463 84 12MB
English Pages [96] Year 1960
THE GIANT GOLDEN BOOK OF
EXPLORING THE WORLD OF NUMBERS AND SPACE
INCLUDING SETS
NUMBER
IndiidtUd
raV
MATH Number Systfiiis
'•x
I
/•
y
/
\
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; 6014879
©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 wellaimed
A man
man\? At "
first this
up with the words
question used to get mi.\ed
cuestion "\\'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
"sevensquared" 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
"twocubed, 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

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
halffuH?
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 halfpast
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 "ropestretching." 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
sixsided
divided into four triangles, so to 4
X
ISO degrees,
to 2
A fivesided 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 fivepointed 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
tai
?
—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®
~
multifi^
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
: