: two three . , . infinity Illustrated $5.00
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THE OF THE
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BIOGRAPHY OF THE EARTH:
Its Past, Present, -and Future
Illustrated
Conijxiss paperbound
'. . . '!'< li A ' Oscillating story of the origin,
sa'-iucy, ami ;^1 ) < : < < i "'^ of the earth . . . has
sekLi t Von told w. 1 '^"i ; 'ive and in
genuity u' ,>'>rase."--KiiPU' ' < u niiKU,
Harvard Uni/ ?-' 'ty
THE VKING PRESS - NvV YORK
IACKE1' DESL-V I,". UOBIK lU *!
APR 6 1932
500
Gam*
One two three . . , Infinity
liiSSl^^
iiiiiSIH^
jllllillllllJ
Iliiililllllllfill^
One two three ... infinity
There was a young fellow from Trinity
Who took V~~
But the number of digits
Gave him the fidgets;
He dropped Math and took up Divinity.
OTHER BOOKS BY GEORGE GAMOW
BIOGRAPHY OF THE EARTH
(Compass edition, revised)
THE BIRTH AND DEATH OF THE SUN
THE CREATION OF THE UNIVERSE
(Compass edition., revised)
A PLANET CALLED EARTH
1 2 3 ... CO
One two three ...
infinity
FACTS & SPECULATIONS OF SCIENCE
by George Garnow
PROFESSOR OF PHYSICS
UNIVERSITY OF COLORADO
ILLUSTRATED BT THE AUTHOR
THE VIKING PRESS * NEW YORK
TO MY SON IGOR
WHO WANTED TO BE A COWBOY
COPYRIGHT 1947, 1961 BY GEORGE GAMOW
ALL RIGHTS RESERVED
REVISED EDITION
PUBLISHED IN 1961 BY THE VIKING PRESS, ING.
625 MADISON AVENUE, NEW YORK 22, N.Y.
SECOND PRINTING OCTOBER 1962
PUBLISHED SIMULTANEOUSLY IN CANADA BY
THE MACMILLAN COMPANY OF CANADA LIMITED
ACKNOWLEDGMENT IS MADE TO PRENTICE-HALL, INC., FOR PERMISSION
TO REPRODUCE THE DRAWINGS ON PAGES 136-37, FROM Matter, Earth
and Sky BY GEORGE GAMOW. 1-958, PRENTICE-HALL, INC., ENGLE-
WOOD CLIFFS, N.J.
PRINTED IN THE U.S.A. BY THE COLONIAL PRESS INC.
t:f ihe time lias come," the Walrus said,
"To talk of many things". . .
LEWIS CABBOLL, Through the Looking-Glass
Preface
... of atoms, stars, and nebulae, of entropy and genes; and whether
one can bend space, and why the rocket shrinks. And indeed, in
the course of this book we are going to discuss all these topics,
and also many others of equal interest.
The book originated as an attempt to collect the most inter
esting facts and theories of modern science in such a -way as to
give the reader a general picture of the universe in its micro
scopic and macroscopic manifestations, as it presents itself to the
eye of the scientist of today. In carrying out this broad plan, I
have made no attempt to tell the whole story, knowing that any
such attempt would inevitably result in an encyclopedia of many
volumes. At the same time the subjects to be discussed have been
selected so as to survey briefly the entire field of basic scientific
knowledge, leaving no corner untouched.
Selection of subjects according to their importance and degree
of interest, rather than according to their simplicity, necessarily
has resulted in a certain unevenness of presentation. Some chap
ters of the book are simple enough to be understood by a child,
whereas others will require some little concentration and study
to be completely understood. It is hoped, however, that the lay
man reader will not encounter too serious difficulties in reading
the book.
It will be noticed that the last part of the book, which discusses
the "Macrocosmos," is considerably shorter than the part on
"Microcosmos." This is primarily because I have already dis
cussed in detail so many problems pertaining to the macrocosmos
in The Birth and Death of tJie Sun, and Biography of the Earth*
and further detailed discussion here would be a tedious repeti-
1 The Viking Press, New York, 1940 and 1941, respectively.
vi Preface
tion. Therefore in this part I have restricted myself to a general
account of physical facts and events in the world of planets, stars,
and nebulae and the laws that govern them, going into greater
detail only in discussing problems upon which new light has been
shed by the advance of scientific knowledge during the last few
years. Following this principle I have given especial attention
to the recent views according to which vast stellar explosions,
known as "supernovae," are caused by the so-called "neutrinos/"
the smallest particles known in physics, and the new planetary
theory, which abolishes the currently accepted views that planets
originated as the result of collisions between the sun and some
other stars, and re-establishes the old half-forgotten views of
Kant and Laplace.
I want to express my thanks to numerous artists and illustrators
whose work, topologically transformed (see Section II, Ch. Ill),
has served as the basis for many illustrations adorning the book.
Above all my thanks are due to my young friend Marina von
Neumann, who claims that she knows everything better than her
famous father does, except, of course, mathematics, which she
says she knows only equally well. After she had read in
manuscript some of the chapters of the book, and told me about
numerous things in it which she could not understand, I finally
decided that this book is not for children as I had originally
intended it to be.
G. GAMOW
December 1, 1946
Preface to the 1961 Edition
All books on science are apt to become out of date a few years
after publication, especially in the case of those branches of
science which undergo rapid development. In this sense, my book
One Two Three . . . Infinity, first published thirteen years ago,
is a lucky one. It was written just after a number of important
scientific advances, which were included in the text, and in order
to bring it up to date relatively few changes and additions were
necessary.
Preface to the 1961 Edition vii
One o the important advances was the successful release of
atomic energy by means of thermonuclear reactions in the form
of H-bomb explosions, and the slow but steady progress toward
the controlled release of energy through thermonuclear processes.
Since the principle of thermonuclear reactions and their applica
tion in astrophysics were described in Chapter XI of the first
edition of this book, man's progress toward the same goal could
be taken care of simply by adding new material at the end of
Chapter VII.
Other changes involved the increase in the estimated age of our
universe from two or three billion years to five or more billion
years, and the revised astronomical distance scale resulting from
explorations with the new 200-inch Hale telescope on Mount
Palomar in California.
Recent progress in biochemistry necessitated re-drawing Fig
ure 101 and changing the text pertaining to it, as well as adding
new material at the end of Chapter IX concerning synthetic pro
duction of simple living organisms. In the first edition I wrote
(p. 266): "Yes, we certainly have a transitional step between
living and non-living matter, and when perhaps in no far-
distant future some talented biochemist is able to synthesize a
virus molecule from ordinary chemical elements, he will be jus
tified in exclaiming: T have just put the breath of life into a
piece of dead matter!' " Well, a few years ago this was actually
done, or almost done, in California, and the reader will find a
short account of this work at the end of Chapter IX.
And one more change: The first printing of my book was dedi
cated "To my son Igor, who wants to be a cowboy.'* Many of
my readers wrote me asking if he actually became a cowboy.
The answer is no; he is graduating this summer, having majored
in biology, and plans to work in genetics.
G. GAMOW
University of Colorado
November 1960
Contents
PART L PLAYING WITH NUMBERS
I. Big Numbers 3
n. Natural and Artificial Numbers 24
PART II. SPACE, TIME 6- EINSTEIN
m. Unusual Properties of Space 41
rv. The World of Four Dimensions 64
v. Relativity of Space and Time 84
PART III. MICROCOSMOS
vi. Descending Staircase 115
vn. Modern Alchemy 149
vm. The Law of Disorder 192,
ix. The Riddle of Life 231
PART IV. MACROCOSMOS
x. Expanding Horizons 269
xi. The Days of Creation 298
Index 337
viii
Illustrations
PLATES
FOLLOWING PAGE 164
i. Hexamethylbenzene Molecule
ii. A. Cosmic Ray Shower
B. Nuclear Disintegration
m. Transformations of Atomic Nuclei
A. A Fast Deuteron Hits Another Deuteron
B. A Fast Proton Hits Boron Nucleus
c. A Neutron Breaks a Nucleus of Nitrogen
iv. Uranium-Nucleus Fission
v. A. and B. Photomicrographs of Fruit-Fly Chromosomes
c. Photomicrograph of Fruit-Fly Female Larva
vi. Particles of Tobacco-Mosaic Virus
vn. A. Spiral Nebula in Ursa Major
B. Spiral Nebula in Coma Berenices
vm. The Crab Nebula
ILLUSTRATIONS IN TEXT
PAGE
1. An ancient Roman tries to write "one million/* 5
2. Grand Vizier Sissa Ben Dahir asks his reward. 8
3. The "End of the World" problem. 10
4. An automatic printing press. 12
5. An African native and Prof. G. Cantor comparing
their counting ability. 15
6. The number of points on a line. 21
7. The number of points in a square. 21
8. The first three infinite numbers. 23
9. Eratosthenes' "sieve." 29
10. Real and imaginary numbers. 34
11. Treasure hunt with imaginary numbers. 30
12. Co-ordinate systems. 42
13. A subdivided sphere transformed into a polyhedron. 44
14. Five regular polyhedrons and one monstrosity. 46
15. Proof of Euler's theorem. 47
x Illustrations
PAGE
16. Two rivals of the ordinary cube. 49
17. Topological maps. 50
18. Double apple eaten by two worms. 54
19. A double apple turned into a doughnut. 56
20. Inside-out universe. 58
21. Right- and left-hand objects alike but different. 59
22. Two dimensional "shadow-creatures" in a plane. 60
23. Surface of Mobius, and Klein's bottle. 62
24. Squeezing a three-dimensional body into a two-dimen
sional surface. 64
25. Two-dimensional creatures look at shadow of three-
dimensional cube. 65
26. A visitor from the Fourth Dimension! 67
27. Plane projection of the globe. 38
28. Space-time cube. 70
29. Space-time span of a man. 71
30. World-lines of sun, earth, and comet. 72
31. Galileo measures velocity of light. 74
32. An "event" in the four-dimensional world. 79
33. Prof. Einstein as magician. 80
34. Axis-cross of two space-coordinates. 84
35. Four-dimensional axis-cross. 85
36. Michelson's apparatus. 93
37. Bodies distorted by space distortions. 97
38. Universal shortening of moving objects. 99
39. Two-dimensional scientists check Euclidian theorem, 103
40. A. Measuring curved space. 106
B. Measuring angles formed by beam of light. 107
41. Two-dimensional analogy to Einstein's curved space
theory. 109
42. Flat and curved space. 110
43. Thin oil layer on a water surface. 122
44. Demonstrating molecular structure of matter. 124
45. Stern's device for studying molecular beam velocity. 126
46. Abbe's mathematical theory of the microscope. 127
47. Water molecule. 130
48. Thomson's conception of the atom. 131
49. Thomson's apparatus for measuring atom's charge/mass
ratio. 132
50. Rutherford's picture of the atom. 134
51. The periodic table of elements, 136
Illustrations xi
PAGE
52. Union of atoms in sodium chloride molecule. 138
53. Electrical and gravitational attraction. 140
54. Electronic motion in the atom. 144
55. Reflection and refraction of light. 146
56. The notion of trajectory. 148
57. Front's hypothesis. 151
58. "Annihilation'* process of two electrons giving rise to an
electromagnetic wave, and "creation" of a pair. 154
59. The origin of a cosmic ray shower. 156
60. Negative and positive beta decay. 159
61. The recoil problem in artillery and nuclear physics. 160
62. Chart of elementary particles of modern physics. 163
63. Explanation of surface-tension forces in a liquid. 164
64. "Deimos." 165
65. Fission and fusion of two droplets. 167
66. Union of carbon and oxygen. 169
67. How the atom was split the first time. 172
68. The scheme of Wilson's cloud-chamber. 174
69. Principle of the electrostatic generator. 175
70. Principle of a cyclotron. 176
71. Principle of a linear accelerator. 177
72. Atomic bombardment. 180
73. Successive stages of the fission process. 183
74. A nuclear chain reaction in a spherical piece of fission
able material. 185
75. Separation of isotopes. 187
76. A uranium pile. 189
77. A bacterium tossed around by molecular impacts. 193
78. Thermal agitation. 195
79. The destructive effect of temperature. 198
80. Drunkard's walk. 200
81. Six walking drunkards. 202
82. Diffusion. 204
83. Four possible combinations in tossing two coins. 207
84. Relative number of tails and heads. 210
85. A flush (of spades). 211
86. Full house. 212
87. Captain Kidd's Message. 216
88. Matches and flag problem. 219
89. Graph of the sine in the match problem. 220
90. Various types of cells. 233
xii Illustrations
PAGE
91. An alcohol molecule as organizer. 238
92. Successive stages of cell division (mitosis). 237
93. Formation of gametes, and fertilization of the egg cell. 241
94. Face value difference between man and woman. 243
95. From egg cell to man. 244
96. Heredity of color blindness. 248
97. Dominant and recessive characteristics. 249
98. A version of Mendel's discovery. 250
99. Chromosome transfer. 252
100. Characteristics of the fruit fly. 253
101. Hereditary "charm bracelet." 258
102. Spontaneous mutation of a fruit fly. 260
103. Comparison between bacteria, viruses, and molecules. 263
104. The world of the ancients. 269
105. An argument against the spherical shape of the earth. 271
106. Eratosthenes measures the Earth. 272
107. Parallactic displacement. 274
108. A naval range finder. 275
109. Parallactic displacement in observing the moon. 276
110. Parallactic displacement in observing 61 Cygni. 279
111. An astronomer looking at 'the stellar system of the
Milky Way. 282
112. The galactic center. 283
113. Looping effect explained by Copernicus. 285
114. Rotation of the Galaxy of stars. 287
115. Various phases of normal galactic evolution. 291
116. The milestones of cosmic exploration. 295
117. Two schools of thought in cosmogony. 301
118. Weiszacker's theory. 306
119. Circular and elliptic motion. 309
120. Dust-traffic lanes in the original solar envelope. 311
121. The cyclic nuclear reaction chain responsible for the
energy generation in the sun. 315
122. The main sequence of stars. 318
123. Giant and supergiant stars* 319
124. White dwarf stars. 320
125. The Urea-process in iron nucleus. 325
126. Stages of a supernova explosion. 326
127. The dots run away from one another on the expanding
balloon. 330
128. An artillery shell explodes in midair. 333
PARTI
Playing with Numbers
^^mi^jis^^i^^^^.
CHAPTER I
Big Numbers
1. HOW HIGH CAN JOU COUNT?
is a story about two Hungarian aristocrats who
-*- decided to play a game in which the one who calls the
largest number wins.
"Well," said one of them, "you name your number first."
After a few minutes o hard mental work the second aristocrat
finally named the largest number he could think of.
"Three/' he said.
Now it was the turn of the first one to do the thuiking, but
after a quarter of an hour he finally gave up.
**YouVe won," he agreed.
Of course these two Hungarian aristocrats do not represent a
very high degree of intelligence 1 and this story is probably just a
malicious slander, but such a conversation might actually have
taken place if the two men had been, not Hungarians, but Hotten
tots. We have it indeed on the authority of African explorers that
many Hottentot tribes do not have in their vocabulary the names
for numbers larger than three. Ask a native down there how many
sons he has or how many enemies he has slain, and if the number
is more than three he will answer "many." Thus in the Hottentot
country in the art of counting fierce warriors would be beaten
by an American child of kindergarten age who could boast the
ability to count up to ten!
Nowadays we are quite accustomed to the idea that we can
write as big a number as we please whether it is to represent
war expenditures in cents, or stellar distances in inches by
1 This statement can be supported by another story of the same collection
in which a group o Hungarian aristocrats lost their way hiking in the Alps.
One of them, it is said, took out a map, and after studying it for a long
time, exclaimed: "Now I know where we axel" "Where?" asked the others.
"See that big mountain over there? We are right on top of it."
4 Playing With Numbers
simply setting down a sufficient number of zeros on the right side
of some figure. You can put in zeros until your hand gets tired,
and before you know it you will have a number larger than even
the total number of atoms in the universe, 2 which, incidentally, is
300,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,-
000,000,000,000,000,000,000,000,000.
Or you may write it in this shorter form: S-10 74 .
Here the little number 74 above and to the right of 10 indicates
that there must be that many zeros written out, or, in other words,
3 must be multiplied by 10 seventy-four times.
But this "arithmetic-made-easy" system was not known in
ancient times. In fact it was invented less than two thousand years
ago by some unknown Indian mathematician. Before his great
discovery and it toas a great discovery, although we usually do
not realize it numbers were written by using a special symbol
for each of what we now call decimal units, and repeating this
symbol as many times as there were units. For example the
number 8732 was written by ancient Egyptians:
whereas a clerk in Caesar's office would have represented it in
this form:
MMMMMMMMDCCXXXII
The latter notations must be familiar to you, since Boman
numerals are still used sometimes to indicate the volumes or
chapters of a book, or to give the date of a historical event on a
pompous memorial tablet. Since, however, the needs of ancient
accounting did not exceed the numbers of a few thousands, the
symbols for higher decimal units were nonexistent, and an ancient
Roman, no matter how well trained in arithmetic, would have
been extremely embarrassed if he had been asked to write "one
million/' The best he could have done to comply with the request,
would have been to write one thousand M 9 s in succession, which
would have taken many hours of hard work (Figure 1).
For the ancients, very large numbers such as those of the stars
2 Measured as far as the largest telescope can penetrate
Big Numbers 5
in the sky, the fish in the sea, or. grains of sand on the beach were
"incalculable/' just as for a Hottentot "five" is incalculable, and
becomes simply "many"!
It took the great brain of Archimedes, a celebrated scientist of
the third century B.C., to show that it is possible to write really
FIGURE
An ancient Roman, resembling Augustus Caesar, tries to write "one
million" in Roman numerals. All available space on the wall-board
hardly suffices to write "a hundred thousand/*'
big numbers. In his treatise The Psammites, or Sand Reckoner,
Archimedes says:
"There are some who think that the number of sand grains is
infinite in multitude; and I mean by sand not only that which
exists about Syracuse and the rest of Sicily, but all the grains of
sand which may be found in all the regions of the Earth, whether
inhabited or uninhabited. Again there are some who, without
regarding the number as infinite, yet think that no number can be
named which is great enough to exceed that which would des-
6 Plat/ing "With Numbers
ignate the number of the Earttis grains of sand. And it is cleat
that those who hold this view, if they imagined a mass made tip
of sand in other respects as large as the mass of the Earth, in
cluding in it all the seas and all the hollows of the Earth filled up
to the height of the highest mountains, would be still more certain
that no number could be expressed which would be larger than
that needed to represent the grains of sand thus accumulated.
But I will try to show that of the numbers named by me some
exceed not only the number of grains of sand which would make
a mass equal in size to the Earth filled up in the way described,
but even equal to a mass the size of the Universe."
The way to write very large numbers proposed by Archimedes
in this famous work is similar to the way large numbers are written
in modem science. He begins with the largest number that existed
in ancient Greek arithmetic: a "myriad/' or ten thousand. Then
he introduced a new number, "a myriad myriad" (a hundred
million), which he called "an octade" or a "unit of the second
class." "Octade octades" (or ten million billions) is called a "unit
of the third class," "octade, octade, octades" a "unit of the fourth
class/ 7 etc.
The writing of large numbers may seem too trivial a matter to
which to devote several pages of a book, but in the time of
Archimedes the finding of a way to write big numbers was a
great discovery and an important step forward in the science of
mathematics.
To calculate the number representing the grains of sand neces
sary to fill up the entire universe, Archimedes had to know how
big the universe was. In his time it was believed that the universe
was enclosed by a crystal sphere to which the fixed stars were
attached, and his famous contemporary Aristarchus of Samos,
who was an astronomer, estimated the distance from the earth to
the periphery of that celestial sphere as 10,000,000,000 stadia or
about 1,000,000,000 miles. 3
Comparing the size of that sphere with the size of a grain of
sand, Archimedes completed a series of calculations that would
give a highschool boy nightmares, and finally arrived at this
conclusion:
3 One Greek "stadium" is 606 ft. 6 in., or 188 meters (m).
Big Numbers 7
"It is evident that the number of grains of sand that could be
contained in a space as large as that bounded by the stellar sphere
as estimated by Aristarchus, is not greater than one thousand
myriads of units of the eighth class." 4
It may be noticed here that Archimedes' estimate of the radius
of the universe was rather less than that of modern scientists. The
distance of one billion miles reaches only slightly beyond the
planet Saturn of our solar system. As we shall see later the uni
verse has now been explored with telescopes to the distance of
5,000,000,000,000,000,000,000 miles, so that the number of sand
grains necessary to fill up all the visible universe would be over:
10 100 (that is, 1 and 100 zeros)
This is of course much larger than the total number of atoms
in the universe, 3 10 74 , as stated at the beginning of this chapter,
but we must not forget that the universe is not packed with
atoms; in fact there is on the average only about 1 atom per cubic
meter of space.
But it isn't at all necessary to do such drastic things as packing
the entire universe with sand in order to get really large numbers.
In fact they very often pop up in what may seem at first sight a
very simple problem, in which you would never expect to find
any number larger than a few thousands.
One victim of overwhelming numbers was King Shirham of
India, who, according to an old legend, wanted to reward his
grand vizier Sissa Ben Dahir for inventing and presenting to him
the game of chess. The desires of the clever vizier seemed very
modest, "Majesty," he said kneeling in front of the king, "give
me a grain of wheat to put on the first square of this chessboard,
and two grains to put on the second square, and four grains to
put on the third, and eight grains to put on the fourth. And so ?
oh King, doubling the number for each succeeding square, give
me enough grains to cover all 64 squares of the board/*
4 In our notation it would be:
thousand myriads 2nd class 3rd class 4th class
(10,000,000) X (100,000,000) X (100,000,000) X (100,000,000) X
5ih class 6tli class 7th class 8tli class
(100,000,000) X (100,000,000) X (100,000,000) X (100,000,000)
or simply:
10 63 (i.e., 1 and 63 zeros)
8 Playing With Numbers
"You do not ask for much, oh my faithful servant/' exclaimed
the king, silently enjoying the thought that his liberal proposal
of a gift to the inventor of the miraculous game would not cost
him much of his treasure. "Your wish will certainly be granted."
And he ordered a bag of wheat to be brought to the throne.
But when the counting began, with 1 grain for the first square,
2 for the second, 4 for the third and so forth, the bag was emptied
FIGURE 2
Grand Vizier Sissa Ben Dahir, a skilled mathematician, asks his reward
from King Shirham of India.
before the twentieth square was accounted for. More bags of
wheat were brought before the king but the number of grains
needed for each succeeding square increased so rapidly that it
soon became clear that with all the crop of India the king could
not fulfill his promise to Sissa Ben. To do so would have required
18,446,744,073,709,551,615 grains! 5
5 The number of wheat grains that the clever vizier had demanded may
be represented as follows:
l+2+2 2 +2 3 +2 4 4- . . . +2 62 +2 08 .
In arithmetic a sequence of numbers each of which is progressively in
creased by the same factor (in this case by a factor of 2) is known as
geometrical progression. It can be shown that the sum of all the terms in
such a progression may be found by raising the constant factor (in this
case 2) to the power represented by the number of steps in the progression
Big Numbers 9
That's not so large a number as the total number of atoms in
the universe., but it is pretty big anyway. Assuming that a bushel
of wheat contains about 5,000,000 grains, one would need some
4000 billion bushels to satisfy the demand of Sissa Ben. Since the
world production of wheat averages about 2,000,000,000 bushels
a year, the amount requested by the grand vizier was that of the
world's wheat production for the period of some two thousand
yearsl
Thus King Shirham found himself deep in debt to his vizier
and had either to face the incessant flow of the latter's demands,
or to cut his head off. We suspect that he chose the latter alter
native.
Another story in which a large number plays the chief role
also comes from India and pertains to the problem of the "End of
the World/' W. W. R. Ball, the historian of mathematical fancy,
tells the story in the following words: 6
In the great temple at Benares beneath the dome which marks
the center of the world, rests a brass plate in which are fixed three
diamond needles, each a cubit high (a cubit is about 20 inches)
and as thick as the body of a bee. On one of these needles, at the
creation, God placed sixty-four discs of pure gold, the largest disc
resting on the brass plate and the others getting smaller and
smaller up to the top one. This is the tower of Brahma. Day and
night unceasingly, the priest on duty transfers the discs from one
diamond needle to another, according to the fixed and immutable
laws of Brahma, which require that the priest must move only
one disc at a time, and he must place these discs on needles so
that there never is a smaller disc below a larger one. When all
the sixty-four discs shall have been thus transferred from the
(in tliis case, 64), subtracting the first term (in this case, 1), and dividing
the result by the above-mentioned constant factor minus 1. It may be
stated thus:
2-1
and writing it as an explicit number:
18,446,744,073,709,551,615.
W. W. R. Ball, Mathematical Recreations and Essays (The Macrnillan
Co., New York, 1939).
10 Playing With Numbers
needle on which, at the creation, God placed them, to one of the
other needles, tower, temple, and Brahmans alike will crumble
into dust, and with a thunderclap the world will vanish.
Figure 3 is a picture of the arrangement described in the story,
except that it shows a smaller number of discs. You can make this
puzzle toy yourself by using ordinary cardboard discs instead of
golden ones, and long iron nails instead of the diamond needles
FIGURE 3
A priest working on the "End of the World" problem in front of a
giant statue of Brahma. The number of golden discs is shown here
smaller than 64 because it was difficult to draw so many.
of the Indian legend. It is not difficult to find the general rule
according to which the discs have to be moved, and when you
find it you will see that the transfer of each disc requires twice as
many moves as that of the previous one. The first disc requires
just one move, but the number of moves required for each suc
ceeding disc increases geometrically, so that when the 64th disc
Big Numbers 11
Is reached it must be moved as many times as there were grains
in the amount of wheat Sissa Ben Dahir requested! 7
How long would it take to transfer all sixty-four discs in the
tower of Brahma from one needle to the other? Suppose that
priests worked day and night without holidays or vacation, mak
ing one move every second. Since a year contains about 31,558,000
seconds it would take slightly more than fifty-eight thousand bil
lion years to accomplish the job.
It is interesting to compare this purely legendary prophecy of
the duration of the universe with the prediction of modern science.
According to the present theory concerning the evolution of the
universe, the stars, the sun, and the planets, including our Earth,
were formed about 3,000,000,000 years ago from shapeless masses.
We also know that the "atomic fuel" that energizes the stars, and
in particular our sun, can last for another 10,000,000,000 or
15,000,000,000 years. (See the chapter on "The Days of Crea
tion.") Thus the total life period of our universe is definitely
shorter than 20,000,000,000 years, rather than as long as the 58,000
billion years estimated by Indian legend! But, after all, it is only
a legend!
Probably the largest number ever mentioned in literature per
tains to the famous "Problem of a Printed Line." Suppose we
built a printing press that would continuously print one line after
another, automatically selecting for each line a different com
bination of the letters of the alphabet and other typographical
signs. Such a machine would consist of a number of separate
discs with the letters and signs all along the rim. The discs would
be geared to one another in the same way as the number discs in
7 If we have only seven discs the number of necessary moves is:
If you moved the discs rapidly without making any mistakes it would
take you about an hour to complete the task. With 64 disks the total nurn*
ber of moves necessary is:
2 _ i = 18,446,744,073,709,551,615
this is the same as the number of grains of wheat required by Sissa Ben
Dahir.
12 Playing With Numbers
the mileage indicator of your car, so that a full rotation of each
disc would move the next one forward one place. The paper as It
comes from a roll would automatically be pressed to the cylinder
after each move. Such an automatic printing press could be built
without much difficulty, and what it would look like is repre
sented schematically in Figure 4.
FIGURE 4
An automatic printing press that has just printed correctly a line from
Shakespeare.
Let us set the machine in action and inspect the endless
sequence of different printed lines that come from the press. Most
of the lines make no sense at all. They look like this:
"aaaaaaaaaaa . . ."
or
"boobooboobooboo . . "
or again:
"zawkporpkossscilm . . ."
But since the machine prints all possible combinations of letters
and signs, we find among the senseless trash various sentences
Big Numbers 13
that have meaning. There are, of course, a lot of useless sentences
such as:
or
"horse has six legs and . . ."
"I like apples cooked in terpentin. . . ."
But a search will reveal also every line written by Shakespeare,
even those from the sheets that he himself threw into the waste-
paper basket!
In fact such an automatic press would print everything that
was ever written from the time people learned to write: every
line of prose and poetry, every editorial and advertisement from
newspapers, every ponderous volume of scientific treatises, every
love letter, every note to a milkman. ... ^
Moreover the machine would print everything that is to be
printed in centuries to come. On the paper coming from the
rotating cylinder we should find the poetry of the thirtieth cen
tury, scientific discoveries of the future, speeches to be made in
the 500th Congress of the United States, and accounts of intra-
planetary traffic accidents of the year 2344. There would be pages
and pages of short stories and long novels, never yet written by
human hand, and publishers having such machines in their base
ments would have only to select and edit good pieces from a lot
of trash which they are doing now anyway.
Why cannot this be done?
Well, let us count the number of lines that would be printed
by the machine in order to present all possible combinations of
letters and other typographical signs.
There are 26 letters in the English alphabet, ten figures (0, 1,
2 ... 9) and 14 common signs (blank space, period, comma,
colon, semicolon, question mark, exclamation mark, dash, hyphen,
quotation mark, apostrophe, brackets, parentheses, braces); al
together 50 symbols. Let us also assume that the machine has
65 wheels corresponding to 65 places in an average printed line.
The printed line can begin with any of these signs so that we
have here 50 possibilities. For each of these 50 possibilities there
are 50 possibilities for the second place in the line; that is, alto
gether 50x50 = 2500 possibilities. But for each given ccftnbina-
14 Playing With Numbers
tion of the first two letters we have the choice between 50 pos
sible signs in the third place, and so forth. Altogether the number
of possible arrangements in the entire line may be expressed as:
65 times
5Qx50x50x ... x 50
or 50 65
which is equal to 10 110
To feel the immensity of that number assume that each atom
in the universe represents a separate printing press, so that we
have 3-10 74 machines working simultaneously. Assume further
that all these machines have been working continuously since the
creation of the universe, that is for the period of 3 billion years
or 10 17 seconds, printing at the rate of atomic vibrations, that is,
10 15 lines per second. By now they would have printed about
3-10 74 xl(Fxl0 15 -3-10 10G
lines which is only about one thirtieth of 1 per cent of the total
number required.
Yes, it would take a very long time indeed to make any kind of
selection among all this automatically printed material!
2. HOW TO COUNT INFINITIES
In the preceding section we discussed numbers, many of tibem
fairly large ones. But although such numerical giants as the num
ber of grains of wheat demanded by Sissa Ben are almost un
believably large, they are still finite and, given enough time, one
could write them down to the last decimal.
But there are some really infinite numbers, which are larger
than any number we can possibly write no matter how long we
work. Thus "the number of all numbers" is clearly infinite, and
so is "the number of all geometrical points on a line." Is there
anything to be said about such numbers except that they are
infinite, or is it possible, for example, to compare two different
infinities and to see which one is 'larger"?
Big Numbers 15
Is there any sense in asking: "Is the number of all numbers
larger or smaller than the number of all points on a line?" Such
questions as this, which at first sight seem fantastic, were first
considered by the famous mathematician Georg Cantor, who
can be truly named the founder of the "arithmetics of infinity. 3 *
If we want to speak about larger and smaller infinities we face
a problem of comparing the numbers that we can neither name
FIGURE 5
An African native and Prof. G. Cantor comparing the numbers beyond
their counting ability.
nor write down, and are more or less in the position of a Hotten
tot inspecting his treasure chest and wanting to know whether he
has more glass beads or more copper coins in his possession. But,
as you will remember, the Hottentot is unable to count beyond
three. Then shall he give up all attempts to compare the number
of beads and the number of coins because he cannot count them?
Not at all. If he is clever enough he will get his answer by com~
paring the beads and the coins piece by piece. He will place one
16 Playing With Numbers
bead near one coin, another bead near another coin, and so on,
and so on ... If he runs out of beads while there are still some
coins, he knows that he has more coins than beads; if he runs
out of coins with some beads left he knows that he has more
beads than coins, and if he comes out even he knows that he has
the same number of beads as coins.
Exactly the same method was proposed by Cantor for com
paring two infinities: if we can pair the objects of two infinite
groups so that each object of one infinite collection pairs with
each object of another infinite collection, and no objects in either
group are left alone, the two infinities are equal. If, however,
such arrangement is impossible and in one of the collections some
unpaired objects are left, we say that the infinity of objects in
this collection is larger, or we can say stronger, than the infinity
of objects in the other collection.
This is evidently the most reasonable, and as a matter of fact
the only possible, rule that one can use to compare infinite quan
tities, but we must be prepared for some surprises when we
actually begin to apply it. Take for example, the infinity of all
even and the infinity of all odd numbers. You feel, of course,
intuitively that there are as many even numbers as there are odd,
and this is in complete agreement with the above rule, since a
one-to-one correspondence of these numbers can be arranged:
1357 9 11 13 15 17 19 etc.
1 1 1 1 1 1 i 1 1 i
2 4 6 8 10 12 14 16 18 20 etc.
There is an even number to correspond with each odd number
in this table, and vice versa; hence the infinity of even numbers
is equal to the infinity of odd numbers. Seems quite simple and
natural indeed!
But wait a moment. Which do you think is larger: the number
of all numbers, both even and odd, or the number of even num
bers only? Of course you would say the number of all numbers is
larger because it contains in itself all even numbers and in addi
tion all odd ones. But that is just your impression, and in order
to get the exact answer you must use the above rule for comparing
Big Numbers 17
two infinities. And if you use it you will find to your surprise that
your impression was wrong. In fact here is the table of one-to-one
correspondence of all numbers on one side, and even numbers
only on the other:
12345678 etc.
1 1 1 1 1 i i i
2 4 6 8 10 12 14 16 etc.
According to our rule of comparing infinities we must say that
the infinity of even numbers is exactly as large as the infinity of
all numbers. This sounds, of course, paradoxical, since even num
bers represent only a part of all numbers, but we must remember
that we operate here with infinite numbers, and must be pre
pared to encounter different properties.
In fact in the world of infinity a part may be equal to the
whole! This is probably best illustrated by an example taken from
one of the stories about the famous German mathematician David
Hilbert. They say that in his lectures on infinity he put this
paradoxical property of infinite numbers in the following words: 8
"Let us imagine a hotel with a finite number of rooms, and
assume that all the rooms are occupied. A new guest arrives and
asks for a room. 'Sony says the proprietor but all the rooms
are occupied/ Now let us imagine a hotel with an infinite number
of rooms, and all the rooms are occupied. To this hotel, too, comes
a new guest and asks for a room.
" "But of course!' exclaims the proprietor, and he moves the
person previously occupying room Nl into room N2, the person
from room N2 into room N3, the person from room N3 into room
N4, and so on. . . . And the new customer receives room Nl,
which became free as the result of these transpositions.
"Let us imagine now a hotel with an infinite number of rooms,
all taken up, and an infinite number of new guests who come in
and ask for rooms.
** "Certainly, gentlemen/ says the proprietor, 'just wait a minute/
"He moves the occupant of Nl into N2, the occupant of N2
into N4, the occupant of N3 into N6, and so on, and so on ...
8 From the unpublished, and even never written, but widely circulating
volume: "The Complete Collection of Hilbert Stories" by R. Courant.
18 Playing With Numbers
"Now all odd-numbered rooms become free and the infinity of
new guests can easily be accommodated in them."
Well, it is not easy to imagine the conditions described by
Hilbert even in Washington as it was during the war, but this
example certainly drives home the point that in operating with
infinite numbers we encounter properties rather different from
those to which we are accustomed in ordinary arithmetic.
Following Cantor's rule for comparing two infinities, we can
also prove now that the number of all ordinary arithmetical frac-
3 735
tions like - or - is the same as the number of all integers. In
fact we can arrange all ordinary fractions in a row according to
the following rule: Write first the fractions for which the sum
of the numerator and denominator is equal to 2; there is only one
such fraction namely: -. Then write fractions with saras equal
21 3 9. 1
to 8: - and -. Then those with sums equal to 4: - 7 -, -. And so
-* " 123
on. In following this procedure we shall get an infinite sequence
of fractions, containing every single fraction one can think of
(Figure 5). Now write above this sequence, the sequence of
integers and you have the one-to-one correspondence between
the infinity of fractions and the infinity of integers. Thus their
number is the same!
"Well, it is all very nice/* you may say, "but doesn't it mean
simply that all infinities are equal to one another? And if that's
the case, what's the use of comparing them anyway?"
No, that is not the case, and one can easily find the infinity
that is larger than the infinity of all integers or all arithmetical
fractions.
In fact, if we examine the question asked earlier in this chapter
about the number of points on a line as compared with the num
ber of all integer numbers, we find that these two infinities are
different; there are many more points on a line than there are
integers or fractional numbers. To prove this statement let us try
to establish one-to-one correspondence between the points on a
line, say 1 in. long, and the sequence of integer numbers.
Each point on the line is characterized by its distance from
Big 'Numbers 19
one end of the line, and this distance can be written in the form
o* an infinite decimal fraction, like 0.7350624780056 . ... or
0.38250375632 . . . . 9 Thus we have to compare the number of all
integers with the number of all possible infinite decimal fractions.
What is the difference now between the infinite decimal fractions,
* 3 8
as given abo 7% and ordinary arithmetical fractions like - or -?
i 2i7 7
You must remember from your arithmetic that every ordinary
fraction can be converted into an infinite periodic decimal fraction.
Thus |=0.66666 . . . . =0.(6), and |=0.428571j4 2S571J4
28571:4 . . . = 0.( 428571). We have proved above that the num
ber of all ordinary arithmetical fractions is the same? as the number
of all integers; so the number of all periodic decimal fractions
must also be the same as the number of all integers. But the
points on a line are not necessarily represented by periodic
decimal fractions, and in most cases we shall get the infinite
fractions in which the decimal figures appear without any
periodicity at all. And it is easy to show that in such case no linear
arrangement is possible.
Suppose that somebody claims to have made such an arrange
ment, and that it looks something like this:
N
1 0.38602563078 ....
2 0.57350762050....
3 0.99356753207...,
4 0.25763200456....
5 0.00005320562....
6 0.99035638567....
7 0.55522730567....
8 0.05277365642.0..
9 All these fractions are smaller than unity, since we have assumed the-
length of the line to be one.
20 Playing With Numbers
Of course., since it is impossible actually to write the infinity
of numbers with, the infinite number of decimals in each, the
above claim means that the author of the table has some general
rule ( similar to one used by us for arrangement of ordinary frac
tions) according to which he has constructed t/ie table, and this
rule guarantees that every single decimal fraction one can think
of will appear sooner or later in the table.
Well, it is not at all difficult to show that any claim of that kind
is unsound, since we can always write an infinite decimal fraction
that is not contained in this infinite table. How can we do it? Oh,
very simply. Just write the fraction with the first decimal dif
ferent from that of Nl in the table, the second decimal different
from that in N? of the table and so on. The number you will get
will look something like this:
_-----
OOOOOOOO ^f^
pjfldaaaaa CIC
0. 52740712
and this number is not included in the table no matter how far
down you look for it. In fact if the author of the table will tell you
that this very fraction you have written here stands under the
No. 137 (or any other number) in his table you can answer imme
diately: "No, it isn't the same fraction because the one hundred
and thirty seventh decimal in your fraction is different from the
one hundred and thirty seventh decimal in the fraction I have in
mind."
Thus it is impossible to establish a one-to-one correspondence
between the points on a line and the integer numbers, which
means that the infinity of points on a line is large?*, or stronger,
tlwn the infinity of all integer or fractional numbers.
We have been discussing the points on a line "1 in. long/' but
it is easy to show now that, according to the rules of our "infinity
arithmetics/' the same is true of a line of any length. In fact,
there is the same number of points in lines one incli, one foot, or
one mile long. In order to prove it just look at Figure 6, which
compares the number of points on two lines AB and AC of dif-
Big Numbers 21
ferent lengths. To establish the one-to-one correspondence be
tween the points of these two lines we draw through each point
on AB a line parallel to BC, and pair the points of intersections as
for example D and D 1 , E and E 1 , F and F 1 , etc. Each point on AB
has a corresp^iding point on AC and vice versa; thus according
to our rulgjfche two infinities of points are equal
A still Jiiore striking result of the analysis of infinity consists in
the statement that: the number of all points on a plane is equal
to the number of all points on a line. To prove this let us consider
the points on a line AB one inch long, and the points within a
square CDEF (Figure 7),
FIGURE 6
FIGURE 7
Suppose that the position of a certain point on the line is given
by some number, say 0.75120386 .... We can make from this
number two different numbers selecting even and odd decimal
signs and putting them together. We get this:
and this:
0.7108
0.5236
Measure the distances given by these numbers in the horizontal
and vertical direction in our square, and call the point so obtained
the "pair-point" to our original point on the line. In reverse, if we
have a point in the square the position of which is described by^
let us say, the numbers:
0.4835
and
0.9907
22 Playing With Numbers
we obtain the position of the corresponding "pair-point" on the
line by merging these two numbers:
0.49893057
It is clear that this procedure establishes the oive-to-one rela
tionship between two sets of points. Every point on fee line will
have its pair in the square, every point in the square will have its
pair on the line, and no points will be left over. Thus according
to the criterion of Cantor, the infinity of all the points within a
square is equal to the infinity of all the points on a line.
In a similar way it is easy to prove also that the infinity of all
points within a cube is the same as the infinity of points within
a square or on a line. To do this we merely have to break the
original decimal fraction into three parts, 10 and use the three new
fractions so obtained to define the position of the "pair-point"
inside the cube. And, just as in the case of two lines of different
lengths, the number of points within a square or a cube will be
the same regardless of their size.
But the number of all geometrical points, though larger than
the number of all integer and fractional numbers, is not the
largest one known to mathematicians. In fact it was found that
the variety of all possible curves, including those of most unusual
shapes., has a larger membership than the collection of all geo
metrical points, and thus has to be described by the third number
of the infinite sequence.
According to Georg Cantor, die creator of the "arithmetics of
infinity," infinite numbers are denoted by the Hebrew letter K
(aleph) with a little number in the lower right corner that indi
cates the order of the infinity. The sequence of numbers (in
cluding the infinite ones! ) now runs:
1. 2. 3. 4. 5 81 82 83
and we say "there are Ki points on a line 77 or "there are Ku
10 For example from
0. 735106822548312 .... etc.
we make
0. 71853 ....
0. 30241 ....
0. 56282 ....
Big Numbers 23
different curves/' just as we say that "there are 7 parts of the
world" or "52 cards in a pack/'
In concluding our talk about infinite numbers we point out
that these numbers very quickly outrun any thinkable collection
to which tr oy can possibly be applied. We know that $ repre
sents the number of all integers, Hi represents the number of all
.<*?
* la^
. THJ-mWv
oil h
line. / < s V 2, etc. are impossible or imaginary numbers, since
they represent roots of negative quantities, and of such numbers
we may truly assert that they are neither nothing, nor greater
than nothing, nor less than nothing, which necessarily constitutes
them imaginary or impossible/'
But in spite of all these abuses and excuses imaginary numbers
soon became as unavoidable in mathematics as fractions, or radi
cals, and one could practically not get anywhere without using
them.
The family of imaginary numbers represents, so to speak, a
fictitious mirror image of the ordinary or real numbers, and,
exactly in the same way as one can produce all real numbers
starting with the basic number 1, one can also build up all
imaginary numbers from the basic imaginary unit V~~X which is
usually denoted by the symbol L
It is easy to see that V r 5= V>< V- r l = 3 ^ V--7= V^'V-"1
= 2.646 ... I etc., so that each ordinary real number has its
imaginary double. One can also combine real and imaginary
numbers to make single expressions such as 5+\ /r ^l5 = 5- J r i \/l5
as it was first done by Cardan. Such hybrid forms are usually
known as complex numbers.
For well over two centuries after imaginary numbers broke
into the domain of mathematics they remained enveloped by a
veil of mystery and incredibility until finally they were given a
simple geometrical interpretation by two amateur mathemati
cians: a Norwegian surveyor by the name of Wessel and a
Parisian bookkeeper, Robert Argand.
According to their interpretation a complex number., as for
example 34-4z, may be represented as in Figure 10 ? in which 3
corresponds to the horizontal distance, and 4 to the vertical,
or ordinate.
Indeed all ordinary real numbers (positive or negative) may
be represented as corresponding to the points on the horizontal
axis, whereas all purely imaginary ones are represented by the
34 Playing With Numbers
points on the vertical axis. When we multiply a real number,
say 3, representing a point on the horizontal axis, by the imagi
nary unit i we obtain the purely imaginary number 3i, which
must be plotted on the vertical axis. Hence, the multiplication
by i is geometrically equivalent to a counterclockwise rotation
by a right angle. (See Figure 10).
FIGURE 10
If now we multiply Si once more by i we must tarn the thing
by another 90 degrees, so that the resulting point is again brought
back to the horizontal axis, but is now located on the negative
side. Hence,
Thus the statement that the "square of i is equal to 1" is a
much more understandable statement than "turning twice by a
right angle (both turns counterclockwise) you will face in the
opposite direction."
The same rule also applies, of course, to hybrid complex num
bers. Multiplying 3+4i by i we get:
(34-41) j =3i+ 4i 2 =3i-4= -4+3*.
And as you can see at once from Figure 10, the point 4+31
corresponds to the point 3+4, which is turned counterclockwise
by 90 degrees around the origin. Similarly the multiplication by
Natural and Artificial Numbers 35
i is nothing but the clockwise rotation around the origin, as
can be seen from Figure 10.
If you still feel a veil of mystery surrounding imaginary num
bers you will probably be able to disperse it by working out a
simple problem in which they have practical application.
There was a young and adventurous man who found among
his great-grandfather's papers a piece of parchment that revealed
the location of a hidden treasure. The instructions read:
"Sail to North latitude and West longitude
where thou wilt find a deserted island. There lieth a large
meadow, not pent ? on the north shore of the island where stand*
eth a lonely oak and a lonely pine. 7 There thou wilt see also an
old gallows on which we once were wont to hang traitors. Start
thou from the gallows and walk to the oak counting thy steps.
At the oak thou must turn right by a right angle and take the
same number of steps. Put here a spike in the ground. Now must
thou return to the gallows and walk to the pine counting thy
steps. At the pine thou must turn left by a right angle and see
that thou takest the same number of steps, and put another spike
into the ground. Dig halfway between the spikes; the treasure
is there."
The instructions were quite clear and explicit, so our young
man chartered a ship and sailed to the South Seas. He found the
island, the field, the oak and the pine, but to his great sorrow the
gallows was gone. Too long a time had passed since the docu
ment had been written; rain and sun and wind had disintegrated
the wood and returned it to the soil, leaving no trace even of the
place where it once had stood.
Our adventurous young man fell into despair, then in an angry
frenzy began to dig at random all over the field. But all his efforts
were in vain; the island was too big! So he sailed back with
empty hands. And the treasure is probably still there.
A sad story, but what is sadder still is the fact that the fellow
might have had the treasure, if only he had known a bit about
6 The actual figures of longitude and latitude were given in the document
but are omitted in this text, in order not to give away the secret.
7 The names of the trees are also changed for the same reason as above.
Obviously there would be other varieties of trees on a tropical treasure
island.
36
Playing With Numbers
mathematics, and specifically the use of imaginary numbers. Let
us see if we can find the treasure for him, even though it is too
late to do him any good.
FIGURE 11
Treasure hunt with imaginary numbers.
Consider the island as a plane of complex numbers; draw one
axis (the real one) through the base of the two trees, and
another axis (the imaginary one) at right angles to the first,
through a point half way between the trees (Figure 11). Taking
one half of the distance between the trees as our unit of length,
Natural and Artificial Numbers 37
we can say that the oak is located at the point 1 on the real
axis, and the pine at the point +1. We do not know where the
gallows was so let us denote its hypothetical location by the
Greek letter r (capital gamma), which even looks like a gallows.
Since the gallows was not necessarily on one of 'the two axes
r must be considered as a complex number: T = a + bi, in which
the meaning of a and & is explained by Figure 11.
Now let us do some simple calculations remembering the rules
of imaginary multiplication as stated above. If the gallows is at r
and the oak at 1, their separation in distance and direction
may be denoted by ( 1) r= (1-f-r). Similarly the separa
tion of the gallows and the pine is 1 r. To turn these two
distances by right angles clockwise (to the right) and counter
clockwise (to the left) we must, according to the above rules
multiply them by i and by i, thus finding the location at which
we must place our two spikes as follows:
first spike: (-<)[-( 1+r)] + l=t(r + l) -1
second spike: ( -H)( 1 r) -l=i(l r) +1
Since the treasure is halfway between the spikes, we must now
find one half the sum of the two above complex numbers. We get:
We now see that the unknown position of the gallows denoted
by r fell out of our calculations somewhere along the way, and
that, regardless of where the gallows stood, the treasure must be
located at the point -H.
And so, if our adventurous young man could have done this
simple bit of mathematics, he would not have needed to dig up
the entire island, but would have looked for the treasure at the
point indicated by the cross in Figure 11, and there would have
found the treasure.
If you still do not believe that it is absolutely unnecessary to
know the position of the gallows in order to find the treasure,
mark on a sheet of paper the positions of two trees, and try to
carry out the instructions given in the message on the parchment
by assuming several different positions for the gallows. You will
38 Plat/ing With Numbers
always get the same point, corresponding to the number +1 on
the complex plane!
Another hidden treasure that was found by using the imaginary
square root of 1 was the astonishing discovery that our ordi
nary three-dimensional space and time can be united into one
four-dimensional picture governed by the rules of four-dimen
sional geometry. But we shall come back to this discovery in one
of the following chapters, in which we discuss the ideas of Albert
Einstein and his theory of relativity.
PA R T II
Space, Time & Einstein
CHAPTER III
Unusual Properties of Space
1. DIMENSIONS AND CO-ORDINATES
WE ALL know what space is, although we should find our
selves in a rather awkward position if we were asked to
define exactly what we mean by the word. We should probably
say that space is that which surrounds us, and through which we
can move forward or backward, right or left, up or down. The
existence of the three independent mutually perpendicular direc
tions represents one of the most fundamental properties of the
physical space in which -we live; -we say that our space is three-
directional or three-dimensional. Any location in space can be
indicated by referring to these three directions. If we are visiting
an unfamiliar city and we ask at the hotel desk how to find the
office of a certain well-known firm, the clerk may say: "Walk five
blocks south, two blocks to the right, and go up to the seventh
floor." The three numbers just given are usually known as co
ordinates, and refer, in this case, to the relationship between the
city streets, the building floors, and the point of origin in the hotel
lobby. It is clear, however, that directions to the same location
can be given from any other point, by using a co-ordinate system,
which would correctly express the relationship between the new
point of origin and the destination, and that the new co-ordinates
can be expressed through the old ones by a simple mathematical
procedure provided we know the relative position of the new
co-ordinate system in respect to the old one. This process is
known as the transformation of co-ordinates. It may be added
here that it is not at all necessary that all three co-ordinates be
expressed by the numbers representing certain distances; and, in
fact, it is more convenient in certain cases to use angular co
ordinates.
Thus, for example, whereas addresses in New York City are
most naturally expressed by a rectangular co-ordinate system
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Space, Time & Einstein
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PoiAR eo-ov
