whether or not the angles are right angles. Triangles have two important characteristics. One is that the three interior angles add up to 180° (the angle which makes a straight line). The other is that, for any triangle, the sum of the lengths of two sides is always greater than the length of the third side. 57

QUADRILATERALS

*A "A are the

A quadrilateral has four vertices, four sides, and four angles. The sum of the interior angles of a quadrilateral is equal to 360°, which is equal to four right angles. A diagonal of a quadrilateral is a line joining opposite corners. A quadrilateral has two diagonals. By looking at the four sides and angles of a quadrilateral, we can find out which kind it is. Let's look around and find all kinds of quadrilaterals. Look at their sides, angles, vertices, and diagonals. Look also for the various types of quadrilaterals.

quadrilateral is a plane figure with four angles and four straight sides. quadrilateral has four vertices and two diagonals. "Quadrilaterals classified into several kinds according to their shapes. "The sum of four interior angles of a quadrilateral is 360°.

Find the quadrilaterals.

VARIOUS TYPES OF QUADRILATERALS

Rectangle.

Trapezium.

Square.

Parallelogram.

Rhombus.

«mmmndbm

Quadrilaterals are divided into types according to their shapes. For example, a parallelogram has its opposite sides parallel. Rectangles, squares, and rhombuses are all parallelograms. Squares and rhombuses have all their sides equal. A trapezium has one pair of opposite sides parallel while a kite-shape has two pairs of equal sides. Atrapezoid is a quadrilateral with no parallel sides. An arrowhead shape is special because one of its interior angles is greater than 180°.

Arrowhead shape.

Trapezoid.

Sfi

Kite shape.

USING PENCILS TO MAKE QUADRILATERALS

PARTS OF A QUADRILATERAL

Trapezoid.

Vertex

Trapezium.

Vertex

Side

Vertex

Square. Rectangle.

Rhombus.

Parallelogram.

PROPERTIES OF QUADRILATERALS LENGTHS OF DIAGONALS AND THE WAYS THEY MEET

Trapezoid.

Trapezium.

For trapezoids and trapeziums, there is no rule as to length of diagonals and the ways they meet.

Diagonals have equal length cut each other in half.

and

THE SUM OF THE FOUR INTERIOR ANGLES OF A QUADRILATERAL IS 360°

Cut a quadrilateral into four pieces and paste them together.

Diagonals have equal length, cut each other in half, and make right angles where they cross.

Parallelogram Rhombus.

Diagonals cut each other in half, but their lengths may be different.

Diagonals cut each other in half, make right angles where they cross, but their lengths may be different.

59

p i p p i

E C

"There are many circular (round) figures around us. "Every point on a circle is equally distant from the centre. We may use this property to draw many different sized circles. "The length of the circumference of a circle is about three times the length of its diameter. "Every regular polygon may be inscribed in a circle, and as the number of their sides gets larger, the regular polygons get closer and closer to the circle.

'

From ancient times, circles have been admired as beautiful shapes. A pair of compasses is used to draw circles. The pointed leg of the compasses is placed at a point (the centre) and the other leg, with a pencil, scribed round the centre making the circumference of a circle. Every point on the circumference of the circle is equally distant from the centre. A line passing through the centre of a circle from one side to the other is called a diameter. A line extending from the centre to a point on the circumference is called a radius. The diameter of a circle is twice as long as its radius. An equilateral triangle, a square, a regular pentagon, and so on may all be inscribed in a circle. As the number of their sides gets larger, the polygons get closer to the circumference. Later we shall learn about n (pronounced 'pie'), a Greek letter standing for the ratio of the circumference of a circle to its diameter. We shall also learn how to find the area of a circle.

HOW TO DRAW CIRCLES

/

v Using the rim of a glass.

PARTS OF A CIRCLE

Using a pair of compasses.

\

Using a needle and a piece of string.

V ^

Using a rod to draw a large circle.

LENGTH OF THE CIRCUMFERENCE AND OF THE DIAMETER

Circumference Roll a coin along a line.

o I ra

Radius

Diameter

The length of the circumference of a circle is just over 3 times its diameter.

Compare the numbers of beads you can put on the circumference and along the diameter.

J

A triangle. (An equilateral triangle.)

Coloured papers. (A square.)

Plum flowers. (A regular pentagon.)

A turtle shell. (A regular hexagon.

An umbrella. (A regular octagon.)

A tyre. (A circle.)

PATTERNS MADE WITH A PAIR OF COMPASSES

We can make pretty patterns using a pair of compasses. Try to make patterns other than those shown here.

A circle sometimes means its circumference, which is an example from the family of curves called conics (others are ellipses, parabolas, and hyperbolas). They can all be produced by the intersection of a plane with a circular cone. The idea of approximating a circle by using regular polygons has been developed since ancient times. It became one of the roots from which the ideas of limits, differentiation, and integration have been developed. They in turn, were needed to define clearly the length of the circumference of a circle and the area of the circle. 61

VARIOUS CURVES

*AII around us are shapes which are bounded by curves. Among such curves are circles, ellipses, ovals, parabolas, hyperbolas, catenaries, involutes, and cycloids.

SECTIONS OF A CARROT

Most curves have their own special names. For instance, there is the perfectly round circle, the slightly 'squeezed' circle which we call an ellipse, and a parabola, which is the shape of the path taken by a ball when it is thrown in the air. The special curve formed by a plane cutting through the side and base of a circular cone is known as a hyperbola. The shape taken up by a chain or rope hanging freely between two points at the same height is known as a catenary. When a piece of string is uncoiled from some fixed curve in the same plane (for example, a length of cotton from a cotton-reel), the end of the piece of string follows a curve which is known as an involute. A cycloid is the name given to the curve traced by a point on the outside edge of a wheel (or circle) rolling along in a straight line. See how many different kinds of curves you can find.

By cutting a carrot in three different ways, we get three different kinds of curves.

A gramophone record. (A circle.)

Orbit of a satellite. (An ellipse.) An egg. (An oval.)

The path of a shell. (A parabola.)

A suspension bridge. (A catenary.)

An involute is traced by the end of a piece of sticky tape as it is unrolled off the spool.

A cycloid is traced by a point on the rim of a bicycle wheel as it rolls along a straight

LINES AND CURVES

Curves formed by straight lines.

A circle formed by straight lines.

FUN WITH SHAPES

/ / / /

ILLUSIONS

/ /

/

/ /

/ / / / / / / '

/ / / /

Because of the arrowheads, the lower line looks longer than the upper one.

A The sloped lines are parts of the same straight line, although they look as if they are not.

/ / / /

ss \ \

\ \ \ \ \ \ \ \ \ \ \ \

1s \ \ V. i* \

\

\

A circle formed by circles.

/ / /

/

/

/

/

/ / / / /

/

/ /

/ / /

/

/

'

/ /

/ /

Three parallel lines look as if they are not parallel.

Perfect squares look as if they are warped. The vertical line looks longer than the horizontal line.

The three columns have exactly the same height.

Is this picture ot two profiles, or a picture of a man and a woman head to head? If we look at the black parts, it looks like two people facing each other. But the white parts look like a woman's head upside-down over a man's.

The same figure may look different when the shapes around it change. We have many examples of this in our daily life. The same person may look fat or slim according to the pattern of the clothes he wears. Some shopkeepers sell carrots or radishes with the tops on so that they look larger. A lion's mane or a peacock's feathers may help to make the male look bigger and more powerful than the female. 63

SOLID SHAPES

"There are many solid shapes (or figures) around us. * A cone is a solid figure with a circular (or other curved) base which gradually tapers to a point. A pyramid has a polygon as its base and triangular sides which meet at a common point called the vertex. * A cylinder is a solid shape such as a roller. A prism is a solid shape such as a cube. * The shadow of a solid shape on a plane is its projection. * By turning a plane shape about an axis we get a solid of revolution. *As the number of sides of a regular polyhedron gets larger, the polyhedron approaches a sphere. A plane figure lies on a plane and has no thickness, whereas a solid figure has length, breadth, and depth. A cone is a solid shape with a circular (or other curved) base and which gradually tapers to a point, whereas a pyramid has a polygon as its base and triangular faces meeting at a common vertex as its sides. A triangular pyramid, quadrangular pyramid, and hexagonal pyramid have as their bases a triangle, quadrilateral, and hexagon respectively. As the number of sides on the base of a regular polygonal pyramid gets larger, the pyramid approaches a circular cone. A pillar or column is an example of a cylinder. A cylinder or a prism is a solid shape with parallel and equal plane shapes as its ends. The sides of the figure are made up of the set of parallel line segments joining opposite points round the sides of the end figures. A triangular prism has equal and parallel triangles as its ends. There are also quadrangular prisms, hexagonal prisms, and so on. Ordinary pencils (the kind with flat sides) are long, narrow hexagonal

A pavilion at Expo'70.

PYRAMIDS AND CONES PROJECTIONS PROJECTION OF A CONE

Side view of a cone.

m

Regular triangular pyramid

Regular quadrangular pyramid

Regular hexagonal pyramid

Regular octagonal pyramid

PRISMS AND CYLINDERS

A projection chart of a solid shape is composed of several projections of the figure onto different planes.

Regular triangular prism

Regular quadrangular prism

Regular hexagonal Prlsm

Regular prism

octagonal

A cone viewed from above or below. By rotating a circular fan we can make a ball shape.

64

prisms. As the number of sides on the base shape of a prism gets larger, and if the base is a regular polygon, the prism approaches a circular

A regular tetrahedron.

cylinder.

A polyhedron is a solid shape with several plane surfaces. The surfaces of a regular polyhedron are equal regular polygons. There are pictures

of

hexahedron regular

a (or

regular a

dodecahedron,

tetrahedron,

cube),

regular

regular

octahedron,

a n d regular

icosahedron

on this page. A circular cone is generated by rotating a right angled triangle about an axis along a side of the triangle making the right angle. Similarly, a cylinder is made by rotating a rectangle about one of its sides. By rotating a semi-circle about its diameter we can make a sphere. Solid shapes such as these, which can be made by rotating plane shapes, are called solids of revolution. Many pieces of pottery and china are such solids. A potter's wheel is a convenient tool to make solids of revolution.

A regular hexahedron (cube).

A regular octahedron.

A regular dodecahedron.

Circular cone

A regular icosahedron.

A calendar. A football.

Circular

cylinder

A sphere.

A sphere is made by rotating a semicircle about its diameter.

65

POSITIONS OF POINTS

*Any two different points have exactly one line passing through them. "The position of each point on a line may be represented by a number. "The position of each point on a plane may be represented by a pair of numbers. "The position of each point in space may be represented by three numbers. "Any shape is a set of points.

To represent the positions of points on a line by numbers, we start by choosing the origin, t h e unit

length,

a n d t h e positive

direction.

In

the picture below, for example, the origin is Bill's house, the unit length is 10 metres, and the positive direction is eastwards. A pair of numbers may be used to represent the position of a point on a plane. To do this, we need to know the origin and two axes (usually called /-axis and /-axis) which meet at the origin and are perpendicular to each other. Now if P is any point on the plane, and if the distance of P from the /-axis is A and the distance of P from the /-axis is B, then the pair of numbers (A,B) represents the position of the point P. Such a pair of numbers is called an ordered pair. The numbers in each ordered pair are called the co-ordinates of the point.

*

_

.

A modern city, seen from the air, looks like a chessboard. The streets run either east to west or north to south. Intersections of the two kinds of streets are used to describe locations in the city.

POSITIONS OF POINTS ON A PLANE The distance between a certain place and Bill's house may be represented by the position of a point on a line. The easterly direction is taken to be positive and the westerly direction negative. The post-box corresponds to the number 30. The bus-stop corresponds to —40.

-50m

-40m

-30m

-20m

20m

I Om

POSITIONS OF POINTS ON A PLANE

On this chessboard, the counter is at the intersection of the sixth vertical line and the fourth horizontal line. The position of the counter can be represented by a pair of numbers (6, 4). The number of the vertical line comes first. The position of any point on the board can also be represented by a pair of numbers (A, B).

Places on a map are given as ( • , 4) and so on, in order that a user can find where he is.

66

30m

40m

50m

POSITION OF A POINT IN SPACE

To define the position of a point in space, we use an ^-axis,' /-axis, and z-axis. These three axes meet at a point of origin 0 and ar3 mutually perpendicular. In the diagram (left), the position of the point P (in blue) is 3 units from the yz-plane, 4 units from the Arz-plane, and 5 units from the jry-plane. Therefore, the position of this point

z-axis

5

I (3, 4, 5)

in space is represented by the (3,4,5).

I

co-crdinates

y-axis

Each line segment in the same colour is the same length.

x-axis VARIOUS SETS OF POINTS

d

M

i #••

>

A

V "

t

/

JJ

(Jf

*

r ' k

0 P W

v.

Shape of a car made by marbles.

Neon signs.

Embroidery.

A stadium filled with spectators watching a display.

67

MATHEMATICAL

* A three-dimensional diagram of a solid figure demonstrates how it is constructed by showing its sides, faces, and vertices. * A solid figure may be changed into a plane figure by cutting some of the edges of the original figure and opening it out flat as a 'fold-out'. * By re-assembling the fold-out of a solid figure and pasting its edges together we get the original shape.

MODELS

REGULAR POLYHEDRONS

Regular

Regular

Regular

68

octahedron

tetrahedron

dodecahedron

Regular

icosahedron

PYRAMIDS AND CONES

(Figure:

Regular triangular

PRISMS AND CYLINDERS

Regular triangular

solid blue. Three-dimensional diagram:

pyramid

blue lines.

Fold-out:

Regular quadrangular

red lines.).

pyramid

Circular

cone

(Figure: solid blue. Three-dimensional diagram: blue lines. Fold-out: red lines.)

prism

Regular pentagonal

A PINHOLE CAMERA

prism'~

Circular cylinder

•

All measurements in cm.

A pinhole camera may be made by folding up and pasting the edges of the cardboard fold-outs. [Below) The inner box is up so that one end is and the other end has a 'window' cut out of it. glue to cover this window tracing paper.

made open small Use with

[Above) The outer box is made up so that one end is open. Make a pinhole in the centre of the other end which is closed.

[Right) Insert the inner box into the outer box. The image mill pass through the hole and appear upsidedown on the tracing paper.

One way to understand the structures of solid figures is to look at their fold-outs, which are plane shapes showing how they are constructed. For example, by looking at the fold-out of a regular tetrahedron we can see that it has four equilateral triangles as its faces; a cube has six squares as its faces; a regular octahedron has eight equilateral triangles as its faces, and so on. By cutting the edges of a cardboard box we can open it up and get a flat shape which is the fold-out of the box (see picture on previous page). To draw a three-dimensional diagram of a solid shape, we usually use dotted lines to show the sides which are hidden, and solid lines to show visible parts. By re-assembling a fold-out of a solid figure and joining its edges we can get a model of the figure. 69

SIMILARITY AND CONGRUENCE

"The Greek mathematician Thales used the idea of similarity to find the height of a pyramid. " T w o shapes are called similar when they have the same shape but not necessarily the same size. * if a shape can be placed upon another so that the two match they are said to be congruent with each other.

FINDING THE HEIGHT OF A PYRAMID About 2 500 years ago a Greek mathematician named Thales surprised people by calculating the height of a pyramid from the length of the shadow of a stick. Thales used the fact that the big triangle ABC, formed by the pyramid and its shadow, and the small triangle DCE, formed by the stick and its shadow, are similar. In this case, by writing AB as the length of the side AB, and so on, we have A B _ DC BC ~ CE' Thales could measure the lengths BC, DC, and CE. So he used this equation to calculate the height AB of the pyramid.

People often enlarge a photograph after developing a film. The enlarged picture and the original scene have the same shape but not the same size. They are called similar. If, on the other hand, two shapes may be placed one upon another so that they coincide exactly in all their parts, they are called congruent. Look around and see how many similar or congruent shapes you can find.

SIMILAR SHAPES A shadow puppet.

CONGRUENT SHAPES fMMMMMMM,MMM

—

Spoons.

t

i

*

i

i

i

A

r"

rjK. M MM mm M

* * * *

A * MM

VIA AIR MAIL

Butterflies.

i.

—*

JF M

* i *

VIA Air' MAIL jr m m m m m m m m m m m

i

. J

Envelopes.

Milk bottles. Books.

A car and a model car.

70

REDUCED COPIES AND ENLARGED COPIES

* A reduced copy of a diagram has the same shape as the original but is smaller in size. "An enlarged copy of a diagram has the same shape as the original but is larger in size. "By enlarging or shrinking the size of a diagram in only the vertical or horizontal direction, we get a distortion of the original diagram.

;>

HOW A MAP IS MADE

%

-Si,

,^ * V

XT'**'** nWl.a-..' » te '- "r*

m u k

w w

m

b

m

Taking aerial photographs. —-•=—— Hospital

©

®

School

©

Factory

fi

Church

4-

Dock

E3 Police-station

©

^f^vji? /A An aerial photograph.

A map is an example of a reduced copy. Aerial photographs and surveying are used to make maps such as the one on the right.

Station Sportsground

Post-office

Surveying directly.

REDUCED COPIES AND ENLARGED COPIES

Taking a photograph is making a reduced copy.

We make an enlargement of the negative.

Enlarging

DISTORTIONS

An enlarged copy is similar to the original figure but larger in size. A reduced copy is also similar to the original figure but is smaller. A photograph may be an enlarged copy of the negative, which is a reduced copy of the scene photographed. Sometimes we make distorted figures by enlarging or contracting a figure in only one direction.

Original

drawing

Q A distorting mirror.

w 71

SYMMETRIES

41 & ted

' T w o 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 — that is, face to face. * There are three kinds of symmetry: symmetry about a point, symmetry about a line, and symmetry about a plane.

VARIOUS SYMMETRIES

A figure can fold half of the right

is symmetric about a line if you it along the line so that the left the figure coincides exactly with half.

A figure is symmetric about a point if it can be rotated 180 degrees around a point and look exactly the same.

A figure is symmetric about a plane if it is divided into two equal parts by a plane and the two halves are mirror images of each other. The mirror image of a face and the face itself are symmetric about the plane of the mirror.

SYMMETRIC SHAPES The sails of a windmill.

The blades of a propeller.

Symmetry about a point

Symmetry about a point

A product symbol.

The symbol of Expo '70.

A Symmetry about a line

The Japanese flag.

Symmetry about a line

72

Symmetry about a line and a point

A butterfly is symmetric about a plane (although its photograph is symmetric about a line).

Two figures are called symmetric if they are mirror images of each other or if they can l)e placed one upon the other and match exactly after 180 degrees of rotation about a point. The three kinds of symmetry are: symmetry about a point, symmetry about a line, and symmetry about a plane. The bodies of most animals are more.or less symmetric about planes. Their left and right sides are almost mirror images of each other. A circle is symmetric about its centre and its diameter. A sphere is symmetric about a plane passing through its centre, about its centre, and about its diameter.

QUANTITIES

We use many kinds of quantities: the length of a ruler, the weight of some apples, the volume of milk in a bottle, the length of time a television programme lasts, and so on. Some quantities, such as length of time, are hard to understand. We use various units to measure quantities. Let's learn how to measure and calculate various quantities.

WAYS OF MEASURING QUANTITIES About 2 000 years ago, a Greek mathematician named Archimedes was ordered by the king to examine his crown to see if it was really made of pure gold. The king insisted that the crown should not be damaged. Archimedes decided to find a way of measuring the exact volume of the crown without melting it down. Then he would be able to tell whether or not the crown was gold by comparing its weight with the weight of a similar volume of pure gold. One day, when he was taking a bath, he realised that if anything is put into a vessel filled with water, the volume of water that overflows is the same as the volume of the object put into the water. Overjoyed by this discovery, he ran naked through the street crying 'Eureka!' ('I have found i t ! ' ) . Using this method, he found that the king's crown was not made of pure gold but was debased with a cheaper metal.

73

LENGTH

* Sometimes we can compare the lengths ot things directly. " by choosing a convenient standard length as a unit, we can measure the lengths of things even when they cannot be compared directly. 'Distance is a kind of length. But the distance between two places along a winding road is longer than a direct line (as the crow flies) between them. *Height and depth are also kinds of length. *As long as we know the scale used in drawing a map, we can tell the distance between two places by measuring the distance on the map. ' T h e length of the circumference of a circle is just over 3 times the length of its diameter.

By lining up pencils side by side, we can compare their lengths directly. Heights can be compared directly.

When their ends are not in line, or when the pencils are not lying straight, their lengths cannot be directly compared.

By choosing a unit of length, we can measure and compare the lengths of things even when we cannot compare them directly.

UNITS OF LENGTH

The distance between the thumb and middle finger (about 15 cm). The circumference of a tree.

The length of a pace (about 60 cm).

The distance to school. The distance between the finger-tips when the arms are stretched out (about 90 cm).

74

The standard metre is a metal rod made of an alloy of platinum and iridium.

1 m (one metre) is a unit of length which is 1 /40 000 000 of the distance along the meridian of the globe (a great circle round the world through the poles). The standard metre is exactly 1 m long. Since 1960, the wavelength of krypton light has been used as the standard unit of length.

Krypton gas gives off light in an electric discharge tube similar to the neon tubes used in advertising signs.

STANDARD UNITS OF LENGTH Sometimes we can compare the lengths of things directly. Even when we cannot compare directly, we can choose a unit of length and then measure the lengths of things and compare them. Our ancestors used the average length of the human foot as the basic unit of length. Now, the units of the metric system are used in most

countries. 1 m (one metre) is an internationally used unit of length which is 1/40000000 of the distance along the meridian of the globe. We also use 1 km (one kilometre) = 1 000 m, 1 cm (one centimetre) = 1/100 m, and 1 mm (one millimetre) = 1/1 000 m. To measure the length of something which is

about the size of a pencil, we use centimetres. Millimetres are used as units to measure smaller things, such as the diameter of a coin. Kilometres are used to measure long lengths such as the distance between two cities. Nowadays, the standard unit of length is the wavelength of krypton light.

TOOLS TO MEASURE LENGTH The tape-measure. The folding ruler.

-WA ih Sliding callipers.

aTF3 ii f

I

J<' "P

-

V*

The curvimeter is used to measure the distance along a curve or on a map.

\

"{MJJI"" h I i !t n >4 Ml <1 < I «i! 1.! 11 .ti, „, i Sliding callipers and micrometers are used to measure exact lengths of small things.

The micrometer.

75

HEIGHT AND DEPTH Distance from the earth to the moon: 384 400 km

Altitude reached by jet planes: 11 000 m

about

Height of Mt. Everest: 8 848 m

The height of a tree or a house is the distance from the ground to the top of the tree or house.

Height of Mt. Fuji: 3 776 m

The he from mountain. The __ r ... . . the distance from sea-level bottom of the ocean.

The height of a triangle is the distance from the base (bottom side) to the opposite vertex.

to

, the

- d -

DISTANCE

The distance between two places may mean two things : either the distance along a road joining them, or the distance along a straight line connecting the places. The latter is usually shorter than the former. The picture (right) shows a zigzag road along a mountain ridge.

76

Depth of nuclear-powered submarines: about 300 m jrrWt

Depth of Mariana Trench: 11 034 m

bfUl M V U K ' MEASURING THE DISTANCE ALONG A

ROAD ON

A MAP.

Heyward Point

Let 1/A be the scale of the map. Lay a piece of. string along the road on the map.

DUNEDIN Measure the length of the string and multiply its length by A. The answer is the distance along the road.

AND

ENVIRONS

1 : 2 0 0

0 0 0

1 7 0 ? JO' The scale of the above map is 1/200 000. How far is the distance along the road coloured in red? Is it longer or shorter than the distance along the blue road?

THE NUMBER Tt

7r =3. 14 I 59265358979323846

Measure the circumference of a plate by rolling it along a table or using a piece of string. Divide this length by the length of the diameter. The answer is the value of 71 .

The number n (the sixteenth letter of the Greek-alphabet, pronounced 'pie') is the ratio of the circumference of a circle to its diameter. It is about " or approximately 3.14, although actually it has an unlimited number of decimal places. Recently, by means ot high-speed computers, it has been possible to calculate n to 100000 decimal places. The ratio of the perimeter (distance round) of a regular polygon to its diameter approaches n as the number of sides increases.

UNITS TABLE To measure height or depth we also use units of length. The height of a tree is the distance from the ground to its top. The depth of the ocean is the distance from sea-level to the bottom. The distance from the station to the park is the distance along the road joining the two places. Usually, the distance along the road is longer than the distance along a straight line connecting the two places.

If we know the scale of a map, we can tell the actual distance between two places marked on it by measuring the distance on the map. For example, if the scale is 1/20000 and two places are 1 cm apart on the map, then in reality they are 200 m apart (1 c m x 20 000). The ratio of the circumference of a circle to its diameter is n . We generally use 3.14 as an approximation for n .

I km

= I

O O O

I m

=

I

I cm

=

n

=

m

O

Ocm

I

O ram

3.14 77

AREA

'Sometimes we can compare two areas directly. ' B y choosing a unit for area we can measure the areas of various things. * Using graph paper, we can find approximate areas of closed plane figures. 'There are formulae for finding the areas of certain figures.

COMPARING AREAS

Direct comparison by placing one upon the other.

. ..iii. WWffyti i —iHTTWirr •smt-

Indirect method using cards. By comparing the numbers of cards it takes to cover various shapes, we can compare their areas indirectly.

UNIT OF AREA 1 cm cm!

1 (one square centimetre) is the area of square whose sides are 1 cm.

X cm H t e l

SPSE •-.v Cmm^w&.^'JmHm^^'iSx^.

Surveying is used to find the areas of tracts of land.

The area of a plane figure is the amount of space enclosed by its sides. In the case of a solid figure, it means the total area of its external surfaces. There are two easy ways of measuring areas. One is direct comparison by placing one area upon another. The other is the indirect method using, for example, cards as units of area. However, there are many things that we cannot measure by either of these methods. In such cases, we use as a unit of area a square of side 1 cm. The area of such a square is 1 cm2 . A larger unit of area, 1 m2 , is the area of a square of side 1 m. We can express areas by a number, so long as we also state the unit of area.

AREAS OF IRREGULAR FIGURES

1 m z (one square metre) is the area of a square whose sides are 1 m. This unit is used to measure large areas.

UNITS TABLE 1 1 1 1

78

km 2 = 100 ha = 1 0 0 0 0 a = 1 0 0 0 0 0 0 m 2 ha (one hectare) = 100 a = 1 0 0 0 0 m 2 a (one are) = 100 m 2 m 2 = 1 0 0 0 0 cm 2

These may he estimated by using graph paper (paper ruled into squares of 1 cm or 1 mm sides).

SURFACE AREA OF SOLID FIGURES

FORMULAE FOR FINDING AREAS S stands for area.

S stands for area.

RECTANGULAR SOLID (CUBOID)

TRIANGLE A triangle is half a rectangle, as shown in the diagram. S = half the base multiplied by the height. _ ( b a s e ) x (height)

o—

S=

Ixaxh

ARALLELOGRAM

S = 2 x ( a x b + b x c + c x a)

A parallelogram has the same area as a rectangle having the same base and height. S = b a s e x height = a x h

CUBE

«

RHOMBUS A rhombus is half the area of the rectangle shown in the diagram. ( d i a g o n a l ) x (diagonal) S=-

2

S = J x ax b

S = 6xa 2 CIRCLE We can divide a circle into several equal parts and make a shape like a parallelogram by rearranging these parts. The base of this parallelogram is about half of the circumference. Its height is approximately equal to the radius r of the circle. Since half of the circumference is equal to n r, the area of the parallelogram is about 7ir. !

CYLINDER End has radius r.

S = 2x(?rr

+jtrxh)

S = r x ( 7 t x r ) = 7i r 2

PYTHAGORAS' THEOREM

1

Pythagoras' theorem states that, in a right-angled triangle, the square on the hypotenuse (the longest side) is equal to

c a

a2

b2

b

mm

S Fig.

1

the sum of the squares of the other two sides. It may be written: a +b = c In the diagrams (left), Fig 1 shows the squares constructed on the sides of a right-angled triangle. By matching corresponding colours in Fig. 2, we can see how the area of the largest square (on the hypotenuse) is equal to the sum of the areas of the smaller squares. 79

VOLUME

* T o measure the volume of a liquid, we sometimes use a cylindrical container with a vertical scale of measurement. *An exact statement of a volume is expressed by using a number and a unit of volume. "If an object is submerged in a vessel filled with water, the volume of the water displaced (which overflows) is equal to the volume of the object. Using this principle, we can measure the volumes of irregular shapes such as an orange or a stone. "There are formulae for finding thff"volumes of various solid figures.

It is not easy to tell merely by looking at them w h i c h container holds the most, because their shapes are different. To compare the volumes of liquids in different containers, we pour them into measures having the same shape, as shown on the right.

UNITS TABLE 1 / = lOd / 1 d / = 100 cm J 1 cm 3 = 1 cc

/ = litre d/ = decilitre cc = cubic centimetre = cm1

MEASURING VOLUME

Take the orange out.

Submerge the orange in filled w i t h water.

basin

Put the water w h i c h overflowed into a measuring cylinder.

Rfl

FORMULAE FOR VOLUMES OF SOLID FIGURES

V stands for volume.

RECTANGULAR SOLID (CUBOID) CUBE

V = sidex sidex side (the length of a side cubed).

V = l e n g t h x w i d t h x height.

PRISM

SPHERE

CYLINDER

Height Height

Many different shapes, all made of the same amount of clay, have the same volume.

V = | x 7t x r 3 (r = radius of sphere). V = area of basex height.

MEASURING TOOLS

A pipette. TRIANGULAR PYRAMID

CONE

\ volumetric flask.

50 j - 30

cup.

A measuring glass. V = 5 x area of basex height.

HOW TO READ THE MEASUREMENT Z

f

V

L

„

^

"""

• • • • h h J

A measuring cylinder.

The volume of a liquid or of a solid figure tells us about its amount or size. We can sometimes compare directly the lengths or areas of two figures, but the volumes of solid figures cannot be compared directly. To compare the volumes of liquids contained in two bottles of different shapes, we have to put them in containers having the same shape. Units of volume are used to measure volumes. The unit 1 cc (one cubic centimetre) is the volume of a cube whose sides are 1 cm, and is generally used to measure small amounts of liquid. It is also written as 1 cm 5 . The unit 1 m3 (one cubic metre) is the volume of a cube whose sides are 1 m. It is used for larger amounts. The units 1 dl (one decilitre) and 1/ (one litre) are also used to measure volumes of liquid (1 d / = 100 cc and 1 / = 10 d / = 1 000 cc). The volume a container will hold is called the capacity of the container. When we put a stone into a basin filled with water, the amount of water that overflows is the same as the volume of the stone. Using this principle, we can measure the volumes of many irregular objects. 81

WEIGHT f

* The weight of a thing cannot be seen directly (as can lengths or areas). * We need tools, such as a balance, to measure weights. * Units of weight include 1 g (one gramme) and 1 kg (one kilogramme). 'The kilogramme is the standard unit of weight in most countries. 'When people sell things in packets, cans, or jars, the weight of the contents is called the net weight while the weight of the container is called the tare. Both weights added together give the gross weight of the container and its contents.

.'ik

r v

^

COMPARING WEIGHTS The board is balanced, because both sides carry the same weight.

A see-saw is balanced when each side carries the same weight. This principle is used to make a balance.

The right side of the see-saw carries a heavier weight than the left side.

Three children on the left weigh more than a boy on the right. As a result the left end stays down.

BALANCES

There are many kinds of balances, each designed for a different purpose.

A A beam balance. The standard kilogramme is a lump of metal made of an alloy of platinum and iridium.

A course chemical balance.

A spring balance. Platform scales.

Dial scales.

To compare weights we use a device similar to a see-saw. An instrument used to measure weights is called a balance or scale. As a unit of weight we use 1 g, which is the weight of 1 cc of pure water at a temperature of 4 degrees centigrade. One litre of pure water at this temperature weighs 1 kg. The standard kg weighs exactly 1 kg. 82

In addition to the above, there are many sensitive chemical balances, which are used to find the exact weights of light objects.

THE WEIGHT OF A DOG The dial showed 35 kg, which is the sum of the boy's weight and the dog's weight.

* <- * ' tw-« ®

John alone weighed 30 kg.

c

John wanted to weigh his dog, but the dog would not stay on the scales. So he thought of a way of weighing the dog without putting it on the scales. First he stood on the scales himself holding the dog.

• ,u • -

%

¥

tk

Subtracting 30 kg from 35 kg, John found that the dog weighed 5 kg.

TARE AND THE WEIGHT

On a coffee jar or a can of fruit we find its net weight, which tells us the weight of the contents.

(Weight of a c o n t a i n e r + w e i g h t of the contents)

(weight of the container)

(weight of the contents)

tare

net weight

t a r e + n e t weight

HOW MUCH DO THEY W E I G H ?

500 g

The weight of the fish tank w i t h a fish swimming in it is the sum of the weights of the tank of water and the fish.

100

600 :

VOLUMES AND WEIGHTS

cotton-woo/

The density of something is the ratio of its weight to its volume. Density = weight-^volume. Specific gravity is the density compared to that of water. Some examples are given in the table below.

Gold

I 9.3

Lead

M .34

Iron

7.8

Ice

0 9I7

Cypress wood

0.49

Cedar wood

0.40

Gross weight is the weight of something and its container. Net weight is the weight of the contents only. Tare is the weight of the container only. Some people are deceived by the relation between weight and size. Some things are light, even though they take up a lot of room. Cotton-wool and feathers are examples. On the other hand, there are heavy things, such as a lump of lead, which are quite small. The relation between weight and size (or volume) is called density. In metric units, density indicates the weight of 1 cm3 of the substance. So, if the density is high, even a small amount of the substance is heavy. If the density is low, the substance is light even in quite large quantities. Specific gravity is the ratio of the weight of 1 cm 3 of a substance to the weight of an equal volume of water. Because 1 cm3 of water weighs 1 g (that is, water has a density of 1), specific gravity and density have the same numerical value.

UNITS OF WEIGHT 1 t (one tonne) = 1 000 kg 1 kg = 1 000 g (tonne = metric ton) 83

TIME

' T i m e may mean a certain distance between two moments, such as time in which to do things. ' T i m e also means a certain moment, like the time indicated by a clock. * Clocks are used to measure time. * Units of time are the second, minute, and hour. ' T h e time given by a clock differs according to where it is in the world.

A length of time is a quantity that we cannot see. We do not know where it begins and where it ends. Suppose you leave the house for the railway station at 7 o'clock and arrive at the station at 8. You take 1 hour for the trip. In this example you can see the difference between time by the clock and the time taken to do things. One day has 24 hours, 1 hour has 60 minutes, and 1 minute has 60 seconds. Hour, minute, and second are the units of time.

ONE DAY OF A SOMMER HOLIDAY

N a i Sleeping.

.

^SP

Having breakfast.

Playing.

ADDING TIME

1 hour

84

David left the house at 2 o'clock to go to the barber's. It took 15 minutes to get there and 1 hour for the hair-cut. It took another 15 minutes for him to walk home. At what time did he get back? To find the answer, simply add the total time he spent to 2 o'clock: 15 m i n + 1 h + 1 5 min = 1 h 30 min 2 h + 1 h 30 min = 3 h 30 min Thus, David got home at 3.30.

The short hand of the clock tells the hour, and goes round once every 12 hours. The long hand tells the minute, and takes 60 minutes (1 hour) to make a full circle. Each number on the face tells the hour. The long hand takes 5 minutes to go from one number to the next.

HOURS, MINUTES, AND SECONDS

1 minute

1 second

UNITS TABLE 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds 60 minutes = 1 hour

60 seconds = 1 minute

Having lunch.

Swimming.

Watching TV.

Going to bed.

SUBTRACTION OF TIME

20 minutes

15 minutes

After leaving his house, Tony had a 20-minute bus journey and then walked for 15 minutes to the town hall where he arrived at 3.15. To find out when he left the house, subtract the time he spent from the time he arrived at the hall: 3 h 15 min—(20 m i n + 1 5 min) = 2 h + 6 0 m i n + 1 5 min—35 min = 2 h + 7 5 min—35 min = 2 h 40 min Thus, it was 2.40 when Tony left the house.

85

A day is divided into 24 hours, an hour into 60 minutes, and a minute into 60 seconds. A day begins at 12 midnight (2400 hours) and ends at 12 midnight of the following day. It is 12 noon (1200 hours) when the sun reaches its highest position in the sky. Forenoon (morning) refers to the 12 hours before 12 noon, and afternoon to the 12 hours after noon. Morning and afternoon together make a day of 24 hours.

TIMES AT VARIOUS PARTS OF THE WORLD WHEN IT IS NOON ON SUNDAY IN TOKYO

Saturday afternoon

Midnight

Sunday noon

Sunday morning

SECONDS. M I N U T E S , ... A N D YEARS

The hands of a'watch tell each second, minute, and hour. A school timetable lists the times of lessons for each weekday. A calendar tells the date of the month. A year has 365 days, except once in every four years, when we have a leap year of 366 days. A month has either 30 or 31 days, except February which has 28 (29 in a leap year). A week has 7 days: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday.

EXPO 7 0 calendar

1 ?. 3 I r> 6 .7 8 II HI II 12 13 14 15 Hi 1? is hi an zi 22 R,\ ?A ". 2B 27 28 20 30 :n

1 2 3 4 0 7 8 II 10 II f • 13 I I 15 Hi 17 .18 J ' 20 21 22 23 2-t 25 i 27 28 2*1 HI 31

2 3 i s « y •I III II 12 13 II

3 I 5 li 7 f 21 < 1(1 II 12 13 II I Ift 17 18 10 211 21 :•: 23 24 25 20 27 28

Hi 17 IS 1!) 211

25 2li 27 28 •»

2 !l Hi 23

A watch that also shows the date.

XfflE

TABLE

3 III 17 24

I II IK 25

7 14 21 28

1 2 3 4 8 0 Ml II 15 10 17 18 22 23 24 25 20 :«>

30 31

!, 12 HI a;

0 7 13 14 Zlt i 2/ 28

i. 7

I M

9 2 34 'I 10 II

#»

II. 17 I

2(1 21 22 ! u :

28 ;»i :w 10

G. 5 •

-


I 5 5 II 7 11 12 13 14 18 10 20 21 .20 27 28

8 'I 15 lii i 22 23 24 20 30 31

11

I 2

:i 4 , 10 II 12 17 18 111 25 2li

1 2 1 4 5 li 7 (i 7 8 0 ; II 10 II 12 13 14 13 14 15 10 15 III 17 18 III 211 21 211 21 22 23 22 23 24 25 20 27 28 27 28 20 SI

I 7 8 14 15 21 22

3 4 5 0 10 II 12 13 17 18 III 20 24 25 20 27

12 2 II Hi 23

28 211 :«i

12 3 1 5 0 7 8 II til II 12 13 14 15 III 17 18 III 2(1 21 22 23 24 25 26 28 29 30 31

jT

A day has 24 hours, an hour has 60 minutes, and a minute has 60 seconds. A calendar. A school timetable.

86

MOTION AND

" T h e speed of a moving object tells us the distance it goes in a certain time. * A s a unit of speed we generally use km/h (kilometres per hour). *To describe high speeds, we use units such as km/min or km/s (kilometres per minute or kilometres per second). "Among various kinds of motion

SPEED

there are parabolic

motion,

falling,

FASTER AND FASTER

uniform

motion,

and

oscillations.

An ostrich. A boy walking.

A skater.

A snail.

97 km/h

3 m/h 4 km/h

45 km/h

300000 km/s

210 km/h

130 km/h

A dove.

Sound waves. Lightwaves

An express train.

VARIOUS KINDS OF M O T I O N

Uniform motion involves a straight path with uniform speed, such as the motion of falling raindrops. Parabolic motion is the path of a ball thrown in the air.

Speed is a kind of quantity. It is defined by the distance travelled in a unit of time. The speed of a man walking is about 65 metres a minute, which is written 65 m/min. When we say the speed of a car is 50 km per hour, it means a car goes a distance of 50 km in one hour. There are two ways of comparing speeds. One is to compare the times taken to go a certain distance. The other is to compare distances travelled in a certain time. We usually express speed as the distance travelled in one hour but for high speeds we can state the speed per second. For instance, sound waves have a speed of 340 m/s, and light has a speed of 300 000 km/s.

Falling is a motion brought about by gravity.

Oscillations are motions such as the swinging of a pendulum.

87

DIRECT

*When two variables, such as the price of fish and the number of fish, are always in the same ratio, we say that one varies directly as the other. So if the price of fish increases, the number of fish also increases. *We can also say that the two variables are in a direct proportion to each other. * The graph for direct proportionality is a straight line sloping up to the right.

PROPORTIONS

TABLE OF COSTS

Number of tins

I

2

3

4

5

Price (dollars)

1.00

2.00

3.00

4.00

5.00

/-axis

Letting jr = the number of tins and y = the price, we can .plot a graph of the equation y=x. It is a straight line sloping upwards to the right.

Certain canned goods cost one dollar per tin. One tin costs one dollar, two tins cost two dollars, and three tins cost three dollars. The price increases regularly with the number of tins.

SPEED AND DISTANCE Man Bicycle

Car

1

2

3

4

5

4

8

12

16

20

Bicycle

20

40

60

80

100

Car

50

100

150

200

250

Time (hours)

4 km/h 20 km/h

50 km/h

E QJ CJ C CO

Man

Putting x = time and / = distance, the graphs of the equation y = \ix, where v stands for the speed, are straight lines. The faster the speed, the steeper the slope of the graph becomes.

Time (hours)

Suppose that a tin of a certain food costs one dollar. If we put x = the number of tins and / = the price in dollars, then x and y vary in direct proportion. For example, if x= 1, then y= 1, and if x = 2, y= 2, and so on. We have y = 1x. To draw the graph of / = 1*, first make a table of values of x and y, as above. This will show, for example, that when x takes the values 1, 2, 3 ... / t a k e s the values 1, 2, 3, ... We can plot the points (1, 1), (2, 2), (3, 3), and so on, on graph paper and then draw a line through them.

INVERSE

*When the product of two variables is constant, they are said to be in an inverse proportion. When one of the variables increases, the other decreases. ' T h e graph for inverse proportionality is a curve sloping down to the right. "The time needed to go a definite distance is inversely proportional to the speed. * If the product of two numbers is fixed, one number is in inverse proportion to the other.

PROPORTIONS

The relation between speed and the time it takes to go a fixed distance.

I 000

Table Time

Speed (km/h) 1000km

Jet fighter

550km

33min

Passenger plane

550km

Ih

Racing car

300km

1h

50min

Express train

I 75 km

3h

lOmin

Car

1 0 0 km

5h

30min

If you travel between two places in vehicles that have different speeds, the time each takes will differ. The graph of speed against time is a hyperbola sloping down to the right. The speed and time are in inverse proportion to each other.

LENGTH AND WIDTH OF BOXES

Suppose we have 12 apples to be packed in a box. We can make long, narrow boxes or nearly square ones, but all must hold exactly 12 apples. The length and width of a box to hold 12 apples are in inverse proportion to each other.

The distance between two cities is 550 km. If we go by car at a speed of 100 km/h, it takes 5^ hours. But if we take a plane flying at 550 km/h, it takes only 1 hour. The faster we go, the less time it takes. The time needed to travel between two places is inversely proportional to the speed. We can find many other things related in inverse proportion. The number of days needed to finish a job varies inversely with the number of workers. The length of a rectangle varies inversely with the width, if the area is kept constant.

RATIO AND PERCENTAGE

*To compare two quantities we use a ratio, which is the first divided by the second. For example, the ratio of 3 to 6 is 3/6 (which is equal to |); it is also written as 3 :6. *A ratio may be expressed as a fraction having 100 as the denominator, such as ^ . In this case we say that the ratio is 50 percent. *Population density tells us how many people live in a certain area.

BATTING AVERAGE

Each red picture shows the time when a baseball batter made a hit, whereas blue pictures show when he failed to hit. Adding them up we see that he had 10 attempts and made 3 hits. In this case, his batting average is 3 : 1 0 , which is equal to 0.3.

RATIO OF YELLOW FLOWERS

There are 7 yellow flowers among 28 flowers in the vase. The ratio of the number of yellow flowers to the total number of flowers is 7 :28, which is g = 0.25. The ratio of the number of red flowers to the total number is 1 — 0.25 = 0.75.

CONCENTRATION

Ann wants to make a fruit drink by diluting concentrated orange squash. She puts 10 cc of squash in a glass and pours 30 cc of water on top of it. She now has 40 cc of liquid in the glass. The concentration of the drink is 1 : 4 , which is equal to 0.25 = 25 percent.

A baseball player made a hit 27 times out of 86 attempts. In this case, his batting average is calculated as 27-^86 = 0.3139, rounded up to 0.314. The concentration of a fruit drink, the density of seats occupied on a train, and so on can be expressed as percentages. If there are only 30 people on a train which has 120 seats, the passenger density is 30-^-120 = 0.25, which is 25 percent. When we water down fruit squash, the concentration of the diluted drink is the same no matter which part of the drink we choose. After mixing, there is the same average concentration throughout the drink. Most people live in cities, but in country areas people are fewer. We express the distribution of population by saying how many people live in a certain area, such as in a square kilometre (1 km2). The figure is called the population density.

90

PASSENGER

DENSITY

jrmrmt

Out of 150 seats, 90 are occupied.

90 people 150 seats

This is the same density as 60 occupied seats out of 100.

60 people 100 seats

There are 90 passengers in a train whose total seating capacity is 150. In this example, the passenger crowded the train is. To get it, we divide the number of passengers by the total number of seats.

POPULATION DENSITY

density is 90 : 1 5 0 = 60 percent. This figure tells us how

Dividing the total population by the area in square kilometres (km 2 ) gives us the population density of a country. The Netherlands has a population density of 375; Jordan, 22. These figures mean that, on the average, 375 people live in every square kilometre of the Netherlands, whereas in Jordan a similar area has only 22 people.

Population density of an island.

LET'S TRY

A

50 people

Blue county

255.8

Pink county

483.5

Orange county

I I 4.4

Green county

I 96.7

1, A baseball player made 8 hits out of 28 attempts. What is his batting average? 2. Judy opened her money-box and found 68 1-cent coins, 24 10-cent coins, and 12 25-cent coins. What is the percentage of each type of coin to the total number of coins?

91

PROBABILITY

' T h e probability that a particular event w i l l happen is expressed as a fraction. *When we throw a die, the probability that it will come up 1 (or any other number) is J . * The permutation number is used to show how many different ways certain things can be arranged. * The combination number shows the number of ways we can pick subsets of B elements from a set w i t h A elements.

THROWING DICE The probability of a die coming up 1 is j . The probability of its coming up any other particular number is also 1.

The probability of its coming up an even number is 3 x 1 = 2 (there are 3 chances out of 6 possibilities).

The probability of its coming up any odd number is 3x S = 2 •

6 ways There are 6 ways in which the sum of two dice can total 7. These 6 ways represent more variations than for any other possible total.

THROWING TWO DICE

4 ways

2 ways

I 2

II

Sum of the two numbers

TOSSING TWO COINS

TOSSING A COIN There are only two possible ways a single coin can land: heads or tails.

Tails

Heads

Heads

Tails

Tails

Tails

Tails

Heads

Heads

The probability of its coming up heads (or tails) is! •

92

Heads

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

James threw a die. Number 1 came up. But the next time he threw, the number 4 came up. We cannot tell beforehand which number will come up. But if James were to throw his die hundreds of thousands of times, number 1 would come up one time in six on average. Other numbers would also come up in the same ratio, 1 : 6 . In this example, the probability that number 1 (or any other named number) of a die comes up is i . The probability that a coin will land heads (or tails) isi.

PAPER, SCISSORS, AND STONE

The sum of probabilities about one series of events is 1. As far as a die is concerned, each number has a probability of J, so i6 +' i6+' i6 +' i6+' i 6 +' 6i = 1.

John and Ann are playing paper, scissors, and stone. Stone (fist) beats scissors (two fingers), which beats paper (open palm), and paper beats stone. There are 9 possible combinations.

•

.

a There are 3 ways out of the 9 in which John can beat Ann.

As for the coin, the probability of heads or tails is \ , so i2 +' 2i = 1 ' 1

J m Jet.

The probability of John beating Ann is 1=1.

CHOOSING STRAWS

Lottery tickets are sold at this stall. In most lotteries the probability of getting a good prize is very small.

There is one short straw among 10. The person who chooses the short straw wins. The probability that the first person to choose will win is ijj . The probability that the second person will win is the probability that the first person will lose multiplied by the probability that the second person will choose the short straw out of the remaining 9. Therefore, it is £ x | = £ . In a similar way, the probability that the third person will win is £ x l x i = n . In this way, we can see that the probability of winning is the same for each person.

There are many things in our everyday life that seem to be controlled by chance. But on a closer look we can find that such chance is controlled by something — a certain principle. This principle is called probability. Keep in mind the theory of probability and try to find what kind of rules govern chances. 93

When we choose a few things from many and arrange them in various ways, such ways of arranging them are called permutations. For instance, we might have three coloured papers — red, blue, and yellow. As shown on the left, there are 6 ways to arrange these coloured papers in a line. The number of ways we can choose and line up B members from the set of A members is called the p permutation number, written A B.

PERMUTATIONS

w

1

COMBINATIONS

When we choose a few things from several without bothering about their order, such ways of choosing are called combinations. For instance, as shown on the left, there are 6 ways (combinations) that we can choose 2 kinds of things from 4 kinds. But there are 12 permutations, because we can replace left-hand things with right-hand ones to give a new order. The number of ways we can choose, in any order, B members from a set of A members is called the combination number, written (' B''

FUN WITH In t h e

FIGURES

Prussian city Konigsberg, there were 7

b r i d g e s over t h e R i v e r P r e g e l .

For a l o n g t i m e

m a n y p e o p l e t r i e d t o t a k e a w a l k w h i c h involved c r o s s i n g a l l t h e b r i d g e s j u s t o n c e , w i t h o u t crossing any of t h e m t w i c e . Leonhard

The

Swiss

Euler ( 1 7 0 7 - 1 7 8 3 )

mathematician

finally solved

the

p r o b l e m by s h o w i n g t h a t it is s i m p l y i m p o s s i b l e t o do it.

He s t u d i e d t h e n a t u r e of

single-line

d r a w i n g s ( f i g u r e s t h a t can be d r a w n w i t h a single u n b r o k e n l i n e w i t h o u t t a k i n g t h e pen f r o m the paper).

TRY TO DRAW THESE FIGURES WITH A SINGLE LINE

94

By using many kinds of graphs and tables we can systematically study the nature of complicated things and events. Statistics is the science of collecting information and classifying it, using tables, graphs, and so on. Here we shall study statistics by considering examples of the statistics of weather; things we like; height, weight, and chest measurements; and so on.

STATISTICS DISTRIBUTION OF POPULATION ncluding Australia and New Zealand) 0.6%

China 37.3%

AfriCa WORLD 3688 million (1971)

Asia 57.6%

ASIA 2112 million (1971)

Others

18.2%

V' India

26%

iyfi

**

NOMBER OF PERSONS PER SQ. KM.

TOWNS AND CITIES j

Uninhabited

50-200

Below 50

Over 200 ^

1-3000000 Over 3 0 0 0 0 0 0

• A 9!

T A D I

CO

MS

"Arranging things is basic to statistics. "Numerical data are arranged in rows and columns in a table. "Tables of weather statistics or heights and weiqhts are common.

ARRANGEMENT AND TABLES

# 1 9 1 1 1 * * £«*

By classifying things and then arranging them according to some order, perhaps in a table, me can sometimes clarify the situation.

m

•c •

20 1 6 FOODS WE LIKE

6

4

14

In the shop on the left, the fruit is not arranged in any order, but in the shop on the right it is arranged in an orderly manner. When the fruit is well arranged, it is easy to make a table showing the numbers of different kinds of fruit in the shop.

Table of foods we like Tea Father Mother Grandfather Sister

• • • •

Me Brother

Fruit juice

Cake

Fruit

• • • • •

•

•

• •

•

4 of us like tea and 2 like fruit juice 3 of us like cake and 5 like fruit 2 of us like both cake and fruit

peopla 4 people

\ 2 people/ \ 3

people.

,5jpeople/

WEATHER snowy

rainy

cloudy

fine

WEATHER IN FEBRUARY

/

2

3

4

%

9

10

11

*

12

7

13

14

MY FAMILY

SUMMARY fine

days

cloudy

days

rainy

A

days

snowy

^

days

Age (years)

Father

Mother

Sister

Mary studied the weather in February. She also gathered information about her family. She classified the weather as fine, cloudy, rainy, or snowy, and represented these on a chart using easy-to-understand symbols. She then made a table using only numbers. Looking at this table, we can see at once a summary of the weather conditions in February. In the family table, she used numbers and pictures so that it could be understood at once. Try to make your own table like this.

Me ,

Brother

s

Height (cm)

Weight (kg)

Favourite things

GRAPHS

"Numerical information expressed in a table can be made into a graph which uses shapes and lines so that the data may be seen clearly. "There are many kinds of graphs: vertical (or horizontal) bar graphs, pictographs, circular distribution graphs, and so on. * Each kind of graph is best suited to a particular type of data. "When we have a series of positive numbers, such as the amount of monthly savings, we may accumulate them by adding them in succession.

BAR GRAPH

numbers

NUMBERS AND KINDS OF VEHICLES PASSING

The height of each column represents the number of the corresponding vehicles passing in a day. A bar graph is convenient when we want to compare such numbers.

lorries

buses

bicycles

motor-cycles

LINEAR D I S T R I B U T I O N GRAPH

MY FATHER'S DAY

resting 2 hours

sleeping

8 hours

working — 7 hours

others 2 hours

travelling 2 hours

eating — 3 hours

HEIGHTS OF SOME BUILDINGS

448 m 400 -

PICTOGRAPH

333m

350

A pictograph is useful for illustrating information, although it is not easy to make.

300 H 250 200 I 50 I 00 -

50 0

I 50m •

I I I

:::

146m

• > •••••IS • • • • • I I

54m

• • • i l i a

S

• • • I I I * • • • • • I B Empire State Building

tuu HiJj Tokyo Tower

A skyscraper

Egyptian Pyramid

M.

Leaning Tower A house of Pisa

98

I0m

We can see the relationship between things in a table, but we must read the numbers one by one. The information can be understood at a glance by representing it in the form of a graph. There are many kinds of graphs. A bar graph is used for comparing the sizes of quantities, a line graph for expressing increases and decreases, and linear distribution graphs, circular distribution graphs, and square distribution graphs for showing how component quantities relate to a whole.

LINE GRAPH Unit: 10 000 tonnes

A line graph is used to show the changes in variable quantities. The graph (right) shows the tonnage of fish caught in Japan each year. Lines sloping upwards to the right show increases. Lines sloping downwards to the right show decreases.

salmon and trout

herring 0 -i

1

1

1

1

1958

1

i

1

1

Vear

r 1^67

CIRCULAR D I S T R I B U T I O N GRAPH TV PROGRAMMES IIM A DAY

We can organise data by making a table. But if we express it by means of a graph, it can be understood more quickly. A pictograph indicates quantities by the sizes or numbers of small pictures. In a vertical bar graph, the length of a bar is used to indicate quantities (the longer the bar, the greater the quantity). When measuring quantities that change with time, such as temperature or weight, a line graph is used. A line graph can be made from a vertical bar graph by connecting the tops of the columns. The line rises when quantities increase, and falls when they decrease, so that we can see the changes at a glance. Circular, linear, or square distribution graphs are used to show the proportions between quantities in relation'to a whole.

ACCUMULATIVE LINE GRAPH This graph shows the sums (in dollars) of savings accumulated each month. The accumulative line graph keeps sloping upwards.

The whole circle represents the total hours used by a TV channel in a day.

100.00 r

ACCUMULATION OF MONTHLY SAVINGS

Table of monthly savings month

savings ( $ )

t o t a l (S)

month

I

I O

I O

5

I O

4 0

2

5

I

6

5

4 5

3

5

2 0

7

2 0

6 5

3 0

8

5

7 0

4

I

0

5

savings ($)

. t o t a l (S)

1

2

3

4

5

6 month

7

8

9

10

11

12

CLASSIFICATION AND ORDERING

u

* The elements of a set, such as the set of books in a library, can be classified into subsets sharing some property, and then ordered within their subsets. "The seat numbers in a theatre, houses along a street, and room numbers in a large hotel are all examples of classification and ordering. " A set may be classified and ordered in various ways.

SEAT NUMBERS IN A THEATRE

Each seat has its number. For example, C-4 means that the seat is the fourth seat in row C, three rows from the left.

In a theatre, the seats are classified and ordered.

100

JAPANESE HOUSE ADDRESSES

CLASSIFYING BOOKS

In Japan, the address of a house is given in terms of a block number, lot number, and house number. Each district of a city is divided into several large numbered blocks. Each block is subdivided into numbered lots, and finally each lot contains several numbered groups of houses.

David and Jane share the use of a bookcase. In a situation such as this, classification and ordering can prevent mistakes. For instance, Jane's atlas might be classified J-3-5, to show that it is hers, that it is located on the third shelf down, and that it is the fifth book from the left.

Block 2 Lot 3 Group i

CLASSIFICATION OF TRAIN TICKETS

•MIWlTMSOr

A computer-controlled ticket office.

When passengers buy tickets for the New Tokaido Line in Japan, the tickets are printed and issued by means of computers. The ticket on the left is number 19 for ShinOsaka on June 19, leaving Tokyo at 9.00 a.m. The coach is the seventh along the train. The seat is number 6-E. Only the person holding this ticket can occupy this seat.

A ticket from Hikari (New Tokaido Line, Japan

On the licence plate of a car, letters and numbers are used to classify and identify the car.

We learned about tables before. This time, let's learn about classification using numbers. Many things are classified in terms of numbers. When you buy a ticket for a reserved seat on a train, there is a number printed on it. For instance, 2-10-A could mean the window seat on the tenth row back in the second coach. Seats in theatres, too, can readily be found so many rows back and so many seats along the row. Japanese houses in the cities have block, lot, and house numbers to identify them. In large apartment buildings, individual flats can readily be indicated by block, floor, and flat numbers. CLASSIFICATION OF APARTMENTS IN A BLOCK OF FLATS

Ann lives in flat 3-203.

The next digit 2 means that she lives on the second floor of the building.

01 02

03

04

05

06

The last two digits 03 mean that her flat is the third one along that floor.

101

Books in this library in an elementary school are classified so that it is easy to find them.

The Dewey decimal system is used to classify and order books in a library.

NO.

CLASSIFICATION

000 100 200 300 400 500 600 700 800 900

general philosophy religion social sciences linguistics natural science applied science arts and recreation literature history and geography

THE DEWEY DECIMAL SYSTEM Sometimes, the Dewey decimal system is used to classify and order the books in a library. The books are classified into 10 classes of the first category. Each class of the first category is again classified into 10 classes of the second category. This process is

repeated several times. Each book carries a number such as 237, which means that the book belongs to the class 2 of the first category, class 3 of the second category, and class 7 of the third category. The table (left) shows classification of the first category.

1 15 1 4 4

12

6

7

9

8 IO 11 5 13 3 2 16 A magic square.

FUN WITH NUMBERS

8

1 6

3

5

7

4

9

2

According to an old Chinese legend, many centuries ago a turtle crawled out of the Yellow River. On its back were strange designs formed by dots. By copying down the number of dots, the people formed a magic square, which you can see in the picture. No matter what direction — horizontal, vertical, or diagonal—you add up the numbers, you always get 15. After that first discovery, people found many ways to make magic squares. There is another example of a magic square above.

FAMOUS PEOPLE IN MATHEMATICS

THALES 624 (?) B.C.-546 B.C. Grcece

THALES Thales was a philosopher. In his day, a philosopher studied mathematics, astronomy, physics, and other sciences. He was born in Greece but went to Egypt to study. 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. One night, staring at the stars while walking,

he fell into a gutter. An old woman servant who saw this said to him, 'My lord, if you can't even see where you are walking, how can you tell anything about the stars?' There are many such stories about him.

the other two sides. The 3 : 4 : 5 right-angled triangle used by the rope-stretchers in Egypt (people who surveyed land area by using knotted ropes) illustrates his theorem ( 3 2 + 4 ! = 5 ! ) . It is said that he discovered this theorem as he was looking at the tiled floor of his friend's house. One day he was passing a blacksmith's shop

and got an idea from the different types of sounds produced by the hammers. He discovered that the shorter the handle of the hammer, the higher the pitch of the note. Using this idea he invented new types of harp and flute. Apart from these achievements, he also put forward many theories about the universe.

PYTHAGORAS Pythagoras was a philosopher. He studied not only mathematics but also music and other subjects. He was born in Greece but went to Egypt and Babylonia to study. He is famous because of the theorem named after him, which states that in a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on

PYTHAGORAS 582 (?) B.C.-493 B.C. Greece

103

EUCLID about 300 B.C., Greece

EUCLID Euclid wrote 13 volumes of geometry books. In them, he started with simple statements (called axioms) such as 'there is one and only one straight line passing through two points' and constructed all his geometry theorems from those axioms.

His books became the most important work in the study of geometry, and have been us-ed throughout the world. For Euclid, mathematics was important as a subject for study and not merely a way to

earn his living. At one time he was lecturing on geometry to a king who asked him: 'Isn't there an easier way for me to understand geometry?' Euclid replied, 'There is no royal way to geometry.' Everyone has to think for himself when studying.

a point to lever it against, then I can even move the earth.' Using a knowledge of density, he found that a crown made for the king was not made of pure gold. He also studied circles and discovered formulae for the circumference and area of a circle.

When Archimedes was old, his country was defeated in a war against the Romans. He was studying a circle he had drawn on the floor when an enemy soldier charged into his room. He shouted, 'Don't step upon my circles!' The soldier stabbed him to death.

ARCHIMEDES Archimedes studied mathematics and physics, and made many inventions. He discovered the principle of the lever, with which he could move heavy weights with only a little effort. He demonstrated this principle by moving an entire ship using a lever. He said, 'If you give me a long enough lever and

ARCHIMEDES 287 (?) B . C . - 2 1 2 B.C., Greece

104

LEONARDO DA VINCI 1452-1519, Italy

LEONARDO DA VINCI From his childhood Leonardo da Vinci showed a special ability in mathematics, music, painting, and other subjects. He especially loved to paint, and took art lessons. He produced masterpieces as a painter and sculptor. He was a noted architect

and left many great works in that field. He also studied geometry, and used a method of making the main parts of a painting fall on an imaginary triangle, a method called pyramidal composition. He used perspective to paint

solid figures on a plane canvas. In all parallel horizontal lines seem to one fixed point. He used this method famous Last Supper. An example of is used in the picture above.

this method be going to to paint the perspective

convinced that the sun is the centre of the universe, and the earth revolves round the sun. This idea was against traditional philosophy and religion. His famous theory was put forward in his book entitled The Revolution of Celestial Bodies. He

was afraid that the publication of the theory would cause him to be persecuted, especially by the church. It was only after his friends' insistence that he agreed to publish the full theory. The book was printed only when its author was dying.

COPERNICUS Copernicus studied astronomy, mathematics, physics, law, and medicine. In his day, it was generally believed that the sun, moon, and stars moved round the earth, which was thought to be the centre of the universe. But Copernicus was

COPERNICUS 1473-1543, Poland

105

GALILEO 1564-1642, Italy

« »

GALILEO Galileo studied mathematics, physics, and astronomy. Before his time, people believed that the speed of a falling body depends on its weight. They thought that a heavy object falls faster than a lighter one. But Galileo believed that the speed of a falling object does not depend on its weight. He is said to have proved this by dropping two metal

balls, one heavier than the other, from the top of the Leaning Tower of Pisa. Although everybody now knows that he was right, his idea and its proof came as a great surprise to the people of his day. At another time, as he was watching a chandelier swinging in a church, he noticed that no matter

how far it swung sideways, the time taken for one oscillation was always the same. He later found that this is a general law, which he called the isochronism of the pendulum. Later in life he was persecuted by the church because he supported Copernicus' idea that the earth revolves round the sun.

translated into statements involving numbers. It is said that he got his idea when he was lying ill in bed. He watched a spider walking on the ceiling and then descending by a spun thread. This led him to the idea of expressing the points

in a space as (A, B, C). He was the first person to have used letters of thfe alphabet such as a, b, c, x, y, z to represent numbers. He also put forward the idea of negative numbers.

DESCARTES It was Rene Descartes who first used the system of two or three numbers, such as (A, B) or (A, B, C), as co-ordinates to represent the points on a phane or in a space. By this means, statements about figures in Euclidean geometry could be

RENE DESCARTES 1596-1650, France

106

BLAISE PASCAL 1623-1662, France

PASCAL Blaise Pascal was a mathematician, physicist, theologian, and man-of-letters. Pascal became interested in mathematics, especially in geometry, when he was 6 or 7 years old. His father took away his mathematics books because he believed that a small child should not study such a difficult

subject. But Pascal continued to study in secret. When he was 12, he discovered for himself that the sum of the interior angles of a triangle is always 180 degrees. He showed this to his father and explained it clearly. His father was so impressed that he allowed him to study mathematics freely. When

he was 19, Pascal invented a calculating machine which used gear wheels. In physics, he discovered a principle about pressure in a liquid, which was later named after him. He left the famous saying: Man is a feeble reed, but he is a thinking reed.

Japanese used Chinese numerals, which were more complicated than the Arabic numerals they use now. They also used special wooden tools (called Sangi) which were originally developed in ancient China. Their mathematics was called Wasan

(Japanese mathematics). Until Western mathematics was introduced into Japan towards the end of the nineteenth century, Wasan used to be popular. Seki was one of the best-known Wasan scholars.

SEKI Seki lived in the same period as Newton and Leibniz, and like them he invented a method of measuring the areas of figures bounded by curves or the volumes of irregular solid figures (the method is now called integration). In his day, the

SEKI TAKAKAZU 1642-1708, Japan

107

ISAAC NEWTOM 1642-1727, England

NEWTON One of the greatest mathematicians, Isaac Newton also studied physics. There is a famous, but apparently fictitious, story about how he discovered the laws of gravity by observing an apple fall from a tree. Gravity is the force by which every

object attracts every other object. The farther apart two things are, the weaker is the force of gravity between them. The motion of the moon round the earth can be explained by the laws of gravity.

Newton also discovered the laws of motion, which are the basis of dynamics. He was also interested in astronomy. He invented a type of reflecting telescope which came to be named after him.

traditional ways of thinking. Both he and Newton formulated the basic ideas of differential calculus. Each claimed he thought of it first. To try to decide who was the originator, they set each other problems in calculus. This was known as the

Mathematical War between Leibniz and Newton. Eventually, they were convinced that neither of them made use of the other's ideas. Leibniz also invented a type of calculating machine.

LEIBNIZ The father of Gottfried Wilhelm Leibniz was a university professor, but he died when Leibniz was only 6 years old. From that time, the young Leibniz was taught by his mother and by himself. Self-education left him free from many of the

GOTTFRIED LEIBNIZ 1646-1716, Germany

108

INO TADATAKA 1745-1818. Japan

INO Ino was the son of a farmer, who belonged to one of the low castes. He did not receive a formal education, but studied on his own. When he was 18, he was adopted by a merchant named Ino, and had to stop studying in order to work.

At the age of 49, he let his own son take over the household duties so that he could take up his studies again, under a tutor, to learn about astronomy, mathematics, history, and surveying. When he was 55, he received government permission

to survey the northern part of Japan. He continued getting information to make maps of the whole country until he died. His maps were used as the basis of official Japanese maps until the end of the nineteenth century.

got the answer in his head by realising the sum could be considered as ( 1 + 4 0 ) + ( 2 + 3 9 ) + . . . + ( 2 0 + 2 1 ) = 4 1 + 4 1 . . . + 4 1 = 4 1 x 2 0 = 820. His father was a stone-mason and could not afford to give him a university education. However, the

king, who was impressed by young Gauss's ability, paid for his education. Gauss later became one of the world's greatest mathematicians. He also left works on astronomy and surveying, and on the study of electromagnetism.

GAUSS Johann Gauss was said to be a genius in arithmetic. When he was 9, a teacher asked his class to add up the series of numbers 1 + 2 + 3 + . . . + 4 0 . Gauss took only a few moments to get the answer (820) without even writing anything down. He

JOHANN GAUSS 1777-1855. Germany

109

TABLE OF UNITS LENGTH 1 km = 1 m

VOLUME

1 000 m

=

1 m3 =

100cm

1 cm =

11

10 mm

AREA

WEIGHT 1 t

1 000!

=

10d/

1 61 =

1 0 0 cc

1 cc =

1 0 0 ha

TIME

1 ha

=

100a

1 day = 24 hours

1 a

=

100m!

1 m!

=

1 000kg 1 000 g

ANGLES 1 ° =

60'

1 ' =

60"

CIRCULAR

1 hour = 60 minutes

10 0 0 0 c m !

=

1 kg =

1 cm 3

1 km 2 =

(tonne = metric ton)

1 minute = 60 seconds

71=

CONSTANT 3.14

F O R M U L A E FOR AREAS A N D V O L U M E S AREAS OF PLANE F I G U R E S square:

(side)

rectangle:

prism:

2

lengthxbreadth

parallelogram:

2 x base a r e a + a r e a of sides

cylinder:

2 x b a s e a r e a + 2 x n x radiusx height

pyramid:

base a r e a + a r e a of sides

cone: base a r e a + slant heightx radiusx 7t

lengthx height

rhombus: b a s e x height = 1 product of two diagonals

sphere: 4 x (radius)

2

xn

V O L U M E S OF S O L I D FIGURES

trapezium: 1 sum of two lengthsx height triangle: 1 basex height

cube:

. , circu ar segment:

rectangular solid (cuboid):

a

, . . , central angle area of the circlex 360°

polygon: divide it into triangles and add up their areas

prism cylinder

2

circle: (radius) x 7t

SURFACE AREAS OF S O L I D F I G U R E S cube:

6x(side)

pyramid cone

2

rectangular solid (cuboid):

2 x sum of the areas of three types of rectangular faces

(side)'

lengthx breadthx height

, , . , : base a r e a x height

, , . , : 5 x b a s e a r e a x height

sphere: 5 x (radius)' x 7t

A N S W E R S TO " L E T ' S T R Y "

•

Page 23 1. 8 6 8 : 4 0 2 : 1 066 2. 9 8 + 2 7 = 125 oranges

•

Page 25 1. 53: 1 1 7 : 2 6 8 : 9 9 2. 4 3 8 - 7 5 = 363 men 3. 3 4 5 - 7 0 = 275 marbles

Page 33 1. 12: 43: 12: 19: 21 with remainder 3: 14 with remainder 29 2. 7 h - 3 = 2 boxes with remainder 1 box: 6 - f - 3 = 2 chocolate bars, so each one gets 2 boxes and 2 chocolate bars

Page 43 1. If the price of a notebook is x, we have 1 0 0 - h 2 = 50 cents, therefore, x < 5 0 cents. 2. If the number of days is x, then 300— (jrx 40) = 20, / = 7 days.

Page 38.

Page 91 1. 8 + 2 8 = 0.2857, so the batting average is about 0.286. 2. The total number of coins is 6 8 + 2 4 + 1 2 = 104. The proportion of 1 -cent coins is 6 8 + 1 0 4 = 0.653 which is about 65 percent. Similarly we find that the proportion of 10-cent coins is about 23 percent, and the proportion of 25-cent coins is about 12 percent.

1. ?

•

110

Page 29 1. 3 9 : 1 5 2 : 3 3 6 : 9 0 : 1 9 8 4 : 5 6 2 8 2. 2 9 x 1 6 = 464 dollars 3. ( 3 7 x 3 ) x 12 = 1 332 cents = $13.32

.,3

15.

=

4 ' "7

_i = 1 - J L ' 16 4'15 12 12

=

19.18 _ 51. H 7 ' 3 3' 3 . 18 _ 30 .. 1 8i o _ 5'24 4 ' 36"

12 12

=

23

INDEX

Abacus, 44 Chinese, 44 Japanese, 44 Addition, 11,22, 23, 26, 38, 41 associative law of, 27 commutative law of, 27 of decimals, 41 of fractions, 38 rules of, 27 Addresses, 100 Angle, 52, 53, 56, 57, 58 exterior, 52 interior, 52 right, 52 straight, 52 Answers to "Let's T r y " , 110

Archimedes, 73,104 Area, 78 unit of, 78 Areas, 78,79,110 comparing, 78 formulae for, 79,110 surface, 79 Astronomy, 105,106,109 Balance, 82 spring, 82 Batting average, 90 Binary system, 18,19, 20 Calculating machine, 44, 45,107 electric, 45 mechanical, 45 Calculations with brackets, 26 Calculator, 44 electric, 45 Calendar, 65, 86 Capacity, 81 Card games, 22, 23, 30 Catenary, 62 Centre, 60 Circles, 50, 51,60, 62,63 Circumference, 60 Classification, 100 Clock, 21,84, 85 Combinations, 94 Compasses, 60 Computer, 21, 44, 45 Cone, 50, 51, 64, 65, 69 circular, 64, 65, 69 Congruence, 47, 70 Co-ordinates, 66,106 Copernicus, 105 Copies, 71 enlarged, 71 reduced, 71 Counting, 12-21 Cube, 64, 65 Curves, 62, 63 Curvimeter, 75 Cycloid, 62, 63

Cylinder, 50, 51, 64, 69 circular, 65, 69 Decimal point, 40 Decimals, 11,18,19, 4 0 - 4 2 mixed, 40 rounding off, 40, 42 Decimals and fractions, 41 Degree, 52 Denominator, 37 lowest common, 38 Density, 83, 90, 91 passenger, 91 population, 90, 91 Depth, 76 Descartes, Rene, 106 Diagram, 68, 69, 79 three-dimensional, 68, 69 Diagonals, 58 Diameter, 60 Dice throwing, 92 Difference, 24 Differential calculus, 108 Direct proportion, 88 Directions, 52, 53 Distance, 76, 77 Distortion, 71 Dividend, 32 Division, 11, 32-34, 39, 42 of decimals, 42 of fractions, 39 Division and multiplication, 32 Division with a remainder, 33 Divisor, 32, 34 common, 38 Dodecahedron, 65 regular, 65 Dozen, 21 Duodecimal system, 19, 20, 21 Dynamics, 108 Egyptians, 13 Ellipse, 61,62 Equal to, 42 Equality, 43 Equations, 43 Euclid, 104 Euler, Leonhard, 94 Factors, 36 common, 36 highest common, 36 Famous mathematicians, 103-109 Fractions, 11, 37, 38, 41 changing a fraction to higher terms, 38 improper, 37, 38 proper, 37 Galilei, Galileo, 106 Gauss, Johann, 109 Geometry, 106

Golden mean, 48 Graph, 78, 95, 98, 99 accumulative line, 99 bar, 98, 99 circular distribution, 98, 99 line, 99 linear distribution, 98, 99 square distribution, 99 Gravity, 87,108 laws of, 108 Greater than, 42 Greeks, 13, 103-104 Half-line, 52 Hexagon, 61 Hexahedron (cube), 65 Hour, 84-85 Hyperbola, 89 Hypotenuse, 54 Icosahedron, 65 regular, 65 Inequality, 43 Ino Tadataka, 109 Integer, 36, 37, 40 positive, 36 Integration, 107 Inverse proportion, 89 Involute, 62, 63 Kilogramme, 82 Kilometre, 87 Leibniz, Gottfried Wilhelm, 108 Length, 74, 75 standard unit of, 75 Leonardo da Vinci, 105 Less than, 42 Line of numbers, 46 Line segment, 52 Lines, 52, 54-55, 63 horizontal, 63 parallel, 5 4 - 5 5 skewed, 55 straight, 52 twisted, 55 vertical, 63 Logarithms, 45 Magic square, 102 Maps, 71,77 Matching, 12 Measuring, 53, 74, 77, 78, 80 angles, 53 distances on a map, 77 volumes, 80 Meridian, 75 Metre, 75 Metric system, 75,110 Micrometer, 75 Minute (angle), 52 Minute (time), 84-85

Motion. 87,108 laws of, 108 parabolic, 87 uniform, 87 Motion and speed, 87 Multiples, 31 common, 31 lowest common, 31 Multiplication, 11, 28-30, 34, 35, 39,41 of decimals, 41 of fractions, 39 rules of, 35 Multiplication table, 30 Newton, Isaac, 108 Numbers, 11-21, 36, 37, 38,92 combination, 92 even, 14 infinite, 17 mixed, 37, 38 natural, 36 negative, 46 odd, 14 permutation, 94 positive, 46 prime, 36 square of, 30 Number game, 23, 24,30 Numerals, 1 1 - 1 4 ancient Egyptian, 13 ancient Greek, 13 Arabic, 1 4 , 1 5 , 1 9 Babylonian, 13 Chinese, 13,14,15 history of, 12 Roman, 13,14,15 Numerator, 37 Octahedron, 65, 69 regular, 65, 69 One-to-one correspondence, 12, 13 Orbit, 62 Oscillation, 87 Oval, 62 Parabola, 61, 62 Parabolic motion, 87 Parallelogram, 58, 59 Pascal, Blaise, 107 Patterns, 61 Percentage, 90 Perimeter, 77 Permutations, 94 Perpendicular lines and planes, 54-55 Perspective, 105 Physics, 105,106 Pi (7r), 77 Pictograph, 98 Plane figure, 64, 78 closed, 78 111

Plumb-line, 55 Point of origin, 17 Points on a line, 66 Polyhedron, 65 regular, 65 Position of a point, 66, 67 Prism, 50, 51, 64 hexagonal, 64 quadrangular, 64 Probability 92, 93 Product of, 35 Projections, 64 Protractor, 53 Pyramid, 51,64, 70, 81 hexagonal, 64 octagonal, 64 quadrangular, 64 triangular, 81 Pythagoras, 103 Pythagoras' theorem, 54, 79, 103 Quadrilateral, 50, 58, 59 Quantity, 43, 73 unknown, 43 Quinary system, 18 Quipu, 13 Quotient, 32 Radius, 60 Ratio, 90

112

Rectangle, 58, 59 Remainder, 3 2 - 3 4 Revolution, 64 Rhombus, 58, 59 Ruler, 75 Scale (map), 77 Scales (weighing), 82 Second (angle), 52 Second (time), 84-85 See-saw, 82 Seki Takakazu, 107 Sets, 7 - 1 0 , 31 complement of, 9 element of, 9 empty, 10 intersection of, 10, 31 relation between, 7 union of, 10 subset, 8 - 9 Set-squares, 52, 53 Sexagesimal system, 18, 19 Shapes, 47, 58, 62, 70 arrowhead, 58 basic, 47 congruent, 70 egg, 62 kite, 58 rectilinear, 52 similar, 70 symmetric, 72

Sides, 56, 57, 58 Similar shape, 70 Similarity, 47, 70 Single-line drawing, 94 Size, 81, 83 Slide-rule, 44, 45 Sliding callipers, 75 Solid figure, 64, 68-69, 78 Solid of revolution, 64 Speed, 87, 88 Sphere, 50, 51, 64, 65 Square, 58, 59 Standard metre, 75 Statistics, 95-96 Subtraction, 11, 24, 26, 38, 41, 85 of decimals, 41 of fractions, 38 of time, 85 Surveying, 71, 78,109 land, 78 Sum, 22 Symmetric, 72 shape, 72 Symmetry, 47, 72 Tare, 82, 83 Tetrahedron, 65, 69 regular, 65, 69 Thales, 70, 103

Time. 21, 84-85 moment of, 84 Transit, 53 Trapezium, 58, 59 Trapezoid, 58,59 Triangle, 50, 54, 56, 57 equilateral, 56 isosceles, 56, 57 isosceles right-angled, 56 right-angled, 54 scalene, 56, 57 Units, 78,85,110 table of, 110 of area, 78 of weight, 83 Vertex, 56, 57, 58, 59 Volume, 80, 81, 110 comparing, 80 formulae for, 81,110 Weight, 82, 83 gross, 82, 83 net, 83 standard unit of, 82 Weights, 82 comparing, 82

Zero, 12,13, 18

TREE OF GEOMETRIC FiGURES AND GRAPHS hyperbola

involute

parabola

octagon

dodecagon

arrowhead shape

pentagon

square

square

parallelogram

rhombus rectangle

trapezium

isosceles trapezium.

trapezoid

isosceles right-angled ^

triangle

>

isosceles triangle scalene triangle .equilateral triangle

ircular d i s t r i b u t i o j

pictograph

V^qiaph

,

[near d i s t r i b u t i o i p o s i t i o n s of p o i n t s V ^ i n a space J*

dodecahedron j o s i t i o n s of p o i n t ; "'•--.on a p l a n e / p o s i t i o n s of p o i n t s t

on a line

j cylinder

tetrahedron

f i e x a g o n a l prism

octahedron

cone

pentagonal

twPrism

^

cuboid hexagonal

pyramid. triangular prism

pentagonal

T h a l e s and a p y r a m i d

.jiyramid^ quadrangular

^.pyramid^

.triangular pyramid Pythagoras' theorem

Euclid's 'Elements'

Euclid's 'Elements', w r i t t e n 2 3 0 0 y e a r s a g o , is one of t h e m o s t imp o r t a n t w o r k s in g e o m e t r y .

Egyptian rope-stretchers