Author(s): Ya. I. Perelman
Publisher: Mir Publishers
Year: 1979
Language: English
City: Moscow
YaLPerelman
Ya. I. Perelman
From the Author’s Preface to the Third Edition
Contents
8
THE FIFTH OPERATION OF MATHEMATICS
The Fifth Operation
Astronomical Numbers
How Much does the Earth’s Atmosphere Weigh?
14
Combustion Without Flames or Heat
15
(-- ■ 1018): (3 • 107) = • 10“ « 1010.
The Changing Weather
A Combination Lock
'fhe Superstitious Cyclist
The Results of Repeated PoubliiiQ
20
Millions of Times Faster
0 O
O 0
22
Ten Thousand Operations per Second
Fig.:3
The Number of All Possible
Chess Games
28
37°=388-3a« 10(34)17^ 10-8017 - 10*817-1017 - 23M018 = 2 (210)5 • 1018 « 2 • 10“ • 1018 = 2.1033 *.
(20 • 20)5. (30-30)35 « 103 * 2.1033.1080 - 2.10116.
The Secret of the Chess Machine
31
The Biggest Number Using Only Three Twos
32
Three Threes
Three Fours
Three Identical Digits
aa_1 > 11.
The Biggest Number Using Four Ones
Four Twos to Make the Biggesi Niiinbei*
2222, 2222, 2222, 2222.
36
2222 « 24 000 000 > 101 200 000
THE LANGUAGE OF ALGEBRA
The Art of Setting up Equations
The Life of Diophantus
40
The Horse and the Mule
41
Four Brothers
42
Two Birds by the Riverside
43
Out for a Stroll
45
IF-TJ^T
Making Hay
46
48
Cows in the Meadow
50
51
52
Newton’s Problem
(3| + i2l):/l8.= “+^
54
Interchanging the Hands of a Clock
55
jy
57
The Hands of a Clock Come Together
58
Guessing Numbers
59
60
61
62
Imaginary Nonsense
The Equation Does the Thinking
Curios and the Unexpected
64
At the Barber’s
Tramcars and a Pedestrian
Rafts and a Steamboat
70
Two Cans of Coffee
J2.02 y + 1.60* = 2,
I y+ t= l.
^=lr=0-95- *=°-05-
A Question of Dancing
Reconnaissance at Sea
74
At the Cycle Track
75
A Competition of Motorcyclists
7G
77
Average Speeds
78
79
High-Speed Computing Machines
80
82
(4) 1,
87
86
Chapter three AS AN AID TO ARITHMETIC
Instantaneous Multiplication
92
93
(3-i)2 = 3.52=12.25 = 12i,
The Digits 13 5, and 6
95
The Numbers 25 and 76
Infinite “Numbers”
Additional Payment
100
Divisibility by 11
101
102
A License Number
Divisibility by 19
A Theorem of Sophie Germain
Composite Numbers
The Number of Primes
109
The Largest Prime Discovered So Far
A Responsible Calculation
Ill
112
« 2 (1 — 0.111 . .. . 10~10) = 2 - 0.0000000000222 ...
When It’s Easier Without Algebra
Chapter four DIOPHANTINE EQUATIONS
Buying a Sweater
116
117
119
8 -f* 5^ < 0, 1 -f- 3ti < 0
— —2, 3, 4,
x = -2, -7, -12,
y = —5, —8, —11.
Auditing Accounts
120
121
Buying Stamps
y = 20 — 11 t9
z 5= 31.
x = 20 + 81.
Buying Fruit
124
f 50x + -f 1 z — 500,
49x + 9 y = 400.
125
Guessing a Birthday
126
12x + 31y = a.
12x2 + 31y2 = a.
Selling Chickens
128
or
130
Two Numbers and Four Operations
131
What Kind of Rectangle?
132
Two Two-Digit Numbers
133
134
Pythagorean Numbers
135
137
An Indeterminate Equation of the Third Degree
140
141
# *= 20r2 + lOrs — 3s2,
y = 12 r2 — lOrs — 5s2,
z = 16r2 + 8 rs + 6s2,
t = —24 r2 — 8rs — 4s2.
One Hundred Thousand for the Proof of a Theorem
143
THE SIXTH MATHEMATICAL OPERATION
The Sixth Operation
146
Which Is Greater?
1/2 >/5
Solve It at a Glance
149
Algebraic Comedies
4 —10 + 6- = 9 —15 + 6-|-.
(2-4)2=(3-4)!-
150
(2-4f=(3-4)2
(-4)2=(4 )*•
151
(4+r=(5-ir-
4 = 5,
SECOND-DEGREE EQUATIONS
Shaking Hands
Swarms of Bees
155
A Troop of Monkeys
156
Farsighted Equations
Eiiler’s Problem
158
second: (100-g)-g = -^(1g=-a;
159
Loudspeakers
160
#2b+ 80# — 2000 = 0.
The Algebra of a Lunar Voyage
163
164
A Hard Problem
166
167
-2, -1, 0, 1, 2.
Fintlincf Numbers
17, 18, 19,
182 - 1749 - 324 - 323 = 1.
Chapter seven LARGEST AND SMALLEST VALUES
Two Trains
170
171
Planning the Site of a Flag Station
172
173
—x + 4 ]/ x2 + 202 == 0.8 m — a.
15z2 - 2kx + 6400 - k2 = 0,
174
inr
An Optimal Highway
175
2 V , /(TO — a)2 — 3d2
y = ~3-(™-0)± — 3^ .
When Is^tlie Pro duct a Maximum?
When Is the Sum a Minimum?
A Beam of Maximum Volume
Two Plots of Lmad
182
Makiiijj a Kite
183
184
Building a House
186
Fencing in a Lot
187
A Trough of Maximum Cross Section
188
189
A Funnel of Maximum Capacity
190
(Ir)V^r
(£)•[*■-(tt)']
191
The Brightest Illumination
192
The Most Ancient Problem Dealing with Progressions
195
(x + (x + y) + (x + 2 y) + (x + 3 y) + (x + 4 y) = 100,
1-|. !0 y, 20, 29-1, 38-.
Aljjebra on Squared Paper
197
Watering the Garden
Feeding Chickens
198
A Team of Diggers
200
Apples
201
(a: + l)(y + -52-+lT+---+^r)z=ir-
Buying a Horse
202
1 + 1+ 1 +2 + 22 + 2*+ ...+22*-*
Paying for the Wounds of a Soldier
204
or
THE SEVENTH MATHEMATICAL OPERATION
The Seventh Operation
206
Rivals of Logarithms
207
The Evolution of Logarithmic Tables
LofyarithmioTable Champions
Logarithms on the Stage
211
Logarithms on a Stock-Raising Farm
213
Logarithms in Music
The $tai'S* Aoise and Lo
Logarithms in Electric Lighting
218
log (1+iJ))=il^ and x=b%•
219
Making out a Will for Hundreds of Years
220
Constant Growth of Capital
The Number e
224
225
A Logarithmic Comedy
Any Number via Three Twos
V V... y i/2,
OTHER MIR TITLES
YaLPerelman
