Mathematics: The Children's Treasury of Knowledge
117
THE CHILDREN'S TREASURY OF KNOWLEDGE Mathematics
Transcript of Mathematics: The Children's Treasury of Knowledge
THE CHILDREN'S TREASURY OF KNOWLEDGE
Mathematics
TREE OF NUMBERS
April weather
l icence plate
c lassi f icat ion and ordering
; ^ ^ e l e c t r o n i c computer
pocket ca lcu la to r
the binary system
abacus
discovery of zero
Roman numerals
Chinese numerals Mayan monuments
Arabic numerals
the Rhind papyrus
the Rosetta stone
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 ROSETTA STONE
THE RHIND PAPYRUS
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.
The Rhind papyrus, wri t ten 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.
MAYAN MONUMENTS
1 2 3 4 5
22-56
THE CHILDREN'S TREASURY OF KNOWLEDGE
Mathematics
Translated from Kodansha's Children's Colour Encyclopaedia
Adapted and edited by the edi tors of FEP Internat ional Ltd.
Distributed by Time-Life Books
Printed in Singapore under the supervision of T ime-L i fe Librar ies (Asia) Pte., Ltd.
Text by Yoshikazu Horiba Teacher of Saginomiya High School
Layout by Mitsumasa Anno
Book design by AD 5
ACKNOWLEDGEMENTS
Photographs, il lustrations, 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 Insti tute; 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 All iance; Pan-Asia Newspaper Al l iance; 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 SETS 7
Making sets; Relationship between sets
NUMBERS 11
The history of numerals; Numerals of today; What numbers stand for ; How to wri te big numbers; Addit ion; Subtraction; Addition and subtraction; Rules of addit ion; Mult ip l icat ion; Mult ipl ication table; Mult iples and common mult iples; Division; Mult ipl ication and division; Rules of mult ipl icat ion; 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 = m centimetre = cm kilometre = km mill imetre = mm
MASS kilogramme = kg gramme = g tonne = t
TIME second = s minute = min hour = h
AREA square metre square centimetre square mill imetre hectare
VOLUME cubic metre cubic centimetre = litre mil l i l i t re =
DENSITY kilogramme per cubic metre = kg/m !
gramme per cubic centimetre = g/cm3
VELOCITY AND SPEED metre per second = m/s kilometre per hour = km/h
POWER watt = W ki lowatt = kW horse power = h.p.
TEMPERATURE Temperature (common) = degree Celsius = °C Absolute temperature = K
m2 PRESSURE (FOR METEOROLOGY) cm2 millibar = mb mm2 bar = b ha
m cm3
I ml
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 wi l l 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 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 wi ldl i fe park, for example, we can say that the giraffes are a subset contained in the set of animals. All the animals kept in the wildl i fe park together form a set.
A set of animals in a w i l d l i f e park.
Animals in a wi ldl i fe 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 f ly 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 f lying 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, ...) c i the set of natural numbers (1, 2, 3, 4, 5, ...). A subset together wi th 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
"The 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 w i th 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 inter-section 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 wrest l ing, 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 wrest l ing and the set of girls skipping shown on this page is an empty set.
The intersection of two 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 f ly ing animals and the set of non-flying animals is the set of all animals, but the intersection of these two sets has nothing in i t . A set that has nothing in it is called an empty set.
10
NUMBERS
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.
I N O R D E R O F O P E N I N G C E R E M O N Y E N T R V 1 H S R 2 R D H 3 flSSU H R S U S
2 SHI? 6BKm 7 B " S F 8 C U S
4 | C U S I 1 0 C B D U 1 1 C D U P 1 2 C N E 15 £S£DU r s u B « G U U C N 16 H K F S
1 7 I N D U 1 8 I U S B I 1 9 I U S B P 2 0 K U S B 2 1 N U S V 2 2 N S S S 2 3 N 2 U S U 2 H O H S 2 5 O I S S U 2 6 S E U 2 7 S f t U L 2 8 S f l I F 2 9 T M T S T 3 0 U f l f l P 3 1 U S N S f l 3 2 U S O F K J 3 3 U S B T 3 H 3 U S B
. . . imm®
The opening ceremony of the Wor ld Student Games.
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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 i t? 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 wi th a mark on the tree. In this way, they could te l l if there was any change in the number of sheep.
Familiar things such as f ingers were used to indicate numbers. The members of a community might agree to use the picture of a l ion'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 f inger stood for one, the third f inger for two, and so on. For eight, they pointed to the elbow. A similar method is s t i l l used by a native people in New Guinea.
USING PARTS OF THE BODY TO TAKE THE PLACE OF NUMBERS USING FAMILIAR THINGS TO TAKE THE PLACE OF NUMBERS
1
2
3
4
12
DISCOVERY OF ZERO The idea of zero was original ly invented in India. It was introduced to Europe and took the form
* we know today. H
Roman numerals.
Chinese numerals.
Ancient Egyptian numerals.
USING KNOTS IN STRINGS TO RECORD NUMBERS
w W
Some primit ive 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.
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
Ancient Japanese numerals (short wooden st icks were used).
NUMERALS OF TODAY
"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 writ ten 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, ...
ODD AND EVEN NATURAL NUMBERS
even
even
14
even
seven
even
ARABIC NUMERALS
CHINESE NUMERALS
W H A T I M I I M R F R Q 'Numbers are used to count and compare, to represent the positions of V V I I r t l l i l U I V I I J I . n o j . 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
NUMBERS MAY BE USED TO COUNT AND COMPARE
3>1
t * j J * . 3 = 3
3<4
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 STAND FOR THE
ORDER OF THINGS IN A LINE
Numbers may also stand for the order of things in a line. The orange coach is third from the left, second from the right.
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The number of apples on a plate may be represented by marking of f distances on a l ine w i t h the numbers 0, 1, 2, 3
POINTS ON A NUMBER LINE
Numbers may stand for volume. 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.
m jfc —M̂Jjm.— At.. 'i
O + * * * * * v * * * 4
Numbers may stand for length or distance.
4 * 4
/ ^
Numbers may stand for t ime.
Numbers may stand for we igh t .
ONE NUMBER COMES AFTER ANOTHER Lights of a l ighthouse.
Skipping.
A metronome. A clock.
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, ... wi th 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 f irst 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.
In a rai lway train, for instance, the number 1 can be matched with the f irst coach, the number 2 with the second coach, and so on. A train has only a l imited number of coaches, but when we match numbers wi th 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 t ime, time of day, and so on. They are used on a metronome for marking exact t ime 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 ^ BIG NUMBERS
- v
* We usually use the decimal system for writ ing 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.
100 10
The numbe r t e n is w r i t t e n 10. Ten 10s make one hundred, wh ich is w r i t t en 100.
Ten 100s make one thousand, which is w r i t t en 1 000.
Zero or 0 f ish means no f ish. But 10 f ish means ten f ish, 100 f ish means a hundred f ish. Using 0 we can wr i te big numbers easi ly.
1fi
ZERO AND BIG NUMBERS
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 wri t ten 100. The number 37 has three tens and seven ones; it is thirty-seven. The number 245 has two hundreds, four tens, and five ones; it is two hundred and forty-five. Ten hundreds is wri t ten 1 000; it is
one thousand. The method of wri t ing 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 gas meter makes use of the decimal system. The cash register makes use of the decimal system.
HOW TO WRITE TWO HUNDRED AMD FORTY-FIVE
In wri t ing 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 wr i t ten 245.
o • o • o o Numbers keep increasing without end.
The commonly used Arabic numerals fol low the decimal system, which is convenient for writ ing 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 writ ing it.
There are many systems other than the decimal system that can be used for writ ing 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 fol low the quinary system. Egyptian numerals and Babylonian numerals fo l low 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 fol lowing numbers:
463 893 4652
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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 pr inted in black, w i t h the i r 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
2 -i I, •
The posit ions 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.
I
5 = 4 + 0 + I 13
1 3 = 8 + 4 + 0 + 1
2 = 2 + 0 6 = 4 + 2 + 0 20
20= I 6 + 0 + 4 + 0 + 0
3 = 2 + I 7 = 4 + 2 + I 29 29= 16+8 + 4 + 0 + I
4 = 4 + 0 + 0 8= 8 + 0 + 0 + 0 31
31= 1 6 + 8 + 4 + 2 + 1
1 1 9 1001 17 10001 25 1 1 0 0 T
2 10 10 1010 18 1 0 0 1 0 26 11010
3 11 11 1011 1 9 1 0 0 1 1 27 11011
4 1 0 0 12 1100 20 1 0 1 0 0 28 1 1100
5 101 13 1 101 21 10101 29 1 1101
6 1 10 14 1 1 10 22 10110 30 1 1 1 1 0
7 1 1 1 15 1 1 1 1 23 10111 31 1 1 1 1 1
8 1000 16 10000 24 11000 32 1 0 0 0 0 0
Counting w i th the f ingers is an example of the quinary system.
The abacus makes use of the quinary system.
The numbers 1 to 32 wr i t t en in the binary system.
These penci ls are sold in a box of 12, an example of the duodecimal system.
20
The (flock makes use of the sexagesimal system. One hour is 60 minutes, one minute is 60 seconds.
The sexagesimal system, w i t h degrees and minutes, is used for f ix ing posit ions on the earth.
The protractor is an example of the sexagesimal system. One degree is 60 minutes, one minute is 60 seconds.
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 writ ten in the decimal system.
Each system of writ ing 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 genera l 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 th is apparent ly humble wish. Well, on the thirt ieth day, Hideyoshi was surpr ised to see Soror i wa i t ing 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, ' i t 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.
The set of four boys playing footba l l and the set of three gir ls playing house have no common part. The union of these sets is the i r sum. The sum has seven chi ldren.
There are four birds on a tree. Three more birds come along and join them. Three added to four makes seven.
The sum of four and three is seven.
SUM AND ADDITION
ADDITION
Four birds on the tree.
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 wri t ten
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 wi l l 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.
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.
DOING ADDITION S U M S
TWO CHILDREN+THREE PUPPIES
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 ari thmetic, the plus sign ( + ) may be put only between two numbers.
ADDITIONS WITHOUT CARRYING
To add 23 and 34, we add the ones' column f i rst and put down 7 in the ones' column, then we add the tens' column and put t iown 5. The sum is 57.
+
34
LET'S TRY
243 + 625
I 04 + 298
478 + 588
1. Suppose we have a lot of oranges. 2. After giving one orange to each of 98
people, we sti l l have 27 oranges left. How many oranges did we start wi th?
57 + 65
I 0 0
I 0 0
I 0 0 io IO
+ 57
65
1 2 2
i o i i
i i
57 | + | 65 | = 122
23
CARD GAME
CARRYING OVER TO THE NEXT COLUMN
To add 57 and 65, f irst 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 f inal ly get the sum 122.
23 23 34
57
23 + 34 = 57
io
io
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 s i t t ing on l i ly-pads. The di f ference 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 subtract ing 2 from 6 we f ind the number of cows le f t in the pen.
6 - 2
The number of the d i f ference 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 subtract ing 2 f rom 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 'dif ference' card. The player who gets the most cards wins the game.
DOING SUBTRACTION SUMS
SUBTRACTION WITHOUT BORROWING
35 13 245 23
35 mmm mnmmm io mmm - 1 3
00 100 10 10 10 10 • • • • • 245 10 10 • • • - 1 2 3
10 10 1 1 22 100 10 10 1 2 2
35 3 = 22 245 23 = 122
SUBTRACTION BY BORROWING
32 8 343 - 154
• • • • • • • • • • 10
_ 10
mm • • •
I H I H
32
10 10 10 10 10
10 10
100 10
8 10 10 IO 10
10 10 10 10
• • • • • mmmmm
• • • 343
• 154
1 1 1 1 14 100 1 0 1 0 1 0
10 10 10 10 10
1 1 1 1 1 1 1 1 189
32 8 = 14 343 154 = 189
LET'S TRY
1 98 I 84 - 4 5 - 67 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,
836 980 so now I have 345 marbles. How 836 980 many did I have in the beginning? -568 - 8 8 I
The four subtraction sums illustrated above show clearly what happens when you subtract one number from another. Try to fol low 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 wi l l have no apples. You wi l l later learn how to subtract 5 from 3 by using something called 'negative numbers'.
25
ADDITION AND SUBTRACTION
"Subtraction is the opposite of addition. "To do calculations like 6—1 + 2—3, we deal with the first pair of numbers first, and then go on from left to right. "However, if there are calculations inside brackets, they must be worked out f irst.
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 H h H ( -
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.
A + B C C - B A C A
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 12—(6+3 ) and 1 5 — ( 2 + 3 + 5 ) .
COMBINATIONS OF ADDITIONS AND SUBTRACTIONS
1 3 - 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 "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
A n n i T l f l l U ^ numbers A, B, and C, we have ( A + B ) + C = A + ( B + C ) . This is called M U U M I U I i l the associative law of addition.
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 f ru i ts .
• + 3 = 3 + 1
+ 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.
COMMUTATIVE LAW OF ADDITION
ASSOCIATIVE LAW OF ADDITION
J i l l f irst bought 4 apples and 3 oranges and then she also bought 5 grapefruit . She bought 12 pieces of f ru i t in al l .
Jane bought 4 apples. She already had 3 oranges and 5 grapefru i t . She f in ished up w i t h 12 pieces of f ru i t in all.
( 4 + 3 ) + 5 : 4 + ( 3 + 5 ) = I 2
( A + B | + H = B + | 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 Two 10s equal 2 times 10, which is 1 0 + 1 0 = 20.
100 100 100
Three 100s equal 1 0 0 + 1 0 0 + 1 0 0 = 300.
2 + 2 = 4 2 + 2 + 2 = 6 2 + 2 + 2 + 2
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 wr i t ten 4 x 2. 4 x 2 is equal to 2 + 2 + 2 + 2 = 8.
4 X 2 = 8
HOW MANY LEAVES?
• I
• •
I I 4 X »
H |
| X 3 = I 5
HOW MANY CHILDRE N? 4 co lumns
r* * * x * * * *
5 chilo * * * * * X *
- * * X * i l i 20
Three times 10 is 1 0 + 1 0 + 1 0 , which is also writ ten 3 x 1 0 . So, 3 x 10 = 30. 2 x 1 0 0 = 100+100 = 200. We also know that 1 0 x 1 0 = 100. It is not so simple to calculate 2 0 x 1 0 by On the next page some examples of multiplication are illustrated. In each adding 10 twenty times, but if we use some simple laws of multiplication which case the number being multiplied is in blue, the number doing the multiplying we shall learn later, we have 2 0 x 10 = ( 2 x 10 )x 10 = 2 x ( 1 0 x 10) = in red, and the answer in yellow. Try to fol low the examples step by step.
MULTIPLICATION
MULTIPLICATION WHICH OOES NOT TAKE A FIGURE UP
x |2
4 x (10 + 2 )
4 X 10 + 4 x 2
/"S ( 1 0 )
( 1 0 )
(10)
G r ( T ) ( j
x 10 + B [ x 2
4 0 + 8
MULTIPLICATION WHICH TAKES A FIGURE UP
x 24 0 • • • •
3 ! x ( 2 0 + 4 )
20 + 3
(To) (m) (To) (To) t~?\t I St I M I \ \ 1 l\ 1 /V L'V 1 '
3 X 20 + 3 x
60
r r \ r r \ / - r \ f r\ l \ 1 / vJ_A 1 i \ J , ' r
MULTIPLICATION (A NUMBER WITH TWO FIGURES) X (A NUMBER WITH TWO FIGURES)
26 x 37
x
4
8
4 0
4 8
x ' I ' L 2 4
I 2
6 0
7 2
26 x ( 3 0 + 7 )
26 x 30 + 26 X 7
2 6 x 3 7 is equal to 2 6 x ( 3 0 + 7 ) . This is an appl icat ion of the distributive iaw of multiplication, which w i l l be explained later. 7 8 0 + 8 2
x I 2
4 8
x 2 4
7 2
2 6 2 6
X 3 0 X 7
I 8 0 4 2
X
2 6
6 0 0 I 4 0 X 3 7
7 8 0 I 8 2 I 8 2
9 6 2
To mult iply numbers accurately and quickly, you need to know the mult ip l icat ion table by heart. The table is given on page 30.
LET'S TRY
1 I 3 X 3
I 8 X 5
38 X 4
248 X 8
28 X I 2
84 X 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 wel l . After mastering multiplication, it is easy to learn about division.
29
MULTIPLICATION TABLE j u F ^
* 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. "To learn the multiplication table by heart, a card game may be helpful.
0 x 0 0 x
0 x | I x |
0 X 2 2 x |
0 x 3 3 x |
0 x 4 4 x |
0 x 5 5 x |
0 x 6 6 x |
0 x 7
0 x 9
0 x 8
7 x
8 x
9 x
2 x 2
0 x
I x 2
5 x 2
4 x 2
3 x 2
6 x 2
9 x 2
8 x
7 x 2
I x 3
3 x 3
2 x 3
0 x
7 x 3
4 x 3
9 x 3
8 x 3
5 x 3
6 x 3
I x 4
3 x 4
0 x
2 x 4
5 x 4
4 x 4
9 x 4
6 x 4
7 x 4
8 x
0 x 0 x 0 x 0 x 0 x
I x 5 I x 6 I x 7 I x 8 I x 9
2 x 5 2 x 6 2 x 7 2 x 8 2 x 9
3 x 5 3 x 6 3 x 7 3 x 8 3 x 9
4 x 5 4 x 6 4 x 7 4 x 8 4 x 9
5 x 5 5 x 6 5 x 7 5 x 8 5 x 9
6 x 5 6 x 6 6 x 7 6 x 8 6 x 9 \
7 x 5 7 x 6 7 x 7 7 x 8 7 x 9
8 x 8 x 8 x 8 x 8 8 x 9
9 x 5 9 x 6 9 x 7 9 x 8 9 x 9
After learning the multiplication table by heart, it wi l l be easy to multiply or divide any numbers. The product of multiplying a number by itself, such as Ox 0, 1 x 1, 2 x 2, and so on, is called the square of the number. You may use a card game to learn the multiplication table. On the faces of the cards, wri te the multiplications 2 x 3 , 4 x 6 , 8 x 8 , and so on. On the backs of the cards
write their products (6, 24, 64, and so on). The dealer calls out a product, while the players try to take as many cards as possible carrying the numbers which multiply together to give this product (shown on the backs of the cards). For example, if the dealer calls 63, the players should look for cards showing 9 x 7 or 7 x 9 .
I 2 3 4 5 6 7 8 9
I I 2 3 4 5 6 7 8 9
2 2 4 6 8 I 0 1 2 1 4 I 6 1 8
3 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7
4 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6
5 5 I 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5
6 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4
7 H H
7 1 4 2 1 2 8 3 5 4 2 ^ H H
4 9 5 6 6 3
8 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4 7 2
9 9 1 8 2 7 3 6 4 5 5 4 6 3 7 2 8 1
30
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.
MULTIPLES OF 2
rprprprpr rpr I 0 I 2 I 4
Sets of t w o f lags, one red and one wh i te , come one af ter another. The mul t ip les of 2 are 2, 4, 6, 8 found by mul t ip ly ing the number series 1 , 2 , 3, 4, . . . b y 2.
MULTIPLES OF 3
m m \ a | | | | • p | | | » p i I : I ! I;
Sets of three f lags are l ined up one after another. The mult ip les of 3 are 3, 6, 9, 12 . . . . found by mul t ip ly ing the number series 1, 2, 3 , 4 , ... by 3.
2 and 3 is 6, which is called the lowest common multiple of 2 and 3.
COMMON MULTIPLES OF 2 AND 3 T h e n u m b e r s on 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
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 f lowers . By dividing them into three equal lots, each of the three chi ldren can have 4 f lowers . 12 contains 4 three t imes. Using the symbol 4 - wh ich means 'd iv ided by' , we may wr i te
* 4
• EALING THE CARDS
Deal a card to each child. [
Deal another card to each chi ld.
There are 15 cards. Af ter dealing them to 5 chi ldren 3 t imes, they are all d is t r ibuted. This means that by mul t ip ly ing 5 by 3 we get 15. We have 3 x 5 = 1 5 . 15 contains 5 three t imes. So, we have 1 5 - ^ 3 = 5 .
Deal another card to each chi ld.
DIVISION AND MULTIPLICATION
I M I I i t . . A I t Twelve t rucks are pul led by a locomotive. The train is div ided into four equal parts each of which contains 3 t rucks. We have 1 2 h - 4 = 3. This also means that by mul t ip ly ing 3 by 4, we get 12.
2 X 3 < I 2 3 X 3 < I 2
5 X 3 > I 2 6 X 3 > | 2
DIVIDING THE FLOWERS
32
DIVISION WITHOUT A REMAINDER
4 2
4 tens divided by 3 gives 1 in the tens ' place and leaves a spare 10 to carry over to the ones.
mm mm m mm
4 m o 1 0 'O'jlia ^ m
10
There are now 12 ones. Dividing by 3, we get 4.
I I I I
0
I 3 2
i 3 2 m^ 100
4 2 ^ 3 ! 4
,1° l 0 Y ' 0 , 0 M ° i f < « ' # . ^ n
$ # Cm m)
• e
Af ter carrying over 100, we have 13 tens. Dividing by 4, we get 3 in the tens ' place, w i t h 10 carried over.
Y
There are now 12 ones. Dividing by 4, we get 3.
10 10 10
I 3 2 = 3 3
y f 3 3
I 3 2 I 2 t
I 2
I 2
0
DIVISION WITH A REMAINDER How can we divide 14 chocolate bars among 4 chi ldren? Af te r giving 3 chocolate bars to each chi ld, we have 2 lef t over. We say that 14 is not exactly div is ib le by 4. By div id ing 14 by 4, we get 3 as the quot ient 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 ) 4 8 9 1 3 8 7 " 1 2 ) 1 4 4
2 6 ) 4 9 4 4 7 8 7 " " 3 1 ) 4 6 3
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 f irst.
DIVISION IS THE OPPOSITE OF MULTIPLICATION
- v V
• •
m :
; i
me.T*-» -
•m
«*
'it '
I X
CALCULATIONS WITH BOTH MULTIPLICATION AND DIVISION
Ann has
2 oranges
Each of her baskets holds 3 oranges
John has
£ 5 I 5 oranges
Adding her baskets and his, we have
Each of his baskets -holds 5 oranges
She needs
4 . baskets
sS m
k i
i ... J - - ' . J f ' l V
He needs
1 = 3 baskets
7 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 A x 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 wi th the remainder D, then D < B, and B x C + D = A.
When we carry out calculations such as 2 x 4 - 6 + 3 ,
we work out the multiplications and the divisions f irst. In our example, we first work out
2 x 4 = 8, and then 6 + 3 = 2. Then we work out the sum
8 - 2 = 6. But if there are brackets in the calculations, we have to work out the calculations inside the brackets first.
34
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
THE COMMUTATIVE LAW Five dishes with three strawberries in each together contain as many strawberries as three dishes with five strawberries.
In general
f x ®
X
Hx |
B X |
The picture of the rabbits can also be used to explain the commutative law
A x B = B x A. There are 4 rows of 3 rabbits each or, looking at them differently, there are 3 columns of 4 rabbits each. So 4 x 3 = 3 x 4 . You may check the law A x B = B x A by looking at the multiplica-tion table. The associate law is
( A x B ) x C = A x ( B x C ) , which we can see b,y counting the ears of the rabbits. The distributive law is
A x (B+C) = A x B + A x C .
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.
THE ASSOCIATIVE LAW
How many rabbi ts ' ears can you f i nd? There are two ways of counting them. One way is to f ind the number
» * l # * ; \ %J- • • ) x / of ears in a row by mul t ip ly ing 2 (the number of ears on a rabbit) by 3 ( = 6), and then mul t ip ly ing the product (6) by 4, wh ich is the number of rows. In th is way, we get
4 x ( 3 x 2 ) = 4 x 6 = 24. * 1 • • 1 The other way is to mult ip ly 2 (the number of ears per
. X rabbi t) by the to ta l number of rabbits, which is 4 x 3 = 1 2 . In th is way, we get
( 4 x 3 ) x 2 = 1 2 x 2 = 24. Comparing these two ways, we f ind that
4 x ( 3 x 2 ) = ( 4 x 3 ) x 2 . > * v »••_ » J
. / • 1 j In fact , we have proved the general statement
w A x ( B x C) = ( A x B ) x C .
' \ / j for any three numbers A, B, and C.
?» «! ; » • J » i * „ ; v * / x . /
THE DISTRIBUTIVE LAW
m m m M M M
m n n MMM i n
V \ t ¥ ¥
How many f lowers can you f ind in the picture? There are 4 x 3 = 12 red f lowers , and 4 x 2 = 8 ye l low f lowers . In al l , there are
4 x 3 + 4 x 2 = 20 f lowers . Or, looking at it another way, each row has 3 + 2 = 5 f l owers , and there are 4 rows.
4 x ( 3 + 2 ) = 20 also. Comparing the t w o calculat ions, we see that
4 x ( 3 + 2 ) = 4 x 3 + 4 x 2 . In general, we always have
A x ( B + C ) = A x B + A x C .
35
FACTORS AND COMMON FACTORS
m z i z
*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.
NATURAL NUMBERS
• m 0 E H m a
FACTORS OF 12 AND FACTORS OF 18
12 = | x • • • • • • 12 may be wr i t ten as the product of three di f ferent pairs of numbers. The numbers in the red boxes and those in the blue boxes are all factors of 12.
1 8 = d x 1 8
! 8 = B x | 9
B = M X I6
18 may also be wr i t ten as the product of three di f ferent pairs of numbers. The numbers in the red and blue boxes are the factors of 18.
^ " N s v S - "s
( \ I \ I \ I •
" i -+4-
• •: * The factors of 12
2 J K J \ v .
I 3 i
J
The factors of 18 18
I 6 I V l
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 (yel low 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, wi th 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 = 2 x 6 , 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
Ml I • 2 6
2 6
i 3
2 i , 6 3 l • 1
6 I 3
1 1 1
1 ^H
CONSTRUCTION OF A FRACTION
1 5
1 5
1 5
1 5
1 5
3 _5_
Numerator
_3 5
Denominator
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-f i f ths, 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.
FRACTIONS ON A LINE
H h J L J L 3 4
J _ 4
3 3
37
MIXED NUMBERS AND IMPROPER FRACTIONS
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 f ract ion t
1. Change the f irst two mixed numbers into improper fractions, and the last two im-proper fractions into mixed numbers:
_ 3 5 1 6 1 3 l n 3 5
2. Reduce these fractions to the lowest terms: A <L ± 8 U L 1 6 1 5 2 4 3 6
3. Change these fractions to the lowest common denominators:
<1 I ) ("
CHANGING A FRACTION TO HIGHER TERMS
Both the numerator and deno-minator of a f ract ion may be mul t ip l ied by the same number w i thou t changing the value of the f ract ion.
Hii 1 2 I 6
CHANGING A FRACTION TO LOWER TERMS
I 0
By div id ing both numerator and denominator of a frac-t ion by a common divisor, we can reduce t he f r a c t i o n to lowest terms w i thou t changing its value.
20
5
ADDITION OF FRACTIONS
i f
SUBTRACTION OF FRACTIONS
I
+ 1
+ _3 5
c mm
5 8_ 6 6
J 3 + + 15
38
MULTIPLICATION OF FRACTIONS DIVISION OF FRACTIONS
X 2_ 5
_4_ 5
A 4
3_ 4
_3 8
<
2 2 X 2 4 b b
I 2 2
X 3
By mul t ip ly ing a number by we get j of the number.
_ 2 _
6
I X 2 2 2
X 3 6
_3 4
_3_ 8
By div id ing a number by \ , we get tw ice (2 t imes) the number.
• I
X _ 6 _
4 _ 3 2
To divide by a f ract ion, you invert the f ract ion and mul t ip ly .
39
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 i l l . 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 wi l l 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?
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 cutt ing pieces of coloured paper into many equal parts, we can compare the sizes of many f ract ions 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 __ ^ i
10 5 .
By dividing the numerator of a fraction by its denominator, we can write the fraction in the form of a decimal.
DECIMALS AND FRACTIONS
A CLOSE LOOK AT DECIMALS
By magni fy ing a part of a ruler, we can clearly see the decimal markings.
DECIMALS AND FRACTIONS
0.2 — I —
5
0.5 H 1-
0.8 — i —
1 5
1.3
1 0
1.5 -H ( —
4
1.8 - i ( —
• I ADDITION OF DECIMALS
0.2 + 0 .5 Look at the number line.
0.2 0.5
0.7
Look at the coloured t i les.
0.2 + 0.5 0 .7
SUBTRACTION OF DECIMALS
1 . 2 ^ " 0 . 8
Look at the number l ine.
0.4
Look at the coloured t i les.
.2
1.2
0.8
0.8 0.4
FIGURES AND THEIR PLACES
7 0
+ 2 30 + 23
4 0 0
+ 2.3
0. I 7
+ 0.2 3
5 0 0
6 0 50
0 . 6
4 0 4.0 0.4 0 4 4 0 4 4
When me add or subtract decimals, we wr i te numbers so that their decimal points come under each other.
4.4
0.5
0.0 6
0.4 4
MULTIPLICATION OF DECIMALS
X
2 x 1 . 2
3.1 x Decimals and mixed decimals sat is fy the com-mutative, associative, and dist r ibut ive laws just as integers do. Using the d is t r ibut ive law, we can w r i t e :
3.1 x 1.2
( 3 + 0 . 1 ) x 1.2
3 x 1.2
3 x |.2 = 3.6
= 3 x 1.2 + 0.1 x 1.2
3 x
0 . 1 .2 = 0.12
3.6 + 0 . 1 2
3.72
.2 = 3 .6
0.7 0.02
3.1 x .2 = 3 . 7 2
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DIVISION OF DECIMALS Actual calculat ion.
4 .4
4 .4 - 2 2.2
2.2
4.4
4 4 0
4 .4 I . I
Divide into 44 equal parts.
4 4 4 4
0
I . I 4 .4
4 .4 = 4
ROUNDING OFF DECIMALS
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.
Discarding decimals.
I M I ^ 1 . 4 |
Raising to a unit.
ft I 1.4 •=> 2
Rounding off decimals to the nearest who le number.
1.3 •=> I
ftftftft »» 1 . 7 2
In th is case, we count any decimal as 0. Count-ing the decimal 0.4 as 0, we discard it f rom 1.4 to leave 1.
In th is case, we consider any decimal as 1. Count-ing the decimal 0.4 as 1, we thus raise 1.4 to 2.
»»» # In th is 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 uni t .
SIZE OF NUMBERS
Greater than or equal to
The natural numbers greater than or equal to 3 are 3, 4, 5, 6
Less than or equal to
The natural numbers less than or equal to 3 are 3, 2, and 1.
Less than
The natural numbers less than 3 are 2 and 1.
42
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.
The relation between the terms represented in an inequality or equation wi l l 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.
"Inequali t ies 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 writ ten in symbols and using the equal sign ( = ) . An equation may contain an unknown.
INEQUALITIES
Is 3 dollars (300 cents) enough to buy four 40-cent note-books, 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, work ing in cents, we have the inequality 4 0 ^ 3 0 0 . Dividing both sides by 40, we get jr«S7.5. Therefore, x. the number of note-books, must be 7.
LET'S TRY
1. Jim gave 1 dollar (100 cents) for two note-books, and got some change. What can you tell about the price of one note-book? Try to express your answer by using inequalities.
2. 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?
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. wr i t ten as Ix. The 3 cats, of course, have 3 x 4 = 12 legs in all.
So we can wr i te the equat ion: 2x+U= 20
Subtracting 12 from both sides of the equation, we get : l x = 8
Dividing both sides by 2, we f inal ly get x = 4. In this way, we have found that there are 4 chickens.
43
EQUALITIES
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 sti l l commonly used in Asia.
The European and oriental abaci have different forms. Japanese abaci were original-ly 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 wi re , two on the upper.
Various parts of a Japanese abacus.
— Marking point
Wire
A Japanese boy using an abacus.
Bead of value 5
HOW TO EXPRESS NUMBERS 759 2863
HOW TO CARRY OUT THE CALCULATION
Addit ion
32 47
794
Subtraction
32
44
• 1
- I 73
To work out a mul t ip l i ca t ion ( 2 x 4), f i r s t sl ide the middle ruler unt i l the 1 marked on the C-scale l ines up w i t h the 2 marked on the D-scale. Then read of f the number on the D-scale which is opposite the 4 marked on the C-scale.
To work out a div is ion (4-h 2), sl ide the middle ruler so that the 4 on the O-scale lines up w i t h the 2 on the C-scale. Then read of f the number on the D-scale which is opposite the 1 on the C-scale.
How to use a sl ide-rule. . The electr ic calculat ing machine works by pushing buttons.
The most recent ly developed calculat ing machine is the computer, wh ich uses the binary number system.
The slide-rule is a convenient too l for carrying out mu l t ip l i ca t ion and division. To use a mechanical calculat ing machine, the operator sets the key and turns the handle.
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 . 5 - 1 - 0 . 5
l u j U X x j e U t
0 . 5
. u u - L L U L U ™
1 . 5
_ L 1 1
Negative numbers are smaller than 0. Positive numbers are larger than 0. Zero (0) is neither posit ive nor negative.
UPSTAIRS AND DOWNSTAIRS +
Let 's call the ground-level step the zero step. Steps leading upwards are posit ive steps, and steps going down-wards are negative steps.
The centigrade thermometer measures temperatures. When the temperature is higher than 0 degrees, it is posit ive; 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.
If the position of the tree is set to be 0, and if positions to the right of the tree are posit ive, then positions to the left of the tree are negative.
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.
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 ...
DEPOSITING AND WITHDRAWING MONEY
If we coun t p u t t i n g money in the bank as posit ive, then drawing money out of the bank is negative.
GOING RIGHT AND GOING LEFT 4 i
SHAPES
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: 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 pavil ions at Expo '70.
The Swiss pavilion. The Ital ian pavilion.
47
INTERESTING SHAPES
* 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.
A seashell has a coiled shape A f lower 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
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.
HOPE
T H E G O L D E N M E A N This mathematical calculat ion
A — D E
was developed by Greek art ists 2500 years ago.
In the above diagram, G is the mid-point of the side BC of the square ABCD. We draw a circle w i th the centre at G and wi th 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).
The design of the Parthenon, bui l t 2 500 years ago, is based on the mathematical calculat ion of the golden mean.
48
These two packets are examples of modern shapes based on the golden mean.
Each surface of th is bui lding, 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 ref lects l ight A pattern of f loor - t i les in a room gives a wel l -balanced to ta l ly . ef fect .
SIMPLE SHAPES ^
0k * 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.
Triangles have three sides and are among the most basic shapes. Their uses include sails, drawing instruments, and the musical instrument called a tr iangle (below).
Quadri laterals are shapes w i th four sides. They include rectangles, squares, and diamonds. Tables, picture-frames, and f lags are all quadri laterals (below).
Circles are the most wel l -balanced shapes made from a curve. Disks, coins, round tables, car tyres, and some watch-faces are circular in shape (below).
TRIANGLES
j H H H k JHHHBHHHHBHHHHm.
V
V * *
CIRCLES QUADRILATERALS
A circular clock-face.
i e a J *
A triangular road sign stands on the right side of the road. Rectangular windows.
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, rectan-gles, and circles. Solid shapes have length, width, and depth. Spheres, cones, pyramids, and prisms are common solid shapes.
SPHERES
Spheres, or balls, are the most wel l -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, tr ipods, and ta l l party-hats (below).
Cylinders are roller-shaped objects. Prisms are solids w i th both ends of the same shape. Most cans, boxes, and buildings (below) are cylindrical or prismatic.
CONES AND PYRAMIDS CYLINDERS AND PRISMS
Spherical sweets. An Egyptian pyramid. Cylindrical and prismatic buildings.
51
LINES AND ANGLES •Recti l inear 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-off ice. Tracing the way he walked (pink line on the street plan) we get a recti l inear shape.
RECTILINEAR SHAPES
Exterior angle
HALF-LINE AND STRAIGHT LINE
A rect i l inear shape is made op of lines, which are called the sides of the f igore. Two sides meet at a vertex, where they make an angle.
A half- l ine is the term osed to describe a l ine stretching from a point (the end) in only one direction.
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 symbols0 , ' , 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 f lat 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°.
52
A straight line stretches in both directions and has no ends.
MEASURING ANGLES
To survey land, engineers use an instrument called a transit to measure horizontal angles.
Angles of small things may be compared direct ly.
A protractor is used to measure angles accurately.
SET-SQUARES
4 5 "
4 5 i k
MAKING ANGLES USING SET-SQUARES
Set-squares come in pairs. One of them has the shape of an isosceles ( two sides equal) r ight-angled triangle. The other has the shape of a r ight-angled tr iangle w i th other angles of 60° and 30°.
Using a pair of set-squares we can make many di f ferent angles. Try to make angles dif ferent from the ones in the picture.
DIRECTIONS POINTS OF A COMPASS
ENE
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 splitt ing these sti l l further, 16 different directions may be defined, as above.
53
145° W 130° E
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. "Two 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 f ield 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. Str ipes on t ies.
II How to draw paral le l l ines.
Stra ight ra i lway tracks.
54
PERPENDICULAR LINES AND PLANES
Fronts of buildings are perpendicular to the road.
A plumb-line.
How to make r ight angles.
Using set-squares, it is easy to make right angles.
PARALLEL AND PERPENDICULAR LINES
These wal l t i les form a pattern of parallel and perpendicular lines.
A Japanese window-frame.
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.
The road and the rai lway tracks never meet. But these two straight lines are not paral lel because they are not on the same plane. Two straight lines are said to be in a twisted, or skewed, posit ion if they do not meet and if they are not paral lel .
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 ra i lway bridge in the picture is bui l t of steel girders bol ted together to make a series of t r iangles.
Tr iangles can be grouped into several kinds according to the relat ions of their three sides and angles. A scalene t r iangle has unequal sides and angles. An isosceles t r iangle has two equal sides, wh i le an equi lateral tr iangle has all i ts sides equal.
56
TYPES OF TRIANGLES
Scalene t r iangle.
Isosceles t r iangle.
Equi lateral t r iangle.
Isosceles r ight-angled t r iangle. Right-angled t r iangle.
PARTS OF A TRIANGLE
Vertex
Vertex Side Vertex
USING PENCILS TO MAKE TRIANGLES
Scalene tr iangle. Isosceles tr iangle.
y Isosceles right-angled tr iangle.
Right-angled tr iangle.
PROPERTIES OF TRIANGLES
The sum of the three interior angles of any tr iangle is 180°.
Base angles of an isosceles triangle are equal. The sum of any two sides of a tr iangle is always greater than the length of the remaining
side. The difference between any two sides of a tr iangle 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 fol lows 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 quadrilateral is a plane figure wi th four angles and four straight sides. " A quadrilateral has four vertices and two diagonals. "Quadrilaterals are classified into several kinds according to their shapes. "The sum of the four interior angles of a quadrilateral is 360°.
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.
Find the quadri laterals.
VARIOUS TYPES OF QUADRILATERALS
Trapezium. Rectangle.
Parallelogram.
« m m m n d b m
Square.
Rhombus.
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 rhom-buses 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 wi th no parallel sides. An arrowhead shape is special because one of its interior angles is greater than 180°.
Trapezoid.
Arrowhead shape.
Kite shape.
Sfi
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 and cut each other in half.
Parallelogram
Diagonals have equal length, cut each other in half, and make right angles where they cross.
Rhombus.
Diagonals cut each other in half, but their lengths may be dif ferent.
Diagonals cut each other in half, make right angles where they cross, but their lengths may be d i f ferent .
THE SUM OF THE FOUR INTERIOR ANGLES OF A QUADRILATERAL IS 360°
Cut a quadri lateral into four pieces and paste them together.
59
PARTS OF A QUADRILATERAL USING PENCILS TO MAKE QUADRILATERALS Trapezoid.
Trapezium.
Square.
Rhombus.
Rectangle.
Parallelogram.
Vertex Side Vertex
Vertex
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, wi th 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 cir-cumference of a circle to its diameter. We shall also learn how to find the area of a circle.
Compare the numbers of beads you can put on the circumference and along the diameter.
Roll a coin along a line.
o I ra The length of the circumference of a circle is just over 3 t imes its diameter.
HOW TO DRAW CIRCLES
/
Diameter
LENGTH OF THE CIRCUMFERENCE AND OF THE DIAMETER
Using the rim of a glass.
PARTS OF A CIRCLE
Circumference
Using a pair of compasses.
Radius
V ^ J Using a rod to draw a large circle.
v \
Using a needle and a piece of str ing.
A turt le shell. (A regular hexagon.
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 circum-ference, 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
A tr iangle. (An equi lateral t r iangle. )
Coloured papers. (A square.)
Plum f lowers . (A regular pentagon.)
An umbrel la. (A regular octagon.)
A tyre. (A circle.)
VARIOUS CURVES *AI I around us are shapes which are bounded by curves. Among such curves are circles, ellipses, ovals, parabolas, hyperbolas, catenaries, involutes, and cycloids.
Most curves have their own special names. For instance, there is the per-fectly 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.
SECTIONS OF A CARROT
By cutting a carrot in three different ways, we get three different kinds of curves.
A gramophone record. (A circle.)
An egg. (An oval.)
Orbit of a satellite. (An ellipse.)
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.
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.
A circle formed by circles.
FUN WITH SHAPES
ILLUSIONS
Because of the arrowheads, the lower line looks longer than the upper one.
The sloped lines are parts of the same straight line, although they look as if they are not.
/ / / / \ \ / / / / s s / / / / \ \ / / / / \ \ / / / / \ \ / / / / \ \ / / / / \ \ / / / ' \ \ / / / / 1 s / / / / \ \ / / / / \ V. / ' / / \ i * / / A \
Three parallel lines look as if they are not parallel.
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.
Perfect squares look as if they are warped.
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SOLID SHAPES "There are many solid shapes (or figures) around us. * A cone is a solid figure wi th 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 f lat sides) are long, narrow hexagonal
A pavilion at Expo'70.
PROJECTIONS
PROJECTION OF A CONE
Side v iew of a cone.
m A projection chart of a solid shape is composed of several projections of the figure onto di f ferent planes.
PYRAMIDS AND CONES
Regular triangular Regular quadrangular Regular hexagonal Regular octagonal pyramid pyramid pyramid pyramid
PRISMS AND CYLINDERS
Regular triangular Regular quadrangular Regular hexagonal Regular octagonal prism prism Prlsm prism
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 cylinder.
A polyhedron is a solid shape wi th several plane surfaces. The surfaces of a regular poly-hedron are equal regular polygons. There are p ic tu res of a regular tetrahedron, regular hexahedron (or a cube), regular octahedron, regular dodecahedron, and 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.
Circular cone
Circular cylinder
A regular tetrahedron.
A regular hexahedron (cube).
A regular octahedron.
A regular dodecahedron.
A regular icosahedron.
A sphere.
A calendar.
A football.
A sphere is made by rotating a semicircle about its dia-meter.
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 he unit length, and the positive direction. In the picture below, for example, the origin is Bil l '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.
- 5 0 m -40m - 3 0 m
POSITIONS OF POINTS ON A PLANE
-20m I O m 20m 30m 40m 50m
Places on a map are given as ( • , 4) and so on, in order that a user can find where he is.
On this chessboard, the counter is at the inter-section 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).
66
POSITION OF A POINT IN SPACE
z-axis 5
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,
I 4 units from the Arz-plane, and 5 units from the ( 3 , 4 , 5 ) jry-plane. Therefore, the position of this point
in space is represented by the co-crdinates I (3,4,5).
y-axis
same colour is the
Neon signs.
A stadium filled with spectators watching a display.
M V "
d i #•• > A
t J J ( J f * r ' k 0 P W
/ v .
Shape of a car made by marbles.
Embroidery.
VARIOUS SETS OF POINTS
x-axis Each line segment in the same length.
67
MATHEMATICAL MODELS
* 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 f lat as a ' fold-out ' . * By re-assembling the fold-out of a solid figure and pasting its edges together we get the original shape.
REGULAR POLYHEDRONS
Regular tetrahedron
Regular icosahedron Regular dodecahedron
Regular octahedron
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P Y R A M I D S A N D C O N E S (Figure: sol id blue. Three-dimensional diagram: blue lines. Fold-out: red lines.).
P R I S M S A N D C Y L I N D E R S (Figure: solid blue. Three-dimensional d iagram: blue lines. Fold-out: red lines.)
One way to understand the structures of solid f igures 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 tr iangles as its faces; a cube has six squares as its faces; a regular octahedron has eight equilateral tr iangles as its faces, and so on. By cutting the edges of a cardboard box we can open it up and get a f lat 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 f igure.
A PINHOLE CAMERA All measurements in cm.
A pinhole camera may be made by folding up and pasting the edges of the cardboard fold-outs.
[Right) Insert the inner box into the outer box. The image mil l pass through the hole and appear upside-down on the tracing paper.
Regular triangular pyramid Regular quadrangular pyramid Circular cone
[Below) The inner box is made up so that one end is open and the other end has a small 'window' cut out of it. Use glue to cover this w indow wi th tracing paper.
[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.
69
Regular triangular prism Regular pentagonal prism'~ Circular cylinder •
SIMILARITY AND CONGRUENCE
"The Greek mathematician Thales used the idea of similarity to find the height of a pyramid. "Two 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.
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.
FINDING THE HEIGHT OF A PYRAMID
About 2 500 years ago a Greek mathematician named Thales sur-prised people by calculating the height of a pyramid from the length of the shadow of a stick.
People often enlarge a photograph after developing a fi lm. 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 con-gruent shapes you can find.
CONGRUENT SHAPES
fMMMMMMM,MMM —
t i * i i i
i A VIA AIR MAIL . rjK. M MM mm M JF M r" —* * * * i * *
* i A VIA Air' MAIL . * MM j r m m m m m m m m m m m J
Envelopes.
Books.
Butterflies.
SIMILAR SHAPES
A shadow puppet.
70
Milk bottles.
Spoons.
A car and a model car.
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
%
Taking aerial photographs.
;> -Si, , ^ * V
nWl.a-.. ' » te '- "r*
XT'**'**
m u k
w b w m m
^f^v j i? /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.
Surveying directly.
—-•=—— Station
Hospital © Sportsground
® School © Factory
fi Church 4- Dock
E3 Police-station © Post-office
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. Some-times we make distorted figures by enlarging or contracting a figure in only one direction.
Original drawing Q w
A distorting mirror.
71
SYMMETRIES
ted 41 & ' 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 is symmetric about a line if you can fold it along the line so that the left half of the f igure coincides exactly with the r ight half.
A f igure is symmetric about a point if i t can be rotated 180 degrees around a point and look exactly the same.
A f igure 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 i tself are symmetric about the plane of the mirror.
SYMMETRIC SHAPES
The sails of a windmi l l .
Symmetry about a point A butterf ly is symmetric about a plane (although its photograph is symmetric about a l ine).
A product symbol.
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.
Symmetry about a line
The blades of a propeller.
Symmetry about a point
The symbol of Expo '70.
A Symmetry about a line
The Japanese flag.
Symmetry about a line and a point
72
QUANTITIES
We use many kinds of quantit ies: the length of a ruler, the weight of some apples, the volume of milk in a bottle, the length of time a television pro-gramme 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 f i l led wi th 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 tel l 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.
Heights can be compared direct ly .
By l in ing up penci ls side by side, we can compare the i r lengths di rect ly .
When the i r ends are not in l ine, or when the pencils are not ly ing s t ra ight , thei r lengths cannot be directly compared.
UNITS OF LENGTH
The distance be tween the thumb and middle f inger (about 15 cm).
The length of a pace (about 60 cm).
The distance be tween the f inger- t ips when the arms are stretched out (about 90 cm).
74
The circumference of a t ree.
By choosing a unit of length, we can measure and compare the lengths of things even when we cannot compare them directly.
The distance to school.
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 mill imetre) = 1/1 000 m.
To measure the length of something which is
about the size of a pencil, we use centimetres. Mil l imetres 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 The curvimeter is used to mea-sure the distance along a curve or on a map.
Sliding callipers. a i f T i F3 - \
I V * J<' "P "{M JJI"" h I i !t n >4 Ml <1 < I «i! 1.! 11 .t i , „, i
Sliding callipers and micrometers are used to measure exact lengths of small things.
75
The micrometer.
HEIGHT AND DEPTH
The height of a tree or a house is the distance from the ground to the top of the tree or house.
- d -The height of a triangle is the distance from the base (bottom side) to the opposite vertex.
Distance from the earth to the moon: 384 400 km
Altitude reached by jet planes: about 11 000 m
The he from mountain. The __ r . . . . . , the distance from sea-level to the bottom of the ocean.
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
Height of Mt. Fuji: 3 776 m
Height of Mt. Everest: 8 848 m
Depth of nuclear-powered submarines: about 300 m j r r W t
Depth of Mariana Trench: 11 034 m
MEASURING THE DISTANCE ALONG A ROAD
A MAP.
Let 1/A be the scale of the map. Lay a piece of. string along the road on the map.
ON
Measure the length of the string and mult iply its length by A. The answer is the distance along the road.
bfUl MVUK'
Hey ward Point
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
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.
7r =3. 14 I 59265358979323846
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 cmx 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 .
UNITS TABLE
I k m = I O O O m
I m = I O O c m
I c m = I O ram
n = 3.14
DUNEDIN A N D E N V I R O N S
1 : 2 0 0 0 0 0
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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.
. . . i i i . - . SPSE WWffyti i —iHTTWirr •smt- • v Cmm^w&.^'JmHm^^'iSx^. Surveying is used to find the areas of tracts of land.
UNIT OF AREA
1 cm 1 c m ! (one square centimetre) is the area of
X cm H t e l square whose sides are 1 cm.
The area of a plane figure is the amount of space enclosed by its sides. In the case of a solid f igure, 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.
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.
AREAS OF IRREGULAR FIGURES
UNITS TABLE
1 km2 = 100 ha = 1 0 0 0 0 a = 1 0 0 0 0 0 0 m2
1 ha (one hectare) = 100 a = 10000 m2
1 a (one are) = 100 m2
1 m2 = 10000 cm2 These may he estimated by using graph paper (paper ruled into squares of 1 cm or 1 mm sides).
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COMPARING AREAS
Direct comparison by placing one upon the other.
Indirect method using cards. By comparing the numbers of cards it takes to cover various shapes, we can compare their areas indirectly.
FORMULAE FOR FINDING AREAS
S = I x a x h
S = b a s e x height = a x h
S = J x a x b
S = r x ( 7 t x r ) = 7i r 2
S stands for area.
TRIANGLE
A t r iang le is half a rectangle, as shown in the diagram. S = half the base mul t ip l ied by the height.
_ ( b a s e ) x (height) o —
SURFACE AREA OF SOLID FIGURES
RECTANGULAR SOLID (CUBOID) S stands for area.
ARALLELOGRAM
A paral le logram has the same area as a rectangle having the same base and height.
S = 2 x ( a x b + b x c + c x a)
CUBE
RHOMBUS
A rhombus is half the area of the rectangle shown in the diagram.
(d i agona l ) x (d iagonal) S = -
2
«
S = 6xa2
CIRCLE
We can div ide a circle into several equal parts and make a shape l ike a para l le logram by rearranging these parts. The base of th is paral le logram 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 para l le logram is about 7ir . !
CYLINDER
End has radius r.
S = 2 x ( ? r r + j t r x h )
PYTHAGORAS' THEOREM
1 c b
a
b2
S mm a2 Fig. 1
Pythagoras ' theorem s ta tes that, in a right-angled triangle, the square on the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. It may be wr i t ten :
a + b = c In the diagrams (left), Fig 1 shows the squares constructed on the sides of a right-angled triangle. By matching corres-ponding colours in Fig. 2, we can see how the area of the largest square (on the hypo-tenuse) is equal to the sum of the areas of the smaller squares.
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VOLUME *To 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. " I f an object is submerged in a vessel f i l led 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 te l l merely by looking at them wh ich container holds the most, because the i r shapes are d i f ferent . To compare the volumes of l iquids in d i f ferent containers, we pour them into measures having the same shape, as shown on the r ight.
UNITS TABLE
1 / = l O d / / = litre 1 d / = 100 cmJ d/ = decilitre 1 cm3 = 1 cc cc = cubic centimetre = cm1
MEASURING VOLUME
Take the orange out.
Submerge the orange in f i l l ed w i t h water .
basin
Put the water wh ich over f lowed into a measuring cyl inder.
Rfl
Many different shapes, all made of the same amount of clay, have the same volume.
FORMULAE FOR VOLUMES OF SOLID FIGURES
RECTANGULAR SOLID (CUBOID)
V stands for volume.
CUBE
V = lengthxwid thx height.
PRISM CYLINDER
V = s idex s idex side (the length of a side cubed).
SPHERE
Height
MEASURING TOOLS A pipette.
\ volumetric flask.
50 j
- 30
cup. A measuring glass.
HOW TO READ THE MEASUREMENT
Z
f L „ ^ V
" " "
• • • • h h J A measuring cylinder.
Height
V = | x 7t x r3 (r = radius of sphere).
V = area of basex height.
TRIANGULAR PYRAMID CONE
V = 5 x area of basex height.
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 writ ten as 1 cm5. 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 d l (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 wi l l hold is called the capacity of the container. When we put a stone into a basin f i l led 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.
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WEIGHT * The weight of a thing cannot be seen directly (as can lengths or areas). f . ' i k * 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
r v while the weight of the container is called the tare. Both weights added ^ together give the gross weight of the container and its contents.
COMPARING WEIGHTS
The standard kilogramme is a lump of metal made of an alloy of platinum and ir idium.
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.
kinds of There are many purpose.
balances, each designed for a different
A course chemical balance.
BALANCES
A spring balance.
Dial scales.
In addition to the above, balances, which are used objects.
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 lef t side.
Three children on the lef t weigh more than a boy on the right. As a result the lef t end stays down.
Platform scales.
A A beam balance.
there are many sensitive chemical to find the exact weights of light
82
THE WEIGHT OF A DOG
The dial showed 35 kg, which is the sum of the * <- * ' boy's weight and the dog's t w - « weight. ®
John alone weighed 30 kg. c • ,u • -
% ¥
tk
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 wi thout putt ing it on the scales. First he stood on the scales himself holding the dog.
Subtracting 30 kg from 35 kg, John found that the dog weighed 5 kg.
TARE AND THE WEIGHT
(Weight of a con ta iner+we igh t of the contents)
On a coffee jar or a can of f ru i t we f ind its net weight, which tel ls us the weight of the contents.
(weight of the container)
t a r e + n e t weight tare
(weight of the contents)
net weight
HOW
500 g 100 600 :
VOLUMES AND WEIGHTS
cotton-woo/
The density of something is the ratio of its we ight to its volume. Density = we igh t -^vo lume. Specific gravity is the density compared to that of water. Some examples are given in the table below.
Gold I 9 . 3
Lead M . 3 4
Iron 7 . 8
Ice 0 9 I 7
Cypress wood 0 . 4 9
Cedar wood 0 . 4 0
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 l ight, 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 l ight even in quite large quantit ies.
Specific gravity is the ratio of the weight of 1 cm3 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
MUCH DO THEY W E I G H ? The weight of the f ish tank w i th a f ish swimming in it is the sum of the weights of the tank of water and the f ish.
TIME
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.
' 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. ' The time given by a clock differs according to where it is in the world.
Playing.
ONE DAY OF A SOMMER HOLIDAY
Sleeping.
N a i . ^SP
Having breakfast.
1 hour
ADDING TIME
David lef t 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 t ime did he get back? To f ind the answer, simply add the total t ime he spent to 2 o 'c lock :
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.
84
The short hand of the clock tells the hour, and goes round once every 12 hours. The long hand tel ls 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.
UNITS TABLE
1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds
Having lunch. Going to bed.
SUBTRACTION OF TIME
Af ter 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 f ind out when he left the house, subtract the t ime he spent from the time he arrived at the ha l l :
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 lef t the house.
85
HOURS, MINUTES, AND SECONDS
60 minutes = 1 hour 60 seconds = 1 minute
20 minutes 15 minutes
Swimming. Watching TV.
1 minute 1 second
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 fo l lowing day. It is 12 noon (1200 hours) when the sun reaches its highest posit ion 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.
SECONDS. MINUTES, ... AND YEARS
EXPO 7 0 calendar
A watch that also shows the date.
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
2 3 i s « y
•I III II 12 13 II
Hi 17 IS 1!) 211 21
23 24 25 20 27 28
2 3 I !, 0 7
!l III II 12 13 14
Hi 17 IK HI Zlt i
23 24 25 a; 2/ 28
30 31
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
3 I 5 li 7 f
< 1(1 II 12 13 II I
Ift 17 18 10 211 21 :•:
25 2li 27 28 •» 9
i. 7
I M 2(1 21 22 !
28 ;»i :w
2 3 4 'I 10 II II. 17 I
# »
The hands of a 'wa tch tel l each second, minute, and hour. A school timetable lists the t imes of lessons for each weekday. A calendar tel ls 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.
u :
XfflE TABLE G. 5
• -
1 2 3 4
<i 7 8 0 Ml II
13 14 15 10 17 18
20 21 22 23 24 25
27 28 20 :«>
I 2 :i 4 , (i 7 8 0
10 II 12 13 14 15 10
17 18 111 211 21 22 23
25 2li 27 28 20 SI
I 2 3 4 5 0
7 8 II 10 II 12 13
14 15 Hi 17 18 III 20
21 22 23 24 25 20 27
28 211 :«i
10 I
5 5 II 7 8 'I
11 12 13 14 15 lii i
18 10 20 21 22 23 24
.20 27 28 20 30 31
11 1 2 1 4 5 li 7
; II 10 II 12 13 14
15 III 17 18 III 211 21
22 23 24 25 20 27 28
1 2 1 2 3 1 5
0 7 8 II til II 12 jT
13 14 15 III 17 18 III
2(1 21 22 23 24 25 26
28 29 30 31
A calendar.
A day has 24 hours, an hour has 60 minutes, and a minute has 60 seconds.
86
A school t imetable.
TIMES AT VARIOUS PARTS OF THE WORLD WHEN IT IS NOON ON SUNDAY IN TOKYO
Saturday afternoon
Midnight
Sunday noon
Sunday morning
MOTION AND SPEED
FASTER AND FASTER
A snail.
3 m/h
"The speed of a moving object tel ls us the distance it goes in a certain t ime. * A s a unit of speed we generally use km/h (ki lometres per hour). *To describe high speeds, we use units such as km/min or km/s (kilometres per minute or ki lometres per second). "Among various kinds of motion there are parabolic motion, falling, uniform motion, and oscillations.
A boy walk ing. A skater.
4 km/h 45 km/h
An ostrich.
97 km/h
Lightwaves
300000 km/s
A dove.
Sound waves. An express train.
210 km/h 130 km/h
VARIOUS KINDS OF M O T I O N
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 t ime.
The speed of a man walking is about 65 metres a minute, which is wr i t ten 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 travel led 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 l ight has a speed of 300 000 km/s.
Oscillations are motions such as the swinging of a pendulum.
Uniform motion involves a straight path wi th uniform speed, such as the motion of fa l l ing raindrops.
Falling is a motion brought about by gravity.
87
DIRECT PROPORTIONS
*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.
TABLE OF COSTS
Number of t ins I 2 3 4 5
Price (dollars) 1.00 2.00 3.00 4.00 5.00
Lett ing jr = the number of t ins and y = the price,
/ -ax is we can .plot a graph of the equation y=x. It is a straight line sloping upwards to the right.
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 wil l 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.
Certain canned goods cost one dollar per t in. One t in costs one dollar, two t ins cost two dollars, and three t ins cost three dollars. The price increases regularly w i th the number of t ins.
SPEED AND DISTANCE
Putting x = t ime 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.
Man 4 km/h
Bicycle 20 km/h
Car 50 km/h
Time (hours)
Time (hours) 1 2 3 4 5
E Man 4 8 12 16 20
QJ CJ C CO
Bicycle 2 0 4 0 60 80 100
Car 50 100 150 2 0 0 2 5 0
INVERSE PROPORTIONS
*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.
I 000
If you travel between two places in vehicles that have different speeds, the time each takes wi l l 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
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 in-versely with the width, if the area is kept constant.
The relation between speed and the time it takes to go a fixed distance.
550km
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 wid th of a box to hold 12 apples are in inverse proportion to each other.
Speed (km/h) Time
Jet f ighter 1000km 33min
Passenger plane 5 5 0 k m I h
Racing car 3 0 0 k m 1 h 5 0 m i n
Express train I 75 km 3h lOmin
Car 1 0 0 km 5h 30min
Table
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.
Each red picture shows the t ime when a baseball batter made a hit, whereas blue pictures show when he fa i led to hit. Adding them up we see that he had 10 attempts and made 3 hits. In th is case, his batting average is 3 :10 , which is equal to 0.3.
BATTING AVERAGE
RATIO
CONCENTRATION
There are 7 yel low f lowers among 28 f lowers in the vase. The ratio of the number of ye l low f lowers to the total number of f lowers is 7 :28 , which is g = 0.25. The ratio of the number of red f lowers to the total number is 1 — 0.25 = 0.75.
Ann wants to make a f ru i t drink by di lut ing 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 l iquid 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 concen-tration 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
OF YELLOW FLOWERS
Blue county 2 5 5 . 8 Pink county 4 8 3 . 5 Orange county I I 4 . 4 Green county I 9 6 . 7
L E T ' S T R Y
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?
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 tota l seating capacity is 150. In th is example, the passenger density is 90 :150 = 60 percent. This f igure tel ls us how crowded the train is. To get it, we divide the number of passengers by the tota l number of seats.
P A S S E N G E R D E N S I T Y
jrmrmt
P O P U L A T I O N D E N S I T Y Dividing the total population by the area in square ki lometres (km2 ) 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.
A 50 people
Population density of an island.
91
PROBABILITY ' T h e probabil i ty that a particular event w i l l happen is expressed as a fract ion. *When we throw a die, the probabil i ty that it w i l l come up 1 (or any other number) is J . * The permutation number is used to show how many di f ferent 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 th A elements.
4 ways
2 ways
I 2 I I Sum of the two numbers
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.
The probability of its coming up an even number is 3 x 1 = 2 (there are 3 chances out of 6 possibilities).
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.
THROWING TWO DICE
The probability of its coming up any odd number is 3x S = 2 •
TOSSING A COIN
There are only two possible ways a single coin can land: heads or tails.
TOSSING TWO COINS
Tails Heads
The probability of its coming up heads (or tails) is! •
Heads
Heads
Heads
Tails
Tails
Tails
Tails
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 .
92
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 wi l l 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 wi l l land heads (or tails) i s i .
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
i + i + i + i + i + i = 1. 6 ' 6 ' 6 ' 6 ' 6 ' 6 As for the coin, the probability of heads or tails is \ , so i + i = 1
2 ' 2 1 '
• . J -m
a J e t .
There are 3 ways out of the 9 in which The probability of John beating Ann is John can beat Ann. 1 = 1 .
CHOOSING STRAWS
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.
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 wil l win is £ x l x i = n . In this way, we can see that the probability of winning is the same for each person.
PAPER, SCISSORS, AND STONE
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.
Lottery tickets are sold at this stall. In most lotteries the probability of getting a good prize is very small.
93
P E R M U T A T I O N S
w
1
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.
C O M B I N A T I O N S
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''
F U N W I T H F I G U R E S
In t he P r u s s i a n c i t y K o n i g s b e r g , t h e r e w e r e 7 b r i d g e s over t he R iver P rege l . For a long t ime m a n y p e o p l e t r i e d to t a k e a w a l k w h i c h invo lved c ross i ng a l l t h e b r i d g e s j u s t once , w i t h o u t c ross ing any of t h e m t w i c e . The S w i s s m a t h e m a t i c i a n L e o n h a r d Euler ( 1 7 0 7 - 1 7 8 3 ) f i n a l l y so lved the p r o b l e m by s h o w i n g t h a t i t is s i m p l y imposs ib le to do i t . He s t u d i e d t h e n a t u r e of s ing le - l i ne 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 s ing le u n b r o k e n l i ne w i t h o u t t a k i n g t he pen f r o m the p a p e r ) .
TRY TO DRAW THESE FIGURES WITH A SINGLE LINE
9 4
STATISTICS
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 l ike; height, weight, and chest measurements; and so on.
DISTRIBUTION OF POPULATION
ncluding Australia and New Zealand) 0.6%
AfriCa
V '
China 37.3%
WORLD 3688 million
(1971)
Asia 57.6% Others
18.2%
ASIA 2112 million
(1971)
India 26%
i y f i
* *
NOMBER OF PERSONS PER SQ. KM.
Uninhabited
Below 50
TOWNS AND CITIES
50-200 j 1 - 3 0 0 0 0 0 0
Over 200 ^ Over 3000000
• A
9 !
T A D I C O "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.
MS 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 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.
FOODS WE LIKE Table of foods we like
Tea Fruit juice Cake Fruit
Father • • Mother • • •
Grandfather • • Sister • •
Me • • Brother • • • 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
4 people \ 2 peop le /
\ 3 people. , 5 j p e o p l e /
peopla
W E A T H E R
snowy
WEATHER IN FEBRUARY
SUMMARY
fine days
cloudy days
rainy A days
snowy ^ days
MY FAMILY
Father
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.
Age (years) Height (cm) Weight (kg) Favourite things
Mother
Sister
Me ,
Brother s
% 9 10 11 1 2 1 3 1 4
/ 2 3 4 * 7
fine cloudy rainy
GRAPHS "Numer ica l 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
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.
numbers
lorries buses bicycles motor-cycles
LINEAR DISTRIBUTION GRAPH MY FATHER'S DAY
NUMBERS AND KINDS OF VEHICLES PASSING
others 2 hours
travelling 2 hours
resting 2 hours
sleeping
HEIGHTS OF SOME BUILDINGS
4 4 8 m
4 0 0 -
3 5 0
3 0 0 H
250
200 -
I 50 -
I 0 0 -
50 -
0
• I I I ::: • > • • • • • I S • • • • • I I
• • • i l i a
• • • I I I * • • • • • I B
Empire State Building
working — 7 hours eating — 3 hours
PICTOGRAPH
A pictograph is useful for i l lustrat ing information, although it is not easy to make.
8 hours
3 3 3 m
I 5 0 m
A skyscraper
5 4 m
S I0m tuu HiJj M.
Leaning Tower A house of Pisa
Tokyo Tower
146m
Egyptian Pyramid
98
month savings ($ ) t o t a l (S) m o n t h s a v i n g s ( $ ) . t o t a l (S)
I I O I O 5 I O 4 0
2 5 I 5 6 5 4 5
3 5 2 0 7 2 0 6 5
4 I 0 3 0 8 5 7 0
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 quantit ies, a line graph for expressing increases and decreases, and linear distr ibution graphs, circular distr ibut ion graphs, and square dis-tr ibut ion graphs for showing how component quantit ies relate to a whole.
LINE GRAPH
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.
0 - i 1 1 1 1 1 i 1 1 r 1 9 5 8 V e a r 1 ^ 6 7
CIRCULAR DISTRIBUTION GRAPH
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
1 2 3 4 5 6 7 8 9 10 11 12 month
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 quantit ies by the sizes or numbers of small pictures. In a vertical bar graph, the length of a bar is used to indicate quanti t ies (the longer the bar, the greater the quantity). When measuring quantit ies 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 quantit ies increase, and fal ls when they decrease, so that we can see the changes at a glance. Circular, l inear, or square distribution graphs are used to show the proportions between quantit ies in relation'to a whole.
TV PROGRAMMES IIM A DAY
Unit: 10 000 tonnes
herring
salmon and trout
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.
In a theatre, the seats are classified and ordered.
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 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. Block 2 Lot 3 Group i
CLASSIFYING BOOKS
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.
100
JAPANESE HOUSE ADDRESSES
CLASSIFICATION OF TRAIN TICKETS
A computer-controlled ticket office. A ticket from Hikari (New Tokaido Line, Japan
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 Shin-Osaka 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.
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 f lat numbers.
On the licence plate of a car, letters and numbers are used to classify and identify the car.
CLASSIFICATION OF APARTMENTS IN A BLOCK OF FLATS
The next digit 2 means that she lives on the second floor of the building.
The last two digits 03 mean that her flat is the third one along that floor.
Ann lives in flat 3-203.
• M I W l T M S O r
101
01 02 03 04 05 06
NO. CLASSIFICATION
000 general 100 philosophy 200 religion 300 social sciences 400 linguistics 500 natural science 600 applied science 700 arts and recreation 800 literature 900 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.
The Dewey decimal system is used to classify and order books in a library. Books in this library in an elementary school are classified so that it is easy to find them.
1 15 14 4 12 6 7 9 8 I O 1 1 5 13 3 2 16
FUN WITH NUMBERS 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 f irst discovery, people found many ways to make magic squares. There is another example of a magic square above.
A magic square.
8 1 6
3 5 7
4 9 2
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 similar i ty, 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 whi le walking,
he fe l l 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 tel l anything about the stars?' There are many such stories about him.
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 tr iangle the square on the hypotenuse is equal to the sum of the squares on
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) i l lustrates his theorem ( 3 2 + 4 ! = 5 ! ) . It is said that he discovered this theorem as he was looking at the t i led f loor of his f r iend'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 f lute. Apart from these achievements, he also put forward many theories about the universe.
PYTHAGORAS 582 (?) B.C.-493 B.C. Greece
1 0 3
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.
about 300 B.C., Greece
EUCLID
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 l i t t le effort. He demonstrated this principle by moving an entire ship using a lever. He said, 'If you give me a long enough lever and
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 287 (?) B .C. -212 B.C., Greece
1 0 4
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 f ield. He also studied geometry, and used a method of
making the main parts of a painting fa l l on an imaginary tr iangle, a method called pyramidal composition. He used perspective to paint
solid f igures on a plane canvas. In this method all parallel horizontal lines seem to be going to one fixed point. He used this method to paint the famous Last Supper. An example of perspective is used in the picture above.
LEONARDO DA VINCI 1452-1519 , Italy
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
convinced that the sun is the centre of the universe, and the earth revolves round the sun. This idea was against t radi t ional philosophy and religion.
His famous theory was put forward in his book enti t led 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 fr iends' insistence that he agreed to publish the ful l theory. The book was printed only when its author was dying.
COPERNICUS 1473-1543 , Poland
105
GALILEO
1564-1642 , Italy « »
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 t ime, 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 osci l lat ion was always the same. He later found that this is a general law, which he called the isochronism of the pendulum.
Later in l i fe he was persecuted by the church because he supported Copernicus' idea that the earth revolves round the sun.
GALILEO
Galileo studied mathematics, physics, and astro-nomy. Before his t ime, people believed that the speed of a fal l ing body depends on its weight. They thought that a heavy object fal ls faster than a lighter one. But Galileo believed that the speed of a fal l ing object does not depend on its weight . He is said to have proved this by dropping two metal
translated into statements involving numbers. It is said that he got his idea when he was
lying ill in bed. He watched a spider walk ing on the ceil ing and then descending by a spun thread. This led him to the idea of expressing the points
DESCARTES
It was Rene Descartes who f i rst 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
in a space as (A, B, C). He was the f i rst 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.
1 0 6
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 diff icult
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.
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
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 mathe-matics 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 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 f ict i t ious, story about how he discovered the laws of gravity by observing an apple fal l 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 ref lecting telescope which came to be named after him.
LEIBNIZ
The father of Gottfr ied Wilhelm Leibniz was a university professor, but he died when Leibniz was only 6 years old. From that t ime, the young Leibniz was taught by his mother and by himself. Self-education left him free from many of the
tradi t ional ways of th inking. Both he and Newton formulated the basic ideas of differential calculus. Each claimed he thought of it f i rst . 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.
GOTTFRIED LEIBNIZ
1646-1716 , Germany
1 0 8
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.
JOHANN GAUSS
1777-1855 . Germany
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 writ ing anything down. He
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.
1 0 9
TABLE OF UNITS
LENGTH V O L U M E W E I G H T (tonne = metric ton)
1 km = 1 0 0 0 m 1 m 3 = 1 0 0 0 ! 1 t = 1 0 0 0 k g
1 m = 1 0 0 c m 11 = 1 0 d / 1 kg = 1 0 0 0 g
1 cm = 10 mm 1 61 = 1 0 0 cc ANGLES
AREA 1 cc = 1 cm 3 1 ° = 6 0 '
1 km2 = 1 0 0 ha T I M E 1 ' = 6 0 "
1 ha = 1 0 0 a 1 day = 24 hours
1 a = 1 0 0 m ! 1 hour = 60 minutes CIRCULAR CONSTANT
1 m ! = 1 0 0 0 0 c m ! 1 minute = 60 seconds 7 1 = 3.14
FORMULAE FOR AREAS A N D V O L U M E S
AREAS OF PLANE F IGURES prism: 2 x base a r e a + a r e a of sides
2 square: (side) cylinder: 2 x b a s e a r e a + 2 x n x radiusx height
rectangle: lengthxbreadth pyramid: base a r e a + a r e a of sides
parallelogram: lengthx height cone: base a r e a + slant heightx radiusx 7t
rhombus: basex height = 1 product of two diagonals 2
sphere: 4 x (radius) xn
trapezium: 1 sum of two lengthsx height V O L U M E S OF SOLID FIGURES
triangle: 1 basex height cube: (side)'
. , , . . , central angle circu ar segment: area of the circlex
a 360° rectangular solid (cuboid): lengthx breadthx height
polygon: divide it into triangles and add up their areas prism , , . , : base a reax height
cylinder 2 circle: (radius) x 7t
prism , , . , : base a reax height
cylinder
SURFACE AREAS OF SOLID F IGURES pyramid , , . ,
: 5 x b a s e a reax height cone 2
cube: 6 x ( s i d e )
pyramid , , . , : 5 x b a s e a reax height
cone
rectangular solid (cuboid): 2 x sum of the areas of three types of rectangular faces sphere: 5 x (radius)' x 7t
ANSWERS 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 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 percent.
Page 33 1. 12: 43: 12: 19: 21 with remainder 3 : 14 with remainder 29 2. 7h-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.
1. .,3 15.
4 ' "7 7 ' 3 ? _ i = 1 - J L = 3 . 18 _ 0 . i o _
' 16 4 ' 1 5 5 ' 2 4
12 12 12 12
= 1 9 . 1 8 _ 5 1 . H = 2 3 3 '
3 . 1 8 4 ' 36"
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
110
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 "Le t 's T ry " , 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 , 40 -42
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-l ine, 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, 54 -55 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
Mult ipl ication, 11, 28-30, 34, 35, 39,41 of decimals, 41 of fractions, 39 rules of, 35
Mult ipl ication 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, 11 -14
ancient Egyptian, 13 ancient Greek, 13 Arabic, 14 ,15 ,19 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
Rectangle, 58, 59 Remainder, 32 -34 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 -10 , 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
112
hyperbola i n v o l u t e
parabola
o c t a g o n
d o d e c a g o n
a r r o w h e a d s h a p e
p e n t a g o n
square
s q u a r e p a r a l l e l o g r a m r h o m b u s
r e c t a n g l e
t r a p e z i u m
isosce les t r a p e z i u m . t r a p e z o i d
i sosce les r i g h t - a n g l e d
^ t r i a n g l e >
i sosce les t r i a n g l e
s c a l e n e t r i a n g l e . e q u i l a t e r a l t r i a n g l e
TREE OF GEOMETRIC FiGURES AND GRAPHS
i rcular d is t r ibu t io j V ^ q i a p h , p ic tograph
[near d is t r ibu t io i
pos i t ions of po in ts V ^ i n a s p a c e J *
d o d e c a h e d r o n jos i t ions of po in t ; "'•--.on a p l a n e /
pos i t ions of po in ts t on a l ine j
cy l inder
t e t r a h e d r o n
f i e x a g o n a l pr ism o c t a h e d r o n
cone
p e n t a g o n a l twPrism ^
cuboid
h e x a g o n a l p y r a m i d .
t r i a n g u l a r pr ism
p e n t a g o n a l . j i y r a m i d ^
T h a l e s and a py ramid
q u a d r a n g u l a r ^ . p y r a m i d ^
. t r i a n g u l a r py ramid P y t h a g o r a s ' t h e o r e m
Euc l id 's
' E l e m e n t s ' Egypt ian r o p e - s t r e t c h e r s
Euc l id 's ' E l e m e n t s ' , w r i t t e n 2 3 0 0 years a g o , is one of t h e most im-p o r t a n t w o r k s in g e o m e t r y .
