1
00:00:00,000 --> 00:00:04,320
On a clay tablet, a scribe works out the length of a diagonal.

2
00:00:04,940 --> 00:00:09,640
Centuries later, a mathematician tries to take the square root of a negative number.

3
00:00:10,340 --> 00:00:13,320
The tools are different, but the impulse is the same.

4
00:00:14,060 --> 00:00:18,000
When an old method reaches its limits, can we find a new way to think?

5
00:00:18,000 --> 00:00:25,120
Across seventeen chapters, we'll follow that question from ancient geometry to the mathematics of modern markets.

6
00:00:26,480 --> 00:00:31,840
Long ago, Babylonian scribes worked out the lengths of diagonals on clay tablets.

7
00:00:32,700 --> 00:00:33,880
Here's a simple example.

8
00:00:34,800 --> 00:00:35,760
Take a right triangle.

9
00:00:36,360 --> 00:00:38,640
Its two shorter sides are 3 and 4.

10
00:00:39,140 --> 00:00:40,400
3 squared is 9.

11
00:00:41,160 --> 00:00:42,260
4 squared is 16.

12
00:00:43,080 --> 00:00:44,820
Add them, and you get 25.

13
00:00:45,760 --> 00:00:47,740
The square root of 25 is 5.

14
00:00:48,200 --> 00:00:50,160
That's the length of the longest side.

15
00:00:50,740 --> 00:00:52,600
This is the Pythagorean theorem.

16
00:00:53,160 --> 00:00:56,140
A squared plus B squared equals C squared.

17
00:00:56,600 --> 00:01:01,900
The rule works for any right triangle, with c as the side opposite the right angle.

18
00:01:02,300 --> 00:01:04,820
It turns two measurements we know into a distance.

19
00:01:05,320 --> 00:01:07,320
We might never need to measure directly.

20
00:01:07,820 --> 00:01:09,000
We can measure that length.

21
00:01:09,500 --> 00:01:12,360
But what if the number under the square root is negative?

22
00:01:13,100 --> 00:01:18,540
In 1545, an Italian mathematician named Cardano published a book on algebra.

23
00:01:19,380 --> 00:01:23,540
In the sixteenth-century search for ways to solve cubic equations,

24
00:01:24,000 --> 00:01:25,280
a strange problem emerged.

25
00:01:25,940 --> 00:01:28,999
A calculation could call for the square root of a negative number,

26
00:01:29,000 --> 00:01:31,480
even when the equation had a real answer.

27
00:01:31,980 --> 00:01:35,179
In 1572, Bombelli published rules

28
00:01:35,180 --> 00:01:37,900
for calculating with these unfamiliar quantities.

29
00:01:38,520 --> 00:01:41,040
In some problems, the troublesome terms canceled,

30
00:01:41,380 --> 00:01:43,620
and the calculation led back to a real answer.

31
00:01:44,160 --> 00:01:47,560
Today, we can see what happens on a diagram called the complex plane.

32
00:01:47,820 --> 00:01:48,660
Start at one.

33
00:01:49,100 --> 00:01:50,080
Multiply by i,

34
00:01:50,440 --> 00:01:52,760
and you turn a quarter turn counterclockwise.

35
00:01:53,080 --> 00:01:54,380
Multiply by i again.

36
00:01:54,940 --> 00:01:57,180
Another quarter turn takes you to negative one.

37
00:01:57,620 --> 00:01:59,380
Two quarter turns make a half turn.

38
00:01:59,540 --> 00:02:01,760
That’s why i squared equals negative one.

39
00:02:02,100 --> 00:02:04,280
In the next chapter, the challenge is different.

40
00:02:04,500 --> 00:02:05,520
The numbers are familiar.

41
00:02:06,540 --> 00:02:08,320
The calculations are just too long.

42
00:02:08,940 --> 00:02:12,660
Imagine working through a table of astronomical measurements by hand.

43
00:02:12,660 --> 00:02:14,940
One multiplication is manageable.

44
00:02:15,740 --> 00:02:20,000
Now imagine a whole page of them, with several digits in every number.

45
00:02:20,000 --> 00:02:24,660
In sixteen fourteen, John Napier published a system of logarithms.

46
00:02:25,020 --> 00:02:27,680
Here's the idea in the familiar base-ten version.

47
00:02:27,767 --> 00:02:29,500
A hundred is ten squared.

48
00:02:29,600 --> 00:02:31,300
A thousand is ten cubed.

49
00:02:31,540 --> 00:02:35,233
Multiply a hundred by a thousand, and you get ten to the fifth power.

50
00:02:35,893 --> 00:02:37,363
The exponents simply add.

51
00:02:37,513 --> 00:02:38,193
That's the rule.

52
00:02:38,293 --> 00:02:44,353
The log of a product equals the sum of the logs, as long as we use the same base and start with positive numbers.

53
00:02:45,153 --> 00:02:50,953
With a table, a calculator could look up the logs, add them, then look up the number that matched the result.

54
00:02:51,553 --> 00:02:53,853
The table took much of the work out of multiplication.

55
00:02:54,413 --> 00:02:57,813
A table helps us calculate with numbers we already have.

56
00:02:58,313 --> 00:03:02,253
How do we measure a quantity that is changing from moment to moment?

57
00:03:02,713 --> 00:03:03,813
Look at a speedometer.

58
00:03:04,453 --> 00:03:06,873
It tells you how fast you're moving right now.

59
00:03:06,873 --> 00:03:12,653
But if speed means distance divided by time, how can you measure it at a single instant?

60
00:03:13,173 --> 00:03:15,493
Start with a short stretch of time.

61
00:03:15,693 --> 00:03:18,653
Divide the change in position by the time that passed.

62
00:03:19,453 --> 00:03:22,313
That's the average velocity over that interval.

63
00:03:22,793 --> 00:03:24,873
Now make the interval shorter and shorter.

64
00:03:25,573 --> 00:03:30,833
If those averages approach a single value, that value is the derivative of position.

65
00:03:31,373 --> 00:03:33,293
It gives the instantaneous velocity.

66
00:03:33,293 --> 00:03:41,213
In the seventeenth century, Isaac Newton and Gottfried Wilhelm Leibniz developed calculus through different approaches.

67
00:03:41,913 --> 00:03:46,423
Leibniz published a method for differential calculus in sixteen eighty-four.

68
00:03:46,573 --> 00:03:54,773
Their work gave mathematicians a way to calculate rates of change, from the slope of a curve to the motion of a planet.

69
00:03:55,313 --> 00:03:57,733
Now we can describe how an object moves.

70
00:03:57,733 --> 00:04:00,953
What makes its path bend in the first place?

71
00:04:01,083 --> 00:04:04,053
Throw a ball, and it curves toward the ground.

72
00:04:04,793 --> 00:04:06,513
The Moon's path curves too.

73
00:04:06,993 --> 00:04:14,033
In sixteen eighty-seven, Newton's Principia set out a theory that could explain both motions through gravitation.

74
00:04:14,513 --> 00:04:24,853
In modern notation, the force between two point masses is the gravitational constant times the product of their masses, divided by the square of the distance between them.

75
00:04:24,853 --> 00:04:28,613
Double the distance, and the force drops to a quarter.

76
00:04:29,293 --> 00:04:34,933
The same distance rule applies outside a spherically symmetric body when we measure from its center.

77
00:04:35,613 --> 00:04:37,653
The Moon has sideways motion as well.

78
00:04:37,953 --> 00:04:43,813
Earth's gravity keeps turning its path inward, so the Moon continually falls around Earth.

79
00:04:44,413 --> 00:04:48,693
The ground beneath our feet and the orbit above us belong to the same story.

80
00:04:49,333 --> 00:04:50,893
An orbit follows a law of motion.

81
00:04:51,613 --> 00:04:55,473
Repeated coin tosses are much harder to predict one at a time.

82
00:04:55,893 --> 00:04:57,213
Can they still reveal a pattern?

83
00:04:57,213 --> 00:04:59,293
Toss a fair coin a hundred times.

84
00:04:59,933 --> 00:05:01,873
Getting close to fifty heads is ordinary.

85
00:05:02,573 --> 00:05:06,013
Getting a hundred heads is possible, but extraordinarily unlikely.

86
00:05:06,793 --> 00:05:15,473
In seventeen thirty-three, Abraham de Moivre showed how a smooth curve could approximate probabilities for a large number of fair coin tosses.

87
00:05:15,473 --> 00:05:20,773
Plot the possible counts, and the pattern rises around the middle and falls away on both sides.

88
00:05:21,313 --> 00:05:25,033
This is an early appearance of the bell curve we now call the normal distribution.

89
00:05:25,633 --> 00:05:26,753
The mean sets the center.

90
00:05:26,933 --> 00:05:29,273
The standard deviation sets the spread.

91
00:05:29,273 --> 00:05:36,733
For a continuous normal distribution, probability is the area under the curve over a range.

92
00:05:37,213 --> 00:05:44,273
The curve gives us a way to estimate how often values fall near the center and how often they stray far from it.

93
00:05:44,273 --> 00:05:47,613
That curve describes the shape of a distribution.

94
00:05:48,593 --> 00:05:51,173
The next curve moves before our eyes.

95
00:05:51,653 --> 00:05:55,113
Pluck a stretched string and watch the disturbance travel.

96
00:05:55,893 --> 00:05:59,613
What links the bend in the string to the way each point moves?

97
00:06:00,493 --> 00:06:05,693
In the seventeen forties, Jean le Rond d'Alembert studied the vibrating string.

98
00:06:06,253 --> 00:06:15,833
In the ideal one-dimensional wave equation, the acceleration of the string at a point equals the wave speed squared times the local curvature of the string's displacement.

99
00:06:16,653 --> 00:06:22,513
Near the top of a bump, the string curves downward, and the acceleration there is downward.

100
00:06:22,513 --> 00:06:27,493
On an ideal string stretching without end, a pulse can travel without changing shape.

101
00:06:27,993 --> 00:06:33,273
On a string fixed at both ends, waves can reflect and combine into standing patterns.

102
00:06:33,693 --> 00:06:38,953
That's how one equation can describe both a traveling pulse and a vibrating musical string.

103
00:06:39,513 --> 00:06:41,053
The shape of the string keeps changing.

104
00:06:41,593 --> 00:06:46,113
Now let's look for a number that stays the same when a shape changes.

105
00:06:46,313 --> 00:06:47,093
Pick up a cube.

106
00:06:47,473 --> 00:06:50,333
Count eight corners, twelve edges, and six faces.

107
00:06:50,953 --> 00:06:53,873
Add the corners and faces, then subtract the edges.

108
00:06:54,053 --> 00:06:54,753
You get two.

109
00:06:55,533 --> 00:06:59,573
Now try a tetrahedron, a solid with four triangular faces.

110
00:07:00,053 --> 00:07:03,093
It has four corners, six edges, and four faces.

111
00:07:03,573 --> 00:07:04,853
Again, the answer is two.

112
00:07:05,793 --> 00:07:10,213
In the eighteenth century, Leonhard Euler studied this relation for polyhedra.

113
00:07:10,213 --> 00:07:16,113
In modern notation, V minus E plus F equals two for any convex polyhedron.

114
00:07:16,573 --> 00:07:19,473
You can even see why a simple change preserves the count.

115
00:07:20,053 --> 00:07:22,933
Draw a diagonal across one square face of the cube.

116
00:07:23,373 --> 00:07:26,093
You've added an edge and split one face into two.

117
00:07:26,193 --> 00:07:29,433
The extra edge and the extra face cancel in the formula.

118
00:07:30,053 --> 00:07:33,253
Euler found a way to read a shape by counting its pieces.

119
00:07:33,933 --> 00:07:38,153
What could we learn by breaking a complicated signal into simple waves?

120
00:07:38,153 --> 00:07:41,433
Listen to a single note played on a violin.

121
00:07:41,913 --> 00:07:45,513
You hear one pitch clearly, but the sound can contain many frequencies.

122
00:07:46,253 --> 00:07:49,733
Their relative strengths help give the violin its particular sound.

123
00:07:50,373 --> 00:07:54,353
In eighteen twenty-two, Joseph Fourier published his work on heat.

124
00:07:54,673 --> 00:07:59,733
It developed the idea of representing functions using combinations of simple waves.

125
00:08:00,333 --> 00:08:06,593
The modern Fourier transform carries that idea into a tool for examining the frequencies in a signal.

126
00:08:06,593 --> 00:08:11,253
For each frequency, it tells us about the strength and phase of that component.

127
00:08:11,413 --> 00:08:14,673
Phase describes where the wave is in its cycle.

128
00:08:15,413 --> 00:08:20,833
Look at a recording in time, and you see how the signal changes from moment to moment.

129
00:08:21,233 --> 00:08:26,033
Look at it in frequency, and you can see which tones contribute to it.

130
00:08:26,573 --> 00:08:29,273
It's the same recording, seen two different ways.

131
00:08:30,013 --> 00:08:32,473
Fourier's work grew out of the movement of heat.

132
00:08:32,473 --> 00:08:35,733
What about the movement of the air and water that carry it?

133
00:08:35,953 --> 00:08:38,093
Watch a thin ribbon of smoke rise.

134
00:08:38,693 --> 00:08:40,313
For a moment it looks smooth.

135
00:08:41,073 --> 00:08:43,653
Then it curls and breaks into swirls.

136
00:08:44,033 --> 00:08:48,813
How do we describe motion that can change so much from one place to another?

137
00:08:49,373 --> 00:08:57,813
The equations that bear the names of Claude-Louis Navier and George Gabriel Stokes developed over the first half of the nineteenth century.

138
00:08:57,813 --> 00:09:02,653
They express how a fluid's motion changes as forces act on it.

139
00:09:03,253 --> 00:09:05,133
A pressure difference can push the fluid.

140
00:09:05,473 --> 00:09:09,673
Viscosity resists differences in motion between neighboring layers.

141
00:09:10,193 --> 00:09:13,173
Gravity can act on the fluid throughout its volume.

142
00:09:13,973 --> 00:09:19,753
For a fluid treated as incompressible, we also require that its motion conserve volume locally.

143
00:09:20,793 --> 00:09:24,653
Together, these rules let us model flows around wings and through pipes.

144
00:09:25,573 --> 00:09:31,653
The same equations can lead to remarkably complicated motion when many swirls interact.

145
00:09:32,533 --> 00:09:34,473
Air and water can carry waves.

146
00:09:35,453 --> 00:09:41,153
In the nineteenth century, another set of equations showed how waves could travel through empty space.

147
00:09:41,933 --> 00:09:48,513
In eighteen sixty-five, James Clerk Maxwell published his theory of the electromagnetic field.

148
00:09:49,273 --> 00:09:54,973
The familiar four equations are a modern way to express the laws brought together by that work.

149
00:09:55,313 --> 00:09:58,073
The first links electric charge to electric fields.

150
00:09:58,933 --> 00:10:03,593
The second says that a closed surface has no net magnetic flux through it.

151
00:10:04,033 --> 00:10:09,373
The third says a changing magnetic field creates a circulating electric field.

152
00:10:09,973 --> 00:10:16,333
The fourth links circulating magnetic fields to electric currents and changing electric fields.

153
00:10:16,333 --> 00:10:19,673
Put the laws together, and a striking prediction emerges.

154
00:10:20,553 --> 00:10:24,513
Electric and magnetic fields can travel as waves, even in a vacuum.

155
00:10:25,013 --> 00:10:27,673
Their predicted speed matches the speed of light.

156
00:10:28,093 --> 00:10:30,833
Light itself is an electromagnetic wave.

157
00:10:31,233 --> 00:10:35,453
Radio waves and visible light occupy different parts of the same spectrum.

158
00:10:36,133 --> 00:10:38,413
A wave carries energy from place to place.

159
00:10:38,873 --> 00:10:43,393
What happens when energy spreads out in a way we never see reverse on its own?

160
00:10:43,893 --> 00:10:45,493
Put a warm cup on a cool table.

161
00:10:45,493 --> 00:10:49,773
Energy flows from the warmer cup toward the cooler surroundings.

162
00:10:50,133 --> 00:10:55,913
You've never seen the cup spontaneously grow hotter by drawing heat from an even colder room.

163
00:10:56,593 --> 00:11:00,733
In eighteen sixty-five, Rudolf Clausius introduced the term entropy.

164
00:11:01,513 --> 00:11:05,893
The second law says that the total entropy of an isolated system cannot decrease.

165
00:11:06,673 --> 00:11:14,433
If we imagine a warm object and a cool object isolated together, the warm object loses entropy as it gives up heat.

166
00:11:14,433 --> 00:11:18,133
The cool object gains more entropy than the warm object loses.

167
00:11:18,653 --> 00:11:19,613
The total goes up.

168
00:11:19,953 --> 00:11:21,793
The energy is still accounted for.

169
00:11:21,973 --> 00:11:23,633
Entropy tells us something else.

170
00:11:23,953 --> 00:11:26,833
It tells us which way a process can go on its own.

171
00:11:27,633 --> 00:11:29,013
Energy can move and change form.

172
00:11:29,513 --> 00:11:35,373
In nineteen oh five, Einstein asked what happens to an object's mass when the object gives up energy.

173
00:11:36,173 --> 00:11:41,633
Imagine an object at rest that sends out two identical pulses of light in opposite directions.

174
00:11:41,633 --> 00:11:44,113
The pulses carry away energy.

175
00:11:44,113 --> 00:11:49,293
Because they go in opposite directions, the object can remain at rest.

176
00:11:49,993 --> 00:11:56,073
In nineteen oh five, Albert Einstein argued that losing energy in this way means losing mass.

177
00:11:56,753 --> 00:12:01,533
Today we write the rest energy of a body as E equals M C squared.

178
00:12:02,113 --> 00:12:06,093
M is its mass, and C is the speed of light in a vacuum.

179
00:12:06,833 --> 00:12:10,613
The speed of light is enormous, and the equation squares it.

180
00:12:10,613 --> 00:12:15,233
That's why even a tiny change in mass can correspond to a large change in energy.

181
00:12:15,233 --> 00:12:25,113
In the Sun, some of the difference in mass between the starting material and the products of fusion corresponds to energy that eventually helps light and warm our world.

182
00:12:25,773 --> 00:12:28,913
The Sun connects mass to energy on an enormous scale.

183
00:12:30,073 --> 00:12:33,913
To understand the energy of a single atom, we need a new kind of equation.

184
00:12:34,773 --> 00:12:37,753
Look at the light from excited hydrogen through a spectroscope.

185
00:12:37,753 --> 00:12:40,113
You see separate bright lines.

186
00:12:40,453 --> 00:12:42,493
Those lines point to something remarkable.

187
00:12:43,393 --> 00:12:45,933
The atom has particular allowed energy levels.

188
00:12:46,613 --> 00:12:50,173
In nineteen twenty-six, Erwin Schrödinger published his wave mechanics.

189
00:12:51,013 --> 00:12:56,533
The time-dependent Schrödinger equation tells us how a quantum wave function changes over time.

190
00:12:56,573 --> 00:12:59,133
The symbol psi stands for the wave function.

191
00:12:59,713 --> 00:13:05,273
Its squared magnitude gives the probability density for finding the particle at a given position.

192
00:13:05,273 --> 00:13:13,413
For a hydrogen atom, the equation for states of definite energy gives discrete energy levels for the bound electron.

193
00:13:13,933 --> 00:13:20,093
When the atom makes a transition between levels and emits light, the photon carries the energy difference.

194
00:13:20,733 --> 00:13:26,013
That helps explain why hydrogen produces a recognizable pattern of spectral lines.

195
00:13:26,813 --> 00:13:30,933
Here, probabilities help us describe possible measurement results.

196
00:13:30,933 --> 00:13:35,973
What if we use probabilities to measure how much a message can tell us?

197
00:13:36,453 --> 00:13:40,513
Imagine someone is about to tell you the result of a fair coin toss.

198
00:13:41,093 --> 00:13:43,133
Heads and tails are equally likely.

199
00:13:43,753 --> 00:13:47,033
Before the answer arrives, there is one bit of uncertainty.

200
00:13:47,693 --> 00:13:52,873
In nineteen forty-eight, Claude Shannon published a mathematical theory of communication.

201
00:13:53,813 --> 00:13:58,233
His entropy formula measures average uncertainty across possible messages.

202
00:13:58,233 --> 00:14:05,013
For each outcome, take its probability times the negative base-two logarithm of that probability.

203
00:14:05,453 --> 00:14:08,173
Add the results, and the answer is in bits.

204
00:14:08,733 --> 00:14:13,773
If the coin almost always lands heads, hearing the outcome tells you less on average.

205
00:14:14,273 --> 00:14:15,953
There is less uncertainty to resolve.

206
00:14:16,573 --> 00:14:25,513
This idea lets us compare how much information different sources produce and helps explain how tightly messages from those sources can be compressed.

207
00:14:25,513 --> 00:14:28,733
Sometimes uncertainty comes from a random source.

208
00:14:29,033 --> 00:14:33,153
Sometimes it grows even when every step follows a fixed rule.

209
00:14:34,213 --> 00:14:44,533
In nineteen sixty-three, Edward Lorenz showed how nearby starting points in a simple model of atmospheric convection could lead to very different behavior.

210
00:14:45,193 --> 00:14:50,093
Let's see how a similar kind of sensitivity can appear in a much smaller model.

211
00:14:50,293 --> 00:14:53,053
Start with a number between zero and one.

212
00:14:53,053 --> 00:14:59,033
Multiply it by one minus itself, then multiply by a fixed number, R.

213
00:14:59,693 --> 00:15:02,893
Use the answer as your new starting value and repeat.

214
00:15:03,493 --> 00:15:05,153
This is the logistic map.

215
00:15:05,153 --> 00:15:07,863
At some settings, the values settle down.

216
00:15:07,863 --> 00:15:10,593
At others, they repeat in a cycle.

217
00:15:11,153 --> 00:15:17,873
At R equal to four, many pairs of nearly identical starting values separate dramatically as we repeat the rule.

218
00:15:18,313 --> 00:15:19,593
Every step is determined.

219
00:15:20,193 --> 00:15:27,713
Yet if we know the starting value only approximately, our ability to predict the distant future can fall apart.

220
00:15:28,073 --> 00:15:30,273
A perfect forecast can be out of reach.

221
00:15:31,053 --> 00:15:38,113
In finance, the next question is how to put a price on a contract whose payoff depends on an uncertain future.

222
00:15:38,993 --> 00:15:43,673
Imagine paying today for the right to buy a share at a fixed price on a future date.

223
00:15:44,233 --> 00:15:47,793
If the share ends up above that price, the right is valuable.

224
00:15:48,453 --> 00:15:52,433
If it ends up below that price, you can let the option expire.

225
00:15:53,013 --> 00:15:55,213
How much should that right cost today?

226
00:15:56,013 --> 00:16:03,573
In nineteen seventy-three, Fischer Black and Myron Scholes published a way to answer that question under a specific model of the market.

227
00:16:04,493 --> 00:16:06,253
Their reasoning uses a hedge.

228
00:16:06,253 --> 00:16:14,653
Hold an amount of the underlying stock chosen to offset small changes in the option's value, and keep adjusting that holding as the price moves.

229
00:16:14,653 --> 00:16:22,133
In the idealized model, the hedged position must earn the risk-free rate to rule out an arbitrage opportunity.

230
00:16:23,133 --> 00:16:30,173
The resulting equation links the option's value to the stock price, time, volatility, and the interest rate.

231
00:16:30,553 --> 00:16:33,453
It gives us a price under the model's assumptions.

232
00:16:34,193 --> 00:16:36,193
Those assumptions are part of the answer.

233
00:16:36,793 --> 00:16:41,973
We started with the length of a diagonal and ended with the price of a promise.

234
00:16:42,473 --> 00:16:49,833
In between, we saw equations describe waves, quantify uncertainty, and show where prediction breaks down.

235
00:16:49,880 --> 00:16:54,413
Every one began with a question we could state in ordinary words.

236
00:16:55,113 --> 00:16:58,933
The equations gave us a way to see how far an answer could take us.