Irving Adler - The Giant Golden Book of Mathematics. Exploring the World of Numbers and Space. 1960

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1

THE GIANT GOLDEN BOOK OF

ATHEMATICS Exploring the World of

by

Numbers and Space

IRVING ADLER, Ph.D.

Digitiz^bf th§'imymm Atehive illustrated 'f)

tOWELL HESS

with a foreword by

HOWARD

F.

FEHR

Viofcssor of MathciiuUics. Tnulicrx CiiHr-^r. Coliimlna Vnhirsili/

GOLDEN PRESS NEW YORK

)OkDl http://www.archive.org/details/giantgoldenbookpOOadle

i

Fourth PrintiDg, 1968 Inc. All rights reserved, including the whole or in part in any lonn. Printed in the U.S.A. Published by Golden Published simultaneously in Canada by the Musson Book Company, Ltd., Toronto. Library of Congress Catalog Card Number; 60-14879

©Cop\Tight 1960, 1958 by Western Publishing Company, right of reproduction in Press,

New York.

9

Foreword

The Science

of Niiml)crs

and Space

10

Mathematics and Civihzation

How We

Numbers and

Write

11

Them

12 14

Standards and Measures

Numbers

CONTENTS i^

We

The Shapes The Puzzle

Cannot

of

16

Split

18

Numbers

of the

-21

Reward

Turns and Spins

22

The Right Angle

23

Triangles and the Distance to the

Figures with

Many

Moon

....

24 25

Sides

and Toothpicks

26

Equal Sides and Equal Angles

27

Circles

28

Square Root Rabbits, Plants and the Golden Section

Getting through the Salt

....

36

and Diamonds

Five

39

Nmnber Systems

Miniature

46

Number Systems

Mathematics Letters for

in

48

Natuie

50

Numbers

Your Number

in

Bridges, Planets

52

Space

and Whispering

30

33

Doorway

Galleries

Shadow Reckoning

...

54 56

Vibrations,

Wheels and Waves

57

Our Home

the Earth

59

62

Navigation Matliematics for a Changing World

63

Infinite

66

Measuring Areas and \'oIumes

67

Surface and Volume in Nature

69

Finite

Fires,

and

Coins and

Pinball Machines

71

Calculating Machines

74

Mathematics and Music

77

Mathematics and Art

79

Fun

81

Mathematics Proving

for

83

It

Three Great Mathematicians

Mathematics Index

in

Use Today

84

90 92

IVJLatlK'inatifS

woiulci— a place

a world ol

is

This book also deals with practical things,

how

make

and perpen-

where, with only a few numbers and points at

such as

our command, the most amazing formulas and

dicular lines,

geometric figures appear as out of a magician "s

scribes die angles that a sur\eyor needs to

know,

and shows how

which

liat.

Mathematics

needs.

When we

many? how

is

also a tool— a ser\ ant to our

wish to

large?

how

know how much? how

fast? in

what direction?

with what chances?— the mathematician gives us a

way

Queen of Knowledge. It has its own logic— that is, a way of thinking. By applying this way of reasoning to numbers and to space, we can come up with ideas and conclusions that only the human mind all

mathematics

is

the

can develop. These ideas often lead us to the

hidden secrets of the ways

in

which nature

works.

I

to discover the

falling or a rocket

to find

at

traveling in space.

how

an area, or a \olume, or

l)y

the

more astounding

the unfolding of seemingly

is

magical numbers for the interpretation of nature

—a

sea shell— a growing tree— a beautiful rec-

tangle—the golden section. The

music

arts of

and painting become the mathematics

of har-

monics and perspective, and the behavior of our entire universe

is

revealed as a mathematical

is

revealed in the pages that follow,

am happy

to invite

you

to a glorious ad-

This

is

a book for inquisitive

words and study the beautiful pictures

ful

in this

delightful book.

metic as you study

it

in school. It

the extraordinary tilings that

what you

study.

struggle to explain

It

does

come from

tell

tlie

you

the use

unfolds the story of

the world in which he

man s

quantitative aspects of

lives. It tells

the story of

episode that,

takes

you from

and even beyond our was

space.

finally

And

it

if

care-

Each

topic

Because

I

only an

is

knowledge

being

built.

teach this subject and train teach-

ers to teach this

mathematics

is

pursued by further study

on which the world of tomorrow

that follow.

It

if

in school or other books, will reveal a

the pleasure

subject; because

to the utmost; it

I

enjoy

and because

I

all

know

gives— I welcome you to the pages

a

point, along a line, into a plane, out into space,

itself

initial

and numbers so

infinitesimal— to the infinite.

of

thought and study, will pay rich dividends

beyond comprehension— from the

exceedingly small numbers, large as to be

new time with more

in intellectual satisfaction.

This book does not teach you ordinary arith-

minds— those

adults also— which

young readers— and bright read and reread, each

space

speed

de-

use of probability to predict ones chances of

venture in numbers and space as you read the

of

is

It

system.

All this

and

How

is

parallel lines

and where they are used.

winning a game, are simply explained. But even

to find the answer.

But above

a stone

to

shows how

conquered by number.

—Howard F. Fehr Teachers College, Cahiiiihia Universitij Professor of Mathcnuitics

page 9

The Science

of

Numbers and Space

we often ha\e to answer "How many?" '"How big?" or "How far?" To answer such questions we have to use numbers. We have to know how numbers are related and how different parts of space fit At work or

at play,

questions like

together.

we

To be

try to think carefull)'.

diings,

we

When we do

these

is

the science in which

at a baseball

of a floor, or decide

buy.

It

we

think

helps us

It

game, measure

which purchase

is

tlie

area

a better

helps the engineer design a machine.

It

helps the scientist explore the secrets of nature. It

supphes us with useful

facts. It

cuts for solving problems.

and puzzles

It

y--

shows us short

helps us under-

we live in. It also gives that we can do for fun.

stand the world

are using mathematics.

Mathematics

page 10

sure our answers are correct,

carefuUy about numbers and space.

keep score

I,

V''

us games

«

Mathematics and Mathematics grew up with

cixilization. It arose

out of practical problems, and

it

helps people

In the days ing,

counts.

To

deal with

all

these activities,

men

invented arithmetic, which studies numbers,

and geometry, which studies space.

solve these problems.

//

Civilization

when men

and gathering wild

got their food by hunt-

fruits, berries,

and

seeds,

To

predict the changes of the seasons, priests

studied the motions of the sun,

moon and

stars.

they had to count to keep track of their supplies.

Navigators looked to the sky, too, for the stars

Counting, measuring, and calculating became more important when people became farmers

that guided

them from place

them

work,

and shepherds. Then people had

which

land and count their

When

to

measure

they had to figure out

dams and

how much

canals,

earth to re-

move, and how many stones and bricks they

would vance

use.

The

how much

overseers

had

to

know

in ad-

food to store up for the work-

ing force. Caqienters and masons had to meas-

To

save time, some people worked out rules for

at once. This

As the centuries went by, men sea, die air,

As

trade

weighed

Tax

grew,

their wares,

To merchants

measured

and counted

collectors figured the tax rate,

their

and

money.

and kept

ac-

many problems

was the beginning of algebra.

and workshops. ple

kings.

To help

spread over the world. The same

doing them, and ways of doing

ure and calculate as they built homes for the

dead

to place.

invented trigonometry,

kinds of calculations often had to be repeated.

people, palaces for their rulers, and great tombs for their

men

relates distances to directions.

Commerce

flocks.

they built irrigation

in this

built

machines

Scientists studied the earth, the

and the

sky. In tliese activities, peo-

worked with things

that mo\'e or change.

think accurately about motion and change,

they invented calculus.

ated

new

New

problems, and

kinds of work cre-

men

invented

new

branches of mathematics to solve them.

6 ^-f^

V page v-^~

1]

Numbers and How We Write Them In the scene abo\

a

e.

has just killed some

The hunters can

rows.

of primiti\e hunters ar-

see at a glance that the

animals killed doesn't match the set of

set of

men

team

game w ith well-aimed

A man

man\? At "

first this

up with the words

question used to get mi.\ed

c|uestion "\\'hat kind?

like couple, pair,

So separate

and brace were used

to

describe different kinds of objects.

without an ani-

But people soon learned that a couple of

mal, so the hunters conclude that there are

people matches a pair of socks or a brace of

more men

ducks, and that the matching has nothing to do

in the

team.

in the

in the catch.

way probabK^

sets.

We

couple.

A

also talk

ducks. pair,

sets of objects in this

led to man's

more and

ha\e many words

grew out

left

team than there are animals

Matching

ideas, the ideas of

We

is

first

mathematical

language that

of our experience with trying to

match

distinguish a single person from

lone wolf

is

different

from a pack.

a

We

about a pair of socks or a brace of

Words

like single,

realized that a couple, a pair,

somediing

Jess.

in our

with the kinds of things that they

couple, lone, pack,

and brace answer the question

"How

in

common

that

are.

They

and a brace have

makes

how

it

possible

numnumber word two to answer the question "How many? for any set that matches a couple, no matter what kind for

them

to

match. This

ber arose. Today

we

is

the idea of

use the

of objects are in the set.

Numbers were used long before there was to write them. The earliest written

anv need

70 IL^ An

early form of the Arabic numerals

numbers we know about

are found in the temple

records of ancient Sumeria. Here priests kept track of the

amount

of taxes paid or

owed, and

of the supphes in the warehouses.

As time passed, men in\ented new and better

ways

of writing

numbers. At

them by making notches the ground.

We

still

first,

men wrote

in a stick, or

use this system

Hues on

when we Ancient records written

write the find

it

and

3.

Roman numerals

I,

II,

and

were joined

as sets of separated strokes.

to

The Arabic numbers ten digits, like.

we can

We do this

groups, just as

in a hurr\',

use only ten symbols, the

write

with these

down any number we

by breaking large numbers

we do

We

into

with money. W'e can sepa-

rate thirteen pennies into groups of ten three.

and

can exchange the ten pennies for a

we have one dime and tJiree penTo write the number thirteen, we write 13.

dime. Then nies.

;

clay

we ha\e

If

poiip of o\er.

only ten pennies, the\' form a

with no additional pennies

ten,

To write the number

ten,

we

put a

left

1 in

the

second space from the right to represent one

each other.

digits 0, 1, 2, 3, 4, 5, 6, 7, S, 9. But,

in

We

hidden, too, in our Arabic numerals 2

They began

Then, when the strokes were written tliey

III.

group of

ten.

second space, first

But

we

to recognize this

nnist write

space as the

something

in the

space, e\ en though there are no additional

We

pennies be)ond the groui^ of ten. to represent

digit

use the s\mbol

"no pennies.

If

'

write the

we

didn't

whole sxstem

in this wa\', the

would not work.

The needed

first

people

who

recognized that

a SNnibol for the numl)er zero

the\'

were the

The 1 written in the second space from the right means one group of ten, just as one dime means

people of ancient India. The Arabs learned

one group of ten pennies.

tem

EGYPTIAN

from the Indians, and then built of written

numbers

that

it

we

it

into the sys-

use todav.

The Egyptians used the royal cubit as a unit of measurement

^

—

y

\\

\

L*—

FOUR DIGITS ONE PALM

f

—

'°-'^"

*\

The Metric System

mass

in

the metric system

was chosen In most countries ot

the world today

the

to

called a ^ram. It

is

be the mass of

1 cc.

of water.

One

pound contains about 454 grams.

standard units of measurement are those of the

The metric system was

metric system.

adopted

France

in

other countries. In

in 1795,

tliis

eartlt

tlie

and

The unit of lengtli is called a meter and was

derived from the distance around the earth in this

way:

A

circle

drawn on

the surface of the

earth through the north and south poles

One

a meridian.

quadrant.

A

to

The length

its

called a

is

how

of

one of these parts

made

measurements

careful

long the meter

is,

bar. This bar, kept in a vault in Paris,

A

standard of length.

meter

is

the

about

is

39.37 inches long. For measuring small distances, parts.

subdivided into one hundred equal

it is

Each part

is

There

called a centimeter.

are about 22 centimeters in an inch.

The

unit of

volume

in the

of a

thimbleful

cube that

is

is

cc.

interval of time

The year

over again. the earth

s

1 cc.

The

of

repeated over and

based on the rhythm of

is

motion around the sun. The month

around the

rhythm

The

earth.

The day

of the earth

smaller units that

is

is

v\

based on the

around

rotation

s

its

axis.

e call an hour, a minute,

are obtained by subdividing the

average length of a day.

We

used to think of the

and

dav'

its

subdivi-

sions as the best standard miits, because th

rotation

of

the

earth

was the most regular

rhythm we knew about. units,

We now

have better

based on very rapid rhythms inside mole-

cules or atoms.

These are used

in

molecular or

atomic clocks. One, the ammonia clock, uses as imit of time the period of a vibration inside

an ammonia molecule. This period

It is

the

that there are 23,870 million vibrations a sec-

A

imit of

ond.

With

y

m^

the

one centimeter high.

nearly equal to

is

Each

in nature,

based on the rh>thm of the moons motion

its

).

different units of time.

based on some rhythm

is

metric system

cubic centimeter (ablireviated as

volume

many

use is

which an

they measured

length between two scratches on a platinum

official

We them

and a second

be a meter.

After scientists to find out

called

quadrant was divided into ten mil-

lion equal parts.

was chosen

fourth of a meridian

is

Time

Units of

to

system, the standard units

are based on measurements of water.

first

and then spread

a molecular clock

we

is

so small

can measure

irregularities in the spinning of the earth. NITROGEN ATOM

i||

Numbers We Cannot can make a "picture" of a whole luimber

Ycni

as

many

A

checkers as the

number

line of four checkers

lines

tells

can be

with two checkers each.

If

you

to.

split into

we put

There

b>-

using a line of checkers. To form the picture, use

two

these

is

number method is a

Split

a smiple is

way

of finding out

called the sieve of Eratostlienes, after

who

the Greek scientist

devised the system, two

centuries before the birtli of Christ. Imagine all

hues under each other, the checkers fonii a rec-

the whole numbers, starting with

tangle. Rectangles can also be formed with

order in a

9 or 10 checkers. So tangle numbers.

we

call these

The rectangle

numbers

for the

ber

is tJic

2x5 =

10.

rec-

number

10 has 2 lines that ha\ e 5 checkers in each Notice that

6, 8,

line.

Every rectangle num-

product of smaller numbers.

in this wa\'.

For example,

we

split

cannot arrange 7

line.

The number

the head of the hne,

count by get.

and

2's,

is

They

are

number

numbers

stands at the head

se\en

in a rectangle. \\'e

lines,

then they are

now

still

in

each

arranged in a single

the line runs up and

down

line.

to left.

in

But

line, onl\'

left,

is

page 16

cross out the

count by 3

15, that

s.

so

Among

is

the ne.xt prime

i 9

10

numbers \ou get

The\' are

numbers

form rectangles with three

like

9

lines.

pic-

prime number.

because they cannot be written as the

product of smaller numbers.

and

Now

}"ou

lines.

and

number 3 now

the

of the line. It

multi-

all

Among the numbers that are left, the number 5 now stands at the head of the line. It is the third

tured as rectangles are called prime numbers.

This

number.

when

and

2.

rec-

instead of going

The number 7 is not a number. Numbers that cannot be

from right tangle

can arrange them

with one checker

Now

number you

like 4. 6, S,

form rectangles with two

the numbers that are

arranged in

a prime number.

^ checkers

2,

wliich stands at

2,

cross out exery

This remo\es the

ples of 2. on, that

There arc some numbers that cannot be

whether

a rectangle or prime number. This

Continue the

number

tiples

of

in this at the

that

way, removing from the line

head

of the line,

and

number. After each

all

mul-

famil\-

of

10 hold

oihe

:

1

1

TRIANGLE NUMBERS

The Shapes of Numbers Numbers,

like people,

Some numbers form form

that

many

in

shapes.

There are others

triangles, squares, or cubes.

Triangle

We

come

rectangles.

Numbers numbers

find the

that form triangles b}'

placing lines of checkers under each other. Put

checker line,

in the first line, 2

3 checkers

get larger

number

and

are

in the third line,

and so

1, 3, 6,

and

on.

larger triangles in this way.

of checkers in a triangle

angle number.

The 10.

number? One way

first

What

1

checkers in the second

is

called a

four triangle is

tri-

numbers

the seventh triangle

to find out

is

to

make

the sev-

enth triangle. Then count the number of checkers in

it.

But there

is

a short cut

drawing on the side shows the with another one

just like

it

we can se\

use.

The

enth triangle,

placed next to

it

upside down. The two triangles together form a rectangle, so the triangle

rectangle

page 18

number

is

®®99##®

We The

half of the

number. The rectangle has se\en

mmmmmmmm mmmmmmmm mmmmmmmm •

•••

and

lines,

tangle

To

eiglit

nnmber

checkers in eacli is

7

X

8, or 56.

the recis

by the

ne.xt

higher num-

and then take half of the product. To

Most whole numbers are not But

e\

third square

itself.

49.

call

it

^

The

it

right

hand corner

k

^

13

bers are related to

them bers. 6;

is

sum

the

"

them

of

fci^

in a

^

few

7

is

to

as 1'.

triangle

is

7

'

7, or

little is

a

two written

way

of

in the

upper

showing that the

be used as a multiplier twice. "Eight-

squared"

is

written as

8',

and means 8

8, or 64.

are relatives of the

odd

numbers (numbers

that cannot

form rectangles

with two lines).

you

odd numbers

order, stop

If

when you

list

like,

the

in

and add those you

few

simple way. Each of

two or three triangle num-

+

10; 14

= 1 -f 10; 12 = 3 + 3 + = 1 + 3 + 10. Find three

numbers

tliat

add up

have fisted, the sum is always a squart' numbei. The drawing abo\e shows you why. Square numbers are also relatives of the tri-

to 48.

Square Numbers

We

angle numbers.

form a square by making a rectangle

which the number

number

of lines

is

the

of checkers in each line.

sc^uare has only

one

line,

same as the The smallest

is

fine.

So the second square number

is

2

X

SQUARE NUMBERS

MULTIPLICATION TABLE The square numbers are found on the diagonal

number

to the

square number. The drawing below shous why.

The

1.

triangle

next higher triangle number. You always get a

with one checker in the

So the smallest square number

Add any

in

next square has two lines, with two checkers in

each

3, or 9. To number by

For example, 11

13 == 3

line.

3 X

"seven-squared" and sometimes

The square numbers 12

is

The seventh square number

We

write

en those that are not triangle num-

number

get a square innnber, multiply any

9.

num-

triangle

The

or 4.

find

the eighth triangle number, take half of S

bers.

28.

number, multiply the number

find a triangle

of lines in the triangle ber,

line. S(i

Half of that

2,

>^

plier three times.

Cubic Numbers

cubed. If

we

we

use blocks instead of checkers,

can

arrange them in hnes to form a square, and pile the squares

When

the

on top of each other

number

of blocks in a line,

The number

number

we have

a cube.

of blocks in a

cube

is

called a

The smallest culiic number is 1. 2 v 2, or 8. We The second cubic number is 2 cubic number.

call

The

it

"two-cubed, and sometimes write

little

it

as

2\

three written in the upper right cor-

ner shows that the 2

is

to

be used

as a multi-

It is

The

ings of 2'

fifth

cubic

written as 5\ and

or 125. \\'hat does 6'

and

number

means 5

mean? Compare

tlie

5

"five-

X

5,

mean-

is

written as

2", is also

called "two raised to the second power.

Two-

cubed, which

"two

is

written as

'2\ is

also called

raised to the third power." In the is

(2

used as a short way of writing

same way,

used as a multiplier four times), and

"two raised out,

we

to the fourth

find that 2^

=

16.

raised to the fifth power,

2*

2X2X2X2 is

called

power." Multiphing

We

read 2' as "two

\Miat does

fl V

is

<

3'.

Two-squared, which

in lasers.

of la\ers equals the

"

4'

it

mean?

The Puzzle

Reward

of the A

wealthy king was t)nee sa\ecl Ironi drown-

ing by a poor farm

To reward the boy, the

l)o\-.

king offered to pay him sums of

two pkrns

clioice of

first

in thirty

of paxnient.

Under pkm number the

money

But he offered the boy a

daily instaUments.

would

the king

1,

pa>-

day, S2 the second day, $3 the third

SI

da\-.

and so on, the payment increasing by Sf each

Under plan number

da\\ !(•

the

first da\', 2'

day, and so on, the

2,

the king

would pay

the second da\', 4< the third

payment doubling each day.

Which plan would reward?

We plan,

can answer the question' by simply writthe thirty installments under each

and then adding them up. But there

is

a

way of getting the answer, too. Under plan number f, the total reward in dollars is the sum of all the whole numbers from 1 to 30. shorter

This

number.

simpl)' the thirtieth triangle

is

According calculate

to the rule given

it

dividing by

on page

by multiplying 30 by 2.

The

19,

31,

reward under

total

we

can

and then this

is

2;

installment

installment

the total

would be

2,

the second installment

the third installment

the fourth installment

last

+

2'

2'^

+

2^

+

2"**

+

2"*^

sum if it

is

1

+

2

+

+

2"

2''

+

2''

+

2"'"*

+

2"''

+

2'^"

subtract, those installments that are

equal to each other cancel. Then is

2^"

ber quickly

b\-

difference

X

32

=

32

—

We

1.

is

is

is

2

a higher

2"''.

A

to write

is

2

•

2 or

2';

each

2

2 or

2';

power

of 2,

and the

short cut for calculating

down what

the reward

were doubled, and then take awa\-

the single reward from the doubled reward;

we

=

noticing that 2'

=

1024; 2'"

1024

Now we

1024 == 1,073,741,824. find the total reward:

see that the

can calculate

this

num-

32; 2"^

=

1024

X

>:

subtract

1 to

1,073,741,823 cents, or

$10,737,418.23.

We

see that an

amount grows

doubled repeatedly. Keep

Under plan number

new

+

When we

plan

would be $465. in cents

2

Single reward:

down

ing

Double reward

gi\e the boy the greatest

you is

try to

placed

amoeba

answer the next puzzle:

in

an empty

splits into

jar.

when it is mind when An amoeba

fast

this in

After one second, the

two amoebas, each

as big as

the mother amoeba. After another second,

daughter amoebas each

new

split

generation

in

splits,

the

the

same

number

w^ay.

of

tlie

As

amoe-

bas and their total bulk douliles each second. In

one hour the

jar is full.

When

Splitting in two, or doubling,

amoebas reproduce

rapidly

is it

half-fuH?

tween them. The angle

needed

to turn

At one

other.

is

is

tliis

is

of rotation

30 degrees. At two

60 degrees. What

between the hands be answer

amount

the

the angle between the

o'clock,

hands of a clock the angle

is

one hand to the position of the

two? The

at half-past

printed upside

down

o'clock,

will the angle

at the

bottom of

page.

The

face of a clock

like a circular race track

is

around which the minute hand and the hour

Turns and Spins

hand race

against each other.

from the same position minute hand moves Tliere are

many

things that turn or spin.

wheel of a moving automobile

turns.

and the minute hand of a elock

around the

faee. Since so

many

often have to measure the

we

hand points

to the zero

from the

to the 12

a protractor.

on the clock,

falls

the minit

it

points out the

through which

it

to reach the

on the clock.

1

has turned.

to reach the 3. It turns

points

full lap

it moves away number of degrees

time not

It is

at the rate of

55 spaces an hour.

A

1=

of an hour, or

1

^

hours.

One

60 minutes, or 5

^

is

minutes. So the

time the hands are together

turns 90 degrees

again

180 degrees, or half a 6.

There are two hands on the face of the

first

contains 60 spaces, so the gap becomes a

full lap after

I

moment

the

eleventh of an hour

complete rotation, to reach the

eacli

is

happens?

turns 30 degrees

It It

happens, the two

this

behind the minute hand, the gap between

them widens

on the protractor. As

12,

When

The face of the clock is divided into 60 spaces. The hour hand moves around the face at a speed of 5 spaces an hour. The minute hand moves at a speed of 60 spaces an hour. The difference between 60 and 5 is 55. So, as the hour hand

In the drawing above, a protractor has been

ute

hour hand,

hard to figure out the answer.

rotation.

When

faster than the

after twelve o'clock that this

of turning.

start

it.

hands are together again. What

degree. There are 360 degrees in one complete

placed over the face of a clock.

They both

12 o'clock. But the

The gap between them hour hand is a full lap behind

ahead of

the minute hand.

An amount of rotation is called an angle. The unit we use for measuring an angle is called a

To measure an angle we use

gets

widens, until the

its

rotates

things turn,

amount

and

So does a

phonograph turntable. The earth spins on a.xis,

A

at

day there

is

oi a clock.

At

an angle be-

is

5

first

^ minutes

j\

after

one

o'clock.

The Right Angle The angle

tliat

90 degrees.

we

We

use most oltcn

call

it

an anisic

is

a right angle.

ol

We make

bricks with right angles in each corner so the\-

Then

piles.

stand up straight instead

leaning over, and

Hoors are

of

In

aiicii'nt

and another

making

Egypt, surxeyors

made

a

right

angle by "rope-stretching." They used a long

a riglit angle

is

to

measure

by

knots.

One man

A

held the two ends of the

out 90 degrees with a protractor. There are

rope together.

other ways of making a

was three spaces from one end. A

a protractor at

line be-

circles.

rope that was divided into twelve equal spaces

le\ el.

One way

tween the centers of the

walls

will stack easily in \ertical ol

then draws a straight line between the points at wliich the circles cross,

A

riglit

angle without using

bricklayer

makes a

second

man

held the knot diat third

man

right

held the knot that was lour spaces from the

angle with strings. lie makes one string hori-

other end. Wlien the rope was stretched tight,

all.

zontal witli the help of a le\el.

other string its

end.

A

\

by hanging

a

the

circles tliat cross

each other. lie

\

^ iM:

^^B

90

a right angle

weight from

draftsman makes a right angle by

drawing two

^^^

ertieal

He makes

The

was formed.

simplest

way

to

to fold a piece of paper. it

again, so the crease

make Fold

falls

it

a right angle

once.

Then

on the crease.

is

fold

A Triangles and the

Moon

Distance to the

Triangles ma\' ha\ e different sizes and shapes,

but the three angles of

up

Place them side

comer, and edge

add up This

to edge.

You

corner to

will see that they

know, because

a useful fact to

you a short cut

tear off the

b\' side,

180 degrees.

to exactly is

this for \ourself,

Then

cut a triangle out of paper. tliree angles.

always add

an\- triangle

same amount. To see

to the

gives

it

for finding the angles of a

tri-

en

if

\ou

measure only two of them. For example,

if

one

angle.

You can

of the angles is

find all three angles, e\

is

40 degrees, and the second one

60 degrees, you can find the number

of de-

grees in the third angle widiout measuring Simpl)-

add 40

to

it.

60 and then subtract the result

from ISO. This short cut

angle that

is

is

especially helpful

the third

if

out of reach. For example, suppose

two men, standing

on

at separate places

The two men and There is nobody on

the earth, look at the moon.

the the

moon form a triangle. moon to measure the

we can

calculate

measure on the

it

angle up there. But

Knowing

earth.

this

important to astronomers, because calculate the distance to the

A

were further awa\- than be smaller.

would be

Oi yy^

If

the

it

it

moon.

moon were

can

angle

is

helps

them

the

moon

If

the angle

is,

would

closer, the angle

The moon

larger.

we

from the angles

approximateh'

is

240.000 miles awa\- from the earth.

Once we know angles A ond

A

6,

=^ angle x -r angle angle w. Angle y and angle

Angle

we can y.

w

x

Angle

z

= =

height of height of

=

angle

z

—

can be calculated from

the positions of the observers, Oi

Angle

calculate angle C.

Angle B

moon above moon above

and

Oi,

on earth.

horizon as seen by Oi

horizon as seen by Oj

Figures

Many Sides

with A

closed figure with straight sides

polygon. the

same

The number as the

ot angles in a

number

of sides.

A

is

called a

polygon

is

polygon widi

One with four sides is The names for some polygons with more than four sides are shown in three sides

is

a triangle.

called a (]uacliilatcral.

the table below. If

two

we

join opposite corners of a quadrilateral,

triangles are fonned. If

of both triangles

we ha\e

we add

the

sum

the angles

of the angles

of the quadrilateral. Since the angles of each

triangle

add up

to

ISO degrees, the angles of

the quadrilateral add or

360 degrees.

into three triangles,

3

X

up

180 degrees.

so

A

its

angles add up to

six-sided

divided into four triangles, so to 4

X

ISO degrees,

to 2

A five-sided figme can be di\ided

its

be

angles add

up

180 degrees. To get the number of

degrees in the

sum

of the angles of

take two less than the

then multiply this

NAME

figure can

any polygon,

number of number by ISO.

NUMBER OF SIDES

sides,

and

Tn "dEgTe^S

''

3.1415926535897932384...

^K.

Circumference Circumference

Area

^

>r

•

= ^

n

diameter

2„

rodius

radius

radius

Circles and Toothpicks

/ We

see circles everywhere.

The wheels

of auto-

mobiles, the rims of cups, and the faces of nick-

and quarters are

els

full

moon

The ter, is

all circles.

you

is

called

Measure the diameter of

will find that

can measure

it is

ure

it

cen-

The

dis-

circumfer-

its

and

a quarter,

the circumference of the quarter,

wind enough

with a

its

about one inch long. You

ruler.

You

around

string

around once. Then unwind the

string,

will find that

the floor. the

The

it is

about

stick

by the number result

is

of times

your value of

For example,

by

if

cumference of any

circle

the diameter. This fixed

is

a fixed

number cannot be

ten exactly as a fraction or decimal, so the Greek letter ^

(pi) to

stand for

it.

It is

writ-

we

use

almost

equal to 3i, or 3.14. Strange as

it

may

is

a

way

of cal-

you drop the

stick

result

get.

is

about

^r.

3.2.

This

more accurate

When you

drop the

turned around

is

a circle. That

is

circle,

why

is

its

t,

^ \ i^^

is

it

its

will

center

center.

it

falls,

When

moves around

related to meas-

also related to the

the stick will cross a crack.

floor

its

center,

which

you

whether or not

and how

it

not a very

a value

stick,

depends on where

a stick turns around

is

The more times you drop

crosses a crack

You can calculate n by dropping toothpicks on a wood

100 times,

The

uring a

seem, there

on a crack. The

fell

62.

the stick, the

The cirnumber times

result.

it

t.

on a crack only 62 times, divide 200

the circumference and diameter of the rim of a

same

and the num-

falls

accurate value of

will get the

it

it

and

three times as long as the diameter. Measure

cup and you

Keep count

times.

you drop

on a crack. Double the num-

falls

it

many

of times

ber of times you drop the stick and then divide

go

to

it

and meas-

number

ber of times

of planks of

a thin stick, such as a tooth-

as long as the planks are wide.

is

Simply drop the of the

made

has to be

floor

same width. Use

pick, that

called the diameter of the circle.

too. First

the

look Hke circles in the sky.

distance across a circle, through

tance around the circle ence.

The sun and

culating the value of t by dropping a stick on

chance that

/

EQUILATERAL

/ \

\ 14

TABLE OF

/

the

column, and the

first

appears

in tlie

the coKimns,

.s(iuiiie

second cohnnn.

it

becomes

If

by dividing 20 into 625.

intercliange

a table of square roots.

square root appears to the right of

it.

left,

But

in

new table, we no longer find every whole number in the first column. The numbers 1, 4, and 9, for example, are listed, but the numbers this

and 8 are

2, 3, 5, 6, 7,

not.

They do not appear

because they are not the squares of whole numbers, or, to say

it

square roots are not whole numbers. These roots that can

be written

appro.ximately as decimal fractions. Since 2

between that

is,

4 and

1

and

4,

between

9,

\/7

lies

between 2 and

1

that

may be

swer from works, is

let

lies

between \/4 and \/9, that

is,

3.

roots.

We

for finding these inshall use a

method

described as "getting the right ana

wrong

us try

the

same

it is

we

get

20.

it

guess."

out

first

To show how

it

on a number that

to find tlie square root of 625.

We

we

take a

li

check our guess

our guess

right,

is

by dividing should conu> out But

as the di\ isor.

it

doesn

t.

It

comes

out about 31 instead. But this gives us a hint

how we can correct om- bad guess. Now we know that the answer should be between 20 and 31. If we try the number 25, we find that it really is the square root of 625. By multiplying on

25 times 25,

Now

let

we

get 625.

same method

us use the

to get

approximate value for the square root of take a guess and say

we

it is

number

3 and

3.3.

good

a guess 3.15

This

is,

is

we

The quotient comes out is

3.

3.16. This

two decimal

places. If

an

We

the average of

is

3.15.

Now,

divide

to test

how

into 10.0000.

it

3.17, so a better guess

of 3.15

the best answer

is

10.

Dividing 3 into 10.0,

get 3.3. So a better guess

would be the average

the square of a whole number. Suppose

want

is

between \/l and \/4, and 2. Since 7 lies between

\/2

There are many methods

between square

the answer

in the opposite direction, their

numbers have square

Now we

guess,

we

Then, for each number that appears on the its

and say

of each numlier

we want

and

we a

3.17,

which

can get with

more accurate

answer with more decimal places, we simply continue the process, checking each

by

di\ iding

it

into 10.

of the ninnbers

from

new

guess

Approximate square roots 1 to

10 are shown in the

third table on the preceding page.

ORIGINAL PAIR OF RABBITS

13

Rabbits, Plants and the Golden Section A man bought The

pair

a pair of

ralsliits,

produced one pair

one montli, and a second pair

and

liied

many new pairs of rabbits did lie get each month? To answer this question, let us write down in

them.

of offspring after of offspring after

a line the

die second month. Tlien they stopped breeding. Eacli

the

new

pair also

produced two more pairs

same way, and then stopped breeding.

number

of pairs in each generation of

rabbits. First write the

number

pair he started with. Next

in

How

for the pair they

Pine cones have Fibonacci ».

8

produced

">_

13

page 30 "'

^

1 for

we write

the

after a

the single

number

1

month. The

i next month, hoth pairs producccl, so the next

number

line: 1, 1, 2.

have three numbers

Each number represents

Now

eration.

the diogrc

We now

is 2.

the

new

in a

a

gen-

The second generation 1 pair) produced The third generation (2 pairs) produced 2 pairs. So the next number we write is 1 + 2, (

1 pair.

Now

taken by

of 91

hie

which

;

generation stopped pro-

first

ducing.

or 3.

1

ch the leaf (he

H

die second generation stopped pro-

The third generation (2 pairs) produced 2 pairs. The fourth generation (3 pairs) produced 3 pairs. So the next number we write is 2 + 3, ducing.

or 5.

Each month, only the

we

produced, so

adding

tlie last

numbers we get numbers. The

two generations

last

can get the next number by

two numbers in this

first

way

twelve of them are:

1, 1, 2, 3, 5, 8,

144

13, 21, 34, 55, 89,

They have very

interesting properties,

popping up

many

Here

is

in

The

in the line.

are called Fihoiuicci

and keep

places in nature

and

art.

one of the curious properties of these

numbers. Pick any three numbers that follow each other

number. The example,

if

Square the middle num-

in the line.

ber and multiply die

first

we

take the

number by

always

results will

the third

by

differ

numbers

3, 5, 8,

1.

For

we

get

X 5 = 25, while 3 X 8 = 24. If we divide each number by its right hand neighbor, we get a series of fractions:

5'

=5

1

2

3

5

8

13

21

34

These fractions are related plants.

When new

leaves

55

89

144

to the

a plant, they spiral around the stem. turns as

one

it

climbs.

The amount

leaf to the next

rotation

is

growth of

grow from the stem

The

of

spiral

of turning from

a fraction of a complete

around the stem. This fraction

is

ohcinjs

one of the Fibonacci fractions. Nature spaces

.^Normal

^J

daisies usually

21 the Fibonacci ratio -r—

34

have

M

e

Count the nu

The construction of the "golden section," with the ratios The lines of the five-pointed star ore |1= |1 £2 (J>.

^

^

broken up

in ratios:

-fj^

show

Living things often

7^=

«!"

^

*t>-

surprising relationships to the

golden section. The diagram of the athlete to the

shows

ratios:

tangles obcd

same

ratios

|£

=g =

-Ji

= -g = -^ ^ -^ =

in

The

the spacing of the knuckles

and

the wrist joint of the average

die leaves in this

way

hand

^

The same

fractions

all

come up

in art.

For

look too long and narrow.

square looks too stubby and

fat.

There

is

between these extremes that looks the

series, the closer

they get to

it.

The

closer to the golden section than

niucli.

ex-

rectangles are equally pleasing to

Some

the eye.

feT"

so that the higher lea\ es

do not shade die lower lea\es too ample, not

Rec-

rectangles."

and wxyz ore "golden

are evident

right

;

To

or —6.

3

=

6,

find out

I

(—3) means start at 3 spaces to the

+

is

number

For example,

of the

is

it

answer.

it is,

multiplier

+ 3. The example (—2) + —2 and move your finger

think of one of

at

to the signs in this

of the multiplier

of the other

to the right.

we

and look

disregard the signs, and multiply as

of die

The answer

is

were working with natural numbers. Then we

t--ai

?

—3, and that

at

do with the other one.

to

sign of the other

and move 5 spaces

integers,

as a multiplier,

of the multipher

T ?

+2, move

So, starting at

You land

left.

subtraction example has an answer.

a positive

as a natural

mo\e

t ?

T

as

5 spaces to the

we

enlarged system,

line.

a negati\e integer,

t

f3

f2

fl

new

The other

follow the old rule of nio\ing to the right. To

add

(

(

?

t

s\s-

which we add a number

by moving along the which

(

the answer. In the system of integers, every

extend the scheme

integer,

+ 2) — + 5). subtracting + 5) is the

to our rule,

T

T

negative integers.

To do addition

on the example we

the natural

rewrite the problem in this way:

to the left of 0, are called

lie

this rule out

in

In this

system, the old natural numbers acquire a

name.

that has the opposite

that lies to the

called the system of integers.

is

dis-

and a minus sign before each number

that lies to the left of 0.

number

the

number sxstem, 2 — 5. The natural number 2 is the same as the positive integer +2, and 5 is the same as +5. So we could not do

us put

let

add

Let us try

sign.

left

with a number attached to each point. To tinguish

a number,

we

suggests an easy

ha\e counted

off

To suhtraet

tion example, accoi'ding to this rule:

picture of the natural nunil)er system

—3 and

answer tion

is

by

look at the signs. ,

so

example

Division

is is

tells

it

attach

+6. With

answer that integers

—

is

in the

it

The

sign of the

us to change the sign

to the answer.

this rule,

Then

the

every multiplica-

system of integers has an

an integer. like

multiplication

done back-

1 1

SYSTEM OF RATIONAL NUMBERS

meaning "one port

This symbol,

by Egyptians to express a combination

in

uted rather thickly along the number

line,

with

each number attached to a point. Does the tional

there So,

ra-

The

number system gi\e us enough numbers number to every point on the line?

no nmnber

is

we ha\ e

to

can attach to

.

.

.,"

found

Over two thousand )ears ago, the mathemati-

numbers can be

another wa\-

in

It

this point.

written.

We

which

in

number rational

can conxert a com-

traction into a decimal fraction

by means I

shown

of dixision, as fraction lie

chain of

an

W'e

3's.

.

.

The

.

.

.,

.

If xxe

decimal

traction

sides has

of

is

an interesting feature

that

mal .49999

\/2.

The diagram abo\e shows how we can number line whose distance

shown

is

equal to

that there

ecjual to

V^-

is

this length.

But

it

So, in the rational

number

.

of

all infinite

ber systen^. In

is

s\stcm.

lier for

we

one an

be

them

.

.

it

as

xvritten as

.

we

get

dix ision, xve find

all.

Each ends up

For example, the deci-

infinite

all

decimals that do not

We get a larger numlier

infinite

decimals, xvhether

they repeat or not. This expanded sxstem,

up

can

repeats the 9 ox er and oxer again.

system by using

can be

no rational number that

.

haxe a repeating pattern.

locate a point on the

from

.

But there are some

We find that the lengtli of the diagonal

34).

in

as a repeating decimal.

It

diagonal by the rule of Pythagoras (see

its

page

aheady understood

we make a square, each of whose length 1, we can figure out the length

does not.

;';

can be xxritten

decimals

infinite

from rational numbers by long

it

like this

\

fraction -^ can

examine the

i

that has an endless

the infinite decimal .15151515.

cians ot ancient Greece

fraction

decimal, too, by xvriting

infinite

.500000.

call a

The

decimal.

i)}fiuitc

as

The

.5.

written as .25. But to xxrite the traction

need the decimal .33333

The

draxving beloxw

in the

can be xxritten as

I

this

made

called the real

num-

system, xve finally get a

num-

decimals,

cmtx' point on

tlu'

is

number

line.

i 0^

.25

7=211.0

==-

4

1.00 I

8

20 20

25 100

wos used

was used a number, as shown below;

clue to this next extension ol the

s\'stem

is

of

fraction.

expand our luunber system again.

to assign a

mon

we

witfi

1

Rotation through plies

Numbers

90®

~

multi-fi^

each number by

'

/

i

^^

multiplying real numbers.

for the Electrician

The by the

electric current

electric

brought

compan\-

to

\ou

produced

is

wire that are rotating in a magnetic stud\- the

Pnd

it

changes

in wires

in coils of

To

field.

in the current, electricians

convenient to use numbers to represent

For example, a rotation of 360 degrees '^ represented by the number 1. A rotation

otations. '.ui

of 180 degrees can be represented b\- the

ber -

1.

other,

(-1) that

Performing two rotations, one after the like

is

:<

(-1)

two

multipKing

=

numbers. Thus,

their

which checks with the

1,

what number can stand

a rotation of 90 degrees? \\'hate\er

multiplied

by

itself,

come out —1, which

is

die

when

multiplied by

a product. This

page 44

is

when

the product should

number

sents a rotation of 180 degrees. ber,

it is.

for

that repre-

But no

itself,

real

can give

num-

—1

as

so because of the rules for

A

times

number

positi\e

e number as a product. number times itself also gi\es a positive number as the product. So no real number times itself can gi\e the negati\e number —1 as a product. This means that there is no real num-

times

A

itself gi\ es a positi\

negati\e

ber that can represent a rotation of 90 degrees.

numnumber system once more. The enlarged number s)stem is called die complex number system. In diis system, the number that stands for a To be able ber

we

to represent e\"er\- rotation b\- a

ha\ e to extend the

90 degree rotation

is

represented by the letter

and has the property

360 degrees.

In this scheme,

is

fact

rotations of 180 degrees are like a

single rotation of

it

num-

The number

as a product.

gixes

itself

complex number plus

i

that

/

/

= —

is

we multipK-

A

/,

Every

number

rotation of

l \/2 + il \ 2. When number by itself, and make use X = \/'2~X \/2^= 2, while

written as

this

of the fact that

we

1.

written as a real

times another real number.

45 degrees

—I,

is

X

find that the product

/

is

i.

i

This checks

widi the fact that two rotations of 45 degrees

Two 45*

"

^rtt

one

combined are SYSTEM OF COMPLEX NUMBERS

To picture (4

M3»

(-2 + 2i)

+ 3i)

on a

line.

ecjiial to

one rotation of 90 degrees.

numbers we used the points

real

To picture complex numbers, we need

the points of a plane.

all

in this

rotations equal

90** rotation

wa\

.

First

draw

The

picture

plane and put the real numbers on line the (ixis of reals.

is

formed

a horizontal line in the

Now

Call this

it.

rotate the line

90 de-

The rotation multiplies number by the number In this way,

grees counterclockwise. -3

-2

-1

e\ery real

we

i.

attach to each point on the vertical line

through

a real

number multiplied by

/.

These

products are called iiiiaginanj numbers, and the \ertical line

Now,

in

in the plane, tical line

is

called the axis of imaginaries.

order to attach a

draw

number

to

a horizontal line

any point

and

a \er-

through that point. The vertical line

points out a real ninriber on the axis of reals.

The horizontal line points out an imaginary' number on the axis of imaginaries. The sum of these two niuubers is the complex number attached to the point.

Now we

ha\e the complete nest of

fi\e

num-

ber systems. Listing them from smallest to largest,

with each number s\stem lying within the

next larger one, the\' are: natural numbers, tegers,

rational

numbers,

real

in-

numbers, and

complex numbers.

page 45

Number Systems

Miniature Now

we ha\e

that

systems,

many

it

will not

more.

Some

seen fi\e distinct

number

be surprising that there are

of

them are

little

toy

number

systems, containing only a few numbers. shall find

natural

number

A System Some

We

examples of them hidden within the system.

with

Two Numbers

natural numbers, like 0, 2,

so on, are divisible

by

4, 6, 8,

and

They make up the

2.

family of even numbers. Those that are not

by

divisible

and so

2, like 1, 3, 5, 7,

6

4

on,

12

10 I

14

1

i

7

,

^

EVEN 5

make

9

ODD

up the family of odd numbers. We can use these two families to build a number system that has only two numbers in family

it.

In this system, each

tliought of as a single

is

have a way of adding two

number, and

we

families, or multi-

plying two famiUes. To add two families, pick

any member

Then

see

of

each family, and add them.

what family the sum belongs

to.

For

example, to add the odd family to the odd family,

add an odd number

result

is

plus the

to

an odd number. The

an even number. So the odd family

odd family equals die even

the same way,

we

find that

family. In

odd plus even equals

odd; even plus odd equals odd; and even plus

even equals even. These

marized

results

can be sum-

in the following addition table.

of

1

sists

when you of

divide by

numbers hke

3.

2, 5,

remainder of 2 wlien you or multiply these families for the

odd and e\en

di\ ide In' o.

families:

Following these

sum

rules,

Number System

We

rules

add used

or multiply

families,

and then

or product l^elongs

we

ivitli

Add

con-

that give a

by the same

any representatives of the find tlie family the

The 2 family S, etc.,

to.

get the tables below.

Only Three Numbers

.n.

Snowflake

crystal

No

coiner.

pohgon

regular

would

sides

more than

witli

contain

more than 120 degrees, and

more

them could not possibly

of

six

do, because then each angle \vt)uld

the 360 degrees around a point. So the only regular polygons that

may

three or

together in

fit

we

see that

ser\e as cells

are equilateral triangles, s(|uares, or regular hex-

Of

agons.

these three possibilities, the regular l*ing

like

It is

a small spaee. In this code,

mean

to

"times,

the letter

We

.v.

show

no SNinbol between them. In

side with

b\-

code, for,

•

(I

multiplication by using

by writing the midtipliers side

a dot instead, or

h means: "the number that

number

midtiplied by the

that

stands

/;

this

stands

ci

tor.

When

the

same multiplier

again,

we

use the same short wa\' of writing the

product that was used

used o\er and o\er

is

nnml^ers and

for scjuare

When we

cubic numbers on pages 19 to 20. write

.v',

we mean

Some

.v.v.v.v.

of the rules

described in earlier sections of this book are

shown

in

Here true:

is -+

.v

code on the preceding page. a statement in code that

+

because 7 3.

A

=

2

not true

5. It is

2

is

not

statement like

is

An X

+

stands for

if

number

to find the

a true statement.

it

The

equation resembles a balance scale. 2

is

7,

stands for

.v

called an ccjiuition.

To solve the equation means which makes

not alwa\'s

is

.v

true

5. It is

this

if

supposed

balance the 5 the w^ay

to

equal weights balance on a scale.

one weight on a

we change

If

we can make

scale,

the weights

balance again by changing the other weight in the

same

wa\'. This

how we can

a hint on

is

solve an equation: Simply

change

lioth sides of

the equation in the same wa\'. Since 5

same .1

+

+

as 3

2

=

+

3

2. If

they will

sides,

way we and

o.v

w^e get

Can you

.v

.v

we

take

and then

+

2

=

is

the

5 says:

away 2 from both

balance each other. In this

=

3

is

=12, we

=

.v

the answer.

To

solve

di\ide both sides

b\'

4 as the answer.

sol\e the equation

find the answer, tion,

still

find that

the equation 3,

the equation

2,

add 4

di\ ide

to

3.v

—

4

=

S? To

each side of the equa-

each side bv

3.

page

.5

Number

Your

Space

in

Many

big cities are divided into blocks

streets

running

crossing

them

can locate

number

nunrbers: the

right

tlie

we

b\' tlie cross b\'

we

if

corner by mentioning two of the street

and the num-

map

and the second

same

classroom

b\"

number and in

can describe the corner marked using the pair of numbers

agree that the

In the

is

first

the street

is

front

we can

locate any seat in a

mentioning two numbers: the row the seat number. In the classroom

and the

back.

to

The

the class, "Raise >our

and

seat

seats are

who

numbered from numbered from

teacher has

hand

number add up

the pupils

(3,4),

number

the avenue number.

wa\'.

the picture, the rows are

right to left,

you

at right angles. In these cities

an\" street

ber of the avenue that cross there. In the

on

In-

one direction, and axenues

in

if

just

said

to

your row number

The

to 5."

locations of

raised their hands are gi\en

by

the pairs of numbers, (1,4), (2,3), (3,2), and (4.1),

where the number. s

first

we

If

number let r

stand for seat number,

tions in the table

in

each pair

is

the

row

stand for row number, and

we can

list

these loca-

shown on the blackboard.

\\'e

can also describe these locations by the equation: r

+

=

s

5.

Notice that the pupils whose

hands are up are arranged equation

is

line. Also,

the fine

pairs described

This

made

is

in a straiglit line.

The

a description of the locations on this is

a picture of the

number

by the equation.

an example of an important discox er>-

in the

seventeenth centur\' by the great

French mathematician and philosopher Rene Descartes.

An

equation with two unknowns can

be i^ictured by means of a

page

.52

line

(straight

or

4

X

i^:

5tb

Avenue

4th

Avenue

^-' 111.-

¥'

3rd Avenue

fe;

ft:

m «5

.».

y 0)

0) a,

0) 0)

i^ _c

i^

(^5

^

-D

t^

CN

^ f^

^

^

*.

12

Avenue

2ncl

(U 0)

J5 st

1

^ "^

^

Avenue

7

6

4

3

8

furved), called a unipli. Also, e\er\' line

means

scribed by

of an equation.

mathematics that grew out

is

de-

The branch

of

of this disco\ery

is

called analytic gcomctnj.

The connection between a line and its equais usually shown in this way: The paper on which the line is drawn is di\ ided into squares

tion

by two streets

sets of lines that cross

each other,

and avenues. To number these

pick out one

line in

each

and

set

call

Then we number the away from each axis,

lines,

like

we

the zero

it

by count-

line, or axis.

lines

ing boxes

in l)oth direc-

tions. In

one direction

we

attach a plus sign to

each number. In the opposite direction

minus

a

sign, so

each intersection bers, telling

down

or

describe

X Y

can

described far

from the axes. .v

number.

the y number. points

tell

that

it is

apart.

b\' a

pair of

right or left

We call We call

NUMBER OF INCHES

IN

WIDTH

NUMBER OF INCHES

IN

LENGTH

X

;

and up left

up or down Iractions

between the

=

-

the

use

Then num-

the right or

Numbers with are

we

them

=-'

Y if

is

you how

numl)er the

number

we

lines.

You con draw an

ellipse with

two pins or

tacks,

^_J^ *

a drawing board, o pencil, and a loop of string I.'—

SUN (FOCUS POINT NO. 2)

I

George Washington bridge

he

City

is

sound coming from one focus and

back

to the other. In

tliis

was scattered and weakened from one focus through

tlie

trated at the other focus its

reflects

The path

it

way, the scjund that as

it

a parabola.

spread out

chamber

and restored

is

way

Here are some more places where we a glass of water, the

of the water surface

a

is

an

ellipse.

boundary

York

is

The

road-

cross-section of a

parabola. is

New

a parabola

the roadway

a parabola.

and a

air

is

used

The same

in solar fur-

naces to catch sunlight faUing on a large area

find

.^ tilt

is

is

type of parabolic reflector

these cur\es:

you

a parabola, too,

searchhght reflector

almost

original loudness.

When

is

roadway

thrown through the

The cable supporting

of a suspension bridge

concen-

to

ot a ball

in

a suspension bridge. Each of the two

cables that support the

f^

and concentrate

it

The shadow on

'T|i lampshade

is

in a small spot.

the wall

made by

a cylindrical

a hyperbola.

LIGHT

SOURCE

PARALLEL RAYS

iFOCUSl

PARALLEL RA>^

A hyperbola

%

cone cut

In

a searchlight, a parabolic reflector

used to send out parallel rays.

In

a

parallel rays

however,

it

is

used to catch

and bring them

to

by the

is

formed where the

from the lamp shade

flat

a focus

many domes

per at

Fi,

1

\

is

i^'Si

surface of the wall

|^

is

reflectIn

ing telescope,

of light

such as

this

one, a whis-

after being reflected twice

by the dome, can be heard clearly

at Fj

3Bf

Shadow Reckoning ©^^3^3

Oi

^^^3

A If

you stand out

you

cast a long

3

15

units

units

in the

sunshine before sunset,

shadow. Your shadow

picture of you, rolled

stretched in the

same way.

fraction -^

We

.

the shadows of

same way, you shadow

leads to a object

it

way

you are 5

feet

is

by writing the

these lengths

get the

same

ratio

when \ou

of anything with the length

casts at the

same time. This

measuring the height of a

by measuring

ratios -^

Since

time that a 2-foot stick casts a 3-foot shadow,

how

its

shadow

and

{_^

high

of

its

.r.

14,

you get

you get

to

2.

70. If

If

they are equal us that sides

=

3*.v

by

we

3,

of the tower

Problem:

ratios.

2

X 8

FEET

7,

you get equal

X

=

fEET

?

by the denominator of the other. If

is §.

^=

3.v .v

§.

to the length

For the

stick,

To show

The

that

rule tells

= 30. Dividing both = 10. So the height

feet.

flagpole casts a

shadow,

casts a 2-foot

tall

\ou multiply the 10 by the

write

find that

10

If a

fact

you multiply the 5 by the

we

shadow

.

shadow

that

is

8 feet long at the same time that a 3-foot stick

are equal, because they

This rule helps us do shadow reckoning.

a

tower casts a shadow 15 feet long at the same

2

page 56

is

it

get die ratio f^

2*15, or

instead.

70, too. In ecfual ratios,

ratio

we

the ratio of height to

products tchcn you multiply the numerator of

one

When we compare

shadow,

3_FEET

both reduce

the tower? Let us call die height of

is

the tower

objects are stretched in the

all

of

tall,

10 feet

Let us note a simple fact about equal

The

at the

call this fraction a ratio.

compare die height of the

If

some moment

at

we compare

long,

you look

If

you see that they are

objects,

and your shadow

Hke a

painted black, and

flat,

stretched out on the ground.

shadows of other

is

FEET

how

high

is

the flagpole?

cr^'^

w

Vibrations, The world

is

lull

mo\e, they move

xibrations

ot

disturbances in space.

When

rolling \

away from

its

which create

these disturbances

form of waves.

in the

a ship glides across a lake,

it

prow.

by

air.

When we

To study these

different kinds of waves, sciena

model wa\e

that

is

produced with the help of a turning

wheel. They find that the simplest

plicated

ot the

waxes

model wave. The more com-

are just like the

waves are often made up of several

simple waves that are combined.

Let us see the

how

model wave.

of the wheel,

a turning First

we

and watch

with a steady speed. tion to the point

Wc

it

wheel can produce

shall

wheel turns

pay special atten-

on the spoke that

P.

As the wheel

above the

this point is

called

is

sometimes

level of tlie center of the wheel,

sometimes below

down

diagram

turns, this point

it.

with a steady rh\ thm. This up and

motion

we

the vibration

is

and

keeps moving up and

It

shall use to

down

form a

model wave. wheel makes one

as the

spoke

horizontal,

is

P

is

center of the wheel. So level

is

0.

As the wheel

at first increases.

full

turn.

When

the

on the same level as the its

height above this

turns, the height of

The height

is

P

greatest after the

wheel has turned 90 degrees, and the spoke

is

Then the height decreases as the wheel some more. When the amount of turning

vertical.

pick out one spoke as the

of the wheel. In the

Let us trace out the motion of the point P

atom.

compare them with

easily

speak, the

sound waves

Light consists of waves sent out

\'ibrations within the

tists

When

sends water waves

ibrations of our \'ocal cords send

out into the

Wheels and Waves

is

at the

rim

turns

reaches 180 degrees, the spoke again,

and the height

P begins

to fall

the wheel.

It

of

P

below the

reaches

its

is

is

horizontal

once more. Then

lev el of

the center of

lowest position

i

when

b

2

U\J

,\

100

N OR NORTH POLE

I

NORTH POLE

/ '^fk:^,

^

LATITUDE CIRCLE

:

/

located.

Home

Our

the Earth

We

can imagine the Greenwich merid-

through which earth

is

We

a sphere.

of any point on

describe the position

siuface by

its

numl:)ers, called the latitude

the point.

They

street

fix

of

and axenue that

numbers

that the

cross each otlier

of a

the

fix

position of a street corner in a big city.

They

identify the place as the intersection of a

lati-

top.

The earth is spinning like a The North Pole and South Pole lie on the

axis

around which

Through every point on the

it

A

spins.

circle

The

ecjuator.

hoops around a

the

same

to

latitude circle are the

of degrees through

would have

move from

on

barrel. Points that are

same distance

from the equator. The latitude of

the earth

called the

latitude circles girdle the earth

like

number

around the

is

a point

which

is

the

a radius of

to turn, north or south,

the equator to the latitude circle.

Meridians The meridians are half-circles that :

join

the Nordi Pole

meridian

is

to the

South Pole. The zero

the one that passes through Green-

wich, England, where a na\'al obserxatory

is

called

is

earth's surface,

one meridian. In one direction,

it

Nortli Pole. This

we

When you can you that a

A

Latitude Circles:

earth, halfway between the poles,

of degrees

to turn

except the North and South Poles, there

is

the direction

tell

which way

is

magnetic compass it

only

is

leads to the call

stand outdoors where you

tion for you, but

tude circle and a meridian.

The number

would have

it

the longitude of that meridian.

two

of

the position of a place on the

same way

earth in the

means

and longitude

MERIDIAN

ian turning east or west to reach the position

of any other meridian.

The

i

North.

how

live,

north? You

may

think

answer

this

ques-

will

won't.

magnetic compass,

e\'en

if it

works without

interference, doesn't point to the North Pole. It

points to the magnetic north pole,

somewhere masses of

in

Hudson Bay.

iron, like die steel

interfere with

its

wrong. The compass nordi only

To

is,

frame of a building,

is

you know what the

so

from

usually

error of the

you can subtract the

of doing this

is

useful for finding true

find the direction of true north,

to get help

way

if

error.

you have

the rotation of die earth. is

is

nearliy

operation. So the direction

pointed out as north by a compass

compass

which

Besides,

to use a

vertically into the ground.

shadow

The

stick,

One

driven

rotation of the

earth makes the sun seem to rise in the east,

^

'^

MORNING SUN -^

v<

AFTERNOON SUN

)?'

^

-^

SHADOW

and

cross the sky,

changes

As the sun

set in the west.

shadow

position in the sky, the

its

shadow

shortest,

is

it

of

When

the stick turns and also changes in length. the

STICK

points to true north.

This happens at about noon.

The noontime shadow changes slowly. So

it is

difficult to locate

length

in

\

ery

north accurately

by watching the noontime shadow. You can do it

more

and afternoon, when

the morning

length

more

rapidly.

shadow

stick, stick.

of the

shadow,

noon,

when

circle.

True north

as

the

end is

make

in the

fairly

in

changes

of a rope

a circle on the

Locate the positions

morning and of the

in the after-

shadow

lies

on the

halfway between these posi-

and can be located with

shown

A

in the

it

With the help

ground around the

tied to the

tions,

shadow

accurately by watching the

a rope

and stakes

about the poles inspired

A

well-known puzzle:

this

hunter lea\es his tent and walks one mile

south and then one mile east.

and then heads north with one mile, he arrives back the color of the bear? to

He

it.

.shoots a

bear

After tra\eling

at his tent.

What

is

The answer was supposed

be "White," because whoe\er invented the

puzzle thought

tlie

only place where a path that

goes one mile south, then one mile east, and then one mile north could form a closed loop

is

near the North Pole, where polar bears are white. (See the diagram.)

It

could also happen

near the South Pole, as shown lielow, except

drawings.

accurate and quick

way

of locating

that there are no land

mammals

in Antarctica.

north uses a small shadow stick and a watch.

Hold the watch

level,

w

ith a diin stick

standing

vertically o\er the center of the watch.

the watch until the hour

hand

is

shadow. Then the direction of north will be

way between

the

Locating north

'%

Turn

under the half-

shadow and the twelve. is

easiest at the

There every direction Pole, every direction

is

is

%

South Pole.

north! At the North

south. This peculiar fact

\ -*•

page 60

SOUTH POIE

'^^.

A JViV^fl^S"^'"^

are examples of such great circles on the earth.

Distances on the Earth

In the days before the airplane, the usual

On

a

flat

two points flat,

suit ace, the shortest path is

eartli,

The

a straight Hne.

so there are

between

earth

no straight paths on

it.

is

On

not the

the shortest path between two points

is

of going to

ship across the Atlantic Ocean. Ship lanes ran

appro.\imately east and west, and

known

as

around

this

we

that

ica

east-west

way

ha\'e airplanes,

of tra\eling.

by following great

and south than

is,

the less

it

larger a circle

cur\es. So the shortest path be-

tween two points on

tlie

earth

is

along the

largest circle that joins them.

Such a

called a great circle. Meridians

and the equator

circle

is

strated

possible to traxel

is

it

east

by means

circle paths.

maps

of

and west. They are demon-

like

North Pole. The circumpolar

seemed

Many

more nearly north

of a circumpolar

shows what the earth looks

places that

built

Now

between Europe and North Amer-

these great circle paths run

The

we used maps

Mcrcator maps which were

shorter routes

the one that curves the least.

way

Europe from North America was by

to

be

map

that

from abo\ e the

map shows

far apart

that

on the old

are really quite close to each other.

The man on

Polaris.

on

the equator sees Polaris

For the others, the

his horizon.

line to Polaris

makes an angle with the horizon. The fmther north the

man

measuring

this

he

Navigation

is,

the larger the angle

angle

tells

above the equator.

is

degrees, then he

him how If

knows he

is

is.

So,

far north

the angle

is

60

somewhere on the

latitude circle 60 degrees above the equator.

A

iia\igat()r lias

One The

is

two kinds

where

to find ont

other

is

of

problems

lie

on the earth.

is

what course

to figure out

Now his ship

circle.

problems are a compass,

tools for soK'ing these

after

way

and an almanac. His compass,

he corrects for is

rectly.

error, tells

its

him which

north, so he can measure directions cor-

With

his

height of the sun, horizon. His clock

se.xtant

he can

moon, or

tlie

tells

him

wich, England. His almanac

measme

a star

tlie

time

tells

to find out

This circle

is

where

lie

on that

is

crossed by the meridian that

passes through Greenwich, England. His clock

should take to go from one place to another. His

a se.xtant, a clock,

he has

to solve.

tells

him the time

at

Greenwich. His almanac

him what the sky looks like there. The sky he sees above him looks different. Compared to tells

the sky as seen from Greenwich,

it

looks as

though

it

were turned through an angle. The

amount

of

tliis

the

above the in

Green-

him how the

sky looks at Greenwich any day of the year, any

time of the day. With

all this

information, he

can figure out the answer to his problems. Let us see

!^6

axis.

The

how he

The

the earth.

eartli

can locate his position on is

a sphere spinning on

its

axis points almost directly to Polaris,

the earth he

is

turning

tells

him how

tar

around

from the place where the merid-

5.

the North Star.

The diagram below shows men

in different positions

on the earth looking

at

ian

through

circle.

Greenwich

This information

crosses

his

latitude

fixes his position.

NORTH POLE

-ariil

60

30 EQUATOR /60

page 62

EQUATOR

Mathematics

for

a Changing World We

%

abont. Animals and plants

When

and numbers.

size

know

often have to It is

steady.

Suppose a car

easy to calculate

moving

the rule: Speed

We

A

height, the longer

4

a

movie

of

the stone

feet

its fall.

at

is

it

by using

onds

64

the faster

it falls.

cliff

The movie shows

any time. After

is

How

feet.

shown

From stone

^5

onds,

16

X

this rule falls in

for 6'

TIME IN

500 feet

1) 6

S

=

far

How

and take

where

us

second, the

1

gets in 3, 4

we can

it

has

and 5

sec-

below. This table

Distance

=

16

figure out

(timef.

how

far the

any number of seconds. In 6

example, 16

OF FALL

SECONDS

it

in the table

gives us the rule:

X

6

too.

in this case?

stone has fallen 16 feet. After 2 seconds, fallen

3

from a great

falls

Suppose we drop a stone from a

LSECONDS

300

falls,

it

speed

its

find

is

speed

its

speed can change,

For example, when something

find

Then

in

= Distance -^ Time. But speeds

are not always steady.

do we

which they

120 miles

at a steady speed.

in

we

the speed

if

it

travels

must be 40 miles an hour. 200 feet

grow

things change,

die speed with

change.

hours,

change. People and

a \vt)ild ol

in

li\'e

move

things

the ;:

6

distance

=

576

feet.

sec-

would be

Distance the stone

Dui

iiig tlif 1st

fell

second the stone

fell

16 iwt.

— 16 = 48 feet. — 64 = 80 feet. — During the 4th second, the stone fell 256 144 = 112 feet. During the 5th second, the stone fell 400 — 256 = 144 feet. 2nd second, the stone

fell

64

During the 3rd second, the stone

fell

144

During

Now

tlie

let

us see

how

far the stone falls during

one second alone. During the 16

fell

64

During

feet.

feet.

To

find out

second alone,

we

take

in the first second:

two seconds

tlie first

how 64

far

second,

first

away the 16

—

16

=

feet

48.

it fell

second

in the

it fell

it

The

it fell

table

shows the calculation for each of the other

sec-

onds. So

we see

tlie first

second, 48 feet during the second sec-

that the stone fell 16 feet during

ond, 80 feet during the third second, and so on. Its

speed increases as

rough answer. During the third second, the

stone

fell

80

feet.

three seconds,

But

we know

its

So our estimate

that our estimate

Distance fallen

is

that, after

speed was 80 feet per second.

cause the speed was changing

in

is

only an aver-

age speed. The stone actually moved more slowly it

than

tliat at

moved

ond.

We

the beginning of the second, and

faster than that at the

end

of the sec-

can get a better estimate by getting

all

is

wrong, be-

through that

almost 3 seconds

its

average speed during a shorter period of time,

when

the speed

had

less of

For example, during the stone the

fell

first

16

X

2i

X

2i

three seconds

a chance to change.

2i seconds, die

first

100

feet, or

144

it fell

feet.

feet.

During

So during

the last half-second of this three-second

it falls.

How fast is the stone falling after three seconds? We may use our table to get an estimate, or

second. Eighty feet per second

the stone

fell

144

our estimate of

its

—

100

feet, or

the same kind of calculation,

three-second

fall,

Now is

88 feet per second.

average speed during the its

feet.

speed after 3 seconds

feet per half-second, or

of

44

fall is

average speed during the

we

last

find that

its

quarter-second

92 feet per second. last

44 B>'

Its

eighth of a second

'^^^;\

NEWTON

ISAAC

SIR

Kepler discovered the laws of planetary motion.

obey

Earth satellites

One

similar lows.

of

that the line joining a satellite to the earth

over equal areas

is

sweeps

equal times. Newton, using

in

calculus, derived the law of gravitation ler's

them

from Kep-

laws. In the orbit shown, the satellite travels at

varying speeds. Distances A, to Bi, A, to Bj, Aj to Bj,

and A4

covered

to Ba are all

in

the

same amount

of time

B,

is

94 feet per second.

Its

a\erage speed during

the last sixteenth of a second

ond. Eacli

new

estimate

is

is

95 feet per

wrong,

wrong than the one before

We

it.

sneak up on the correct answer.

be 96

feet per second.

It

of sneaking

turns out to

It

Newton and

up on the

shows that the speed of a

times the

number

of seconds

it

has

body

is

3

>C

satellite,

began

When

flying

Sputnik, the

=

around the earth,

used calculus to figure out

how

moving. They needed calculus to do

Sputniks speed was changing culus

omers,

is

first

all

fast it

32

This

fallen.

The branch of mathematics that uses method of Newton and Leibnitz is called ferential calculus.

the effi-

right answer.

falling

checks with our result, because 32

tists

less

gradually

philosopher Leibnitz inxented an

method

cient

sec-

is

it

During the seventeenth

century, the English scientist

German

l)ut

96.

die dif-

earth scien-

it

was

because

the time. Cal-

used every day by physicists, astron-

and engineers whenever they study

changes in which the change

itself is

A

changing.

rocket launching a satellite

page 65

and

Finite when we set,

we

Infinite

count the objects

ing each

number

To count the

\

owels O,

3;

/,

say that the set

that counting

it

comes

to

in the set.

we we

in the alphabet,

4;

run througli the complete

we

an object

in turn to

set of

say, "A, 1; £, 2;

way,

in a collection or

say the natural numbers in order, assign-

V, 5." Because

set of is

\

owels in

an end.

this

This means

finite.

We

can try to

count the set of even natural numbers by assigning natural numbers to them, in this wa\': "2, 4, 2; 6, 3; 8, 4;

"

and

1;

so on. In this case, the count-

ing never comes to an end, because no matter

how many even numbers we count, always more left oxer. So we say that e\en numbers

is

infinite.

Infinite sets differ

from

finite sets in

portant way. If you remo\e a set,

what

is

left will

For example, the

there are

the set of

not

set of

match the complete

member

match the

\owels

c,

/,

an im-

of a finite

original set. o,

n does not

set of vowels, a, e,

ijjuj>fcw