1
00:00:01,505 --> 00:00:04,338
♪ ♪

2
00:00:06,271 --> 00:00:11,271
TALITHIA WILLIAMS:
We live our lives surrounded
by numbers.

3
00:00:11,271 --> 00:00:14,138
REPORTER:
Tipping the scales at a
whopping 14 pounds...

4
00:00:14,138 --> 00:00:15,805
The price? $400.

5
00:00:15,805 --> 00:00:17,671
... at 145,000 new infections...

6
00:00:17,671 --> 00:00:20,638
WILLIAMS:
But they didn't all arrive
at once.

7
00:00:20,638 --> 00:00:22,405
Why did it take so long...

8
00:00:22,405 --> 00:00:25,471
PATRICK KIMANI:
Its utility in mathematics
is undisputed.

9
00:00:25,471 --> 00:00:27,838
WILLIAMS:
...for one number in
particular...

10
00:00:27,838 --> 00:00:29,305
It's more like a concept
than a number.

11
00:00:29,305 --> 00:00:32,071
WILLIAMS:
...to gain full "numberhood"?

12
00:00:32,071 --> 00:00:34,238
I'd call it a
very significant number.

13
00:00:34,238 --> 00:00:35,571
WILLIAMS:
What's so scary...

14
00:00:35,571 --> 00:00:36,705
VINODH CHELLAMUTHU:
You divide a number by it,

15
00:00:36,705 --> 00:00:37,938
you blow up.

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00:00:37,938 --> 00:00:41,138
WILLIAMS:
...about zero?

17
00:00:41,138 --> 00:00:42,438
In science and mathematics,

18
00:00:42,438 --> 00:00:44,138
the simplest ideas end up

19
00:00:44,138 --> 00:00:47,338
the most influential,
the most profound.

20
00:00:47,338 --> 00:00:51,071
WILLIAMS:
From zero,
where do numbers lead?

21
00:00:51,071 --> 00:00:52,271
Can we follow them

22
00:00:52,271 --> 00:00:53,971
all the way to infinity?

23
00:00:53,971 --> 00:00:55,405
AISHA ARROYO:
Infinity and zero

24
00:00:55,405 --> 00:00:56,871
are two sides to the same coin.

25
00:00:56,871 --> 00:01:01,038
WILLIAMS:
Can one infinity be
bigger than another?

26
00:01:01,038 --> 00:01:02,105
EUGENIA CHENG:
How much there is

27
00:01:02,105 --> 00:01:04,405
to understand, that's where

28
00:01:04,405 --> 00:01:06,405
all the amazingness
of infinity is!

29
00:01:06,405 --> 00:01:10,038
WILLIAMS:
What happens when
mathematicians mix

30
00:01:10,038 --> 00:01:13,305
the clout of zero
with the power of infinity?

31
00:01:13,305 --> 00:01:15,738
STEVE STROGATZ:
It's all one big principle.

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WILLIAMS:
Nothing less than our
modern world.

33
00:01:20,605 --> 00:01:23,905
Come join me,
Talithia Williams,

34
00:01:23,905 --> 00:01:27,005
as we dance with two
of the strangest beasts

35
00:01:27,005 --> 00:01:29,938
in all of mathematics.

36
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It's nothing... and everything.

37
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"Zero to Infinity,"

38
00:01:35,405 --> 00:01:38,805
right now, on "NOVA."

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♪ ♪

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ANNOUNCER:
Major funding for "NOVA"
is provided by the following:

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♪ ♪

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WILLIAMS:
Imagine if you had to explain
how we keep track of time

43
00:02:19,438 --> 00:02:22,838
to an alien.

44
00:02:22,838 --> 00:02:26,338
♪ ♪

45
00:02:27,938 --> 00:02:29,105
Since they are an alien,

46
00:02:29,105 --> 00:02:32,938
you start with how long it takes
for Earth

47
00:02:32,938 --> 00:02:35,205
to travel around the sun.

48
00:02:35,205 --> 00:02:37,671
One year.

49
00:02:37,671 --> 00:02:38,738
So far, so good.

50
00:02:38,738 --> 00:02:42,038
Then you explain we break
a year

51
00:02:42,038 --> 00:02:44,605
down into 12 months,

52
00:02:44,605 --> 00:02:47,971
though they don't fit exactly.

53
00:02:47,971 --> 00:02:52,605
And we break months
into four weeks.

54
00:02:52,605 --> 00:02:54,938
Though that's not an
exact fit, either.

55
00:02:54,938 --> 00:03:00,638
At this point, the alien
might think, "One, 12, four.

56
00:03:00,638 --> 00:03:02,505
Is there a pattern forming?"

57
00:03:02,505 --> 00:03:05,105
But then you go on and explain

58
00:03:05,105 --> 00:03:07,538
a week is made up of
seven days.

59
00:03:07,538 --> 00:03:12,705
And that a day is made
of 24 hours.

60
00:03:12,705 --> 00:03:16,971
And an hour is made up
of 60 minutes.

61
00:03:16,971 --> 00:03:20,971
So that's groups of one, 12,
four, seven, 24, and 60.

62
00:03:20,971 --> 00:03:23,138
That's the "system."

63
00:03:23,138 --> 00:03:27,605
Even the alien's buddies
can't figure it out.

64
00:03:27,605 --> 00:03:32,471
Maybe they can wrap their heads
around another number--

65
00:03:32,471 --> 00:03:34,338
a dance number!

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00:03:34,338 --> 00:03:38,905
♪ ♪

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00:03:38,905 --> 00:03:40,338
It's easy to imagine

68
00:03:40,338 --> 00:03:43,205
that the real
universal language

69
00:03:43,205 --> 00:03:45,205
should be mathematics.

70
00:03:45,205 --> 00:03:48,005
And maybe it is.

71
00:03:48,005 --> 00:03:50,005
Though on Earth,

72
00:03:50,005 --> 00:03:52,805
over the course of our history,

73
00:03:52,805 --> 00:03:57,005
how we represent numbers
has been anything but universal.

74
00:03:57,005 --> 00:03:58,305
Over thousands of years,

75
00:03:58,305 --> 00:04:01,538
we humans have tried out
a lot of systems,

76
00:04:01,538 --> 00:04:04,871
but there is one that
many of us use today.

77
00:04:04,871 --> 00:04:07,705
With just ten numerals--

78
00:04:07,705 --> 00:04:09,471
zero through nine--

79
00:04:09,471 --> 00:04:14,205
we can, in principle,
write out any number we want,

80
00:04:14,205 --> 00:04:15,938
however large or small.

81
00:04:15,938 --> 00:04:17,271
Though writing out some

82
00:04:17,271 --> 00:04:22,305
may require an eternity--
I'm looking at you, pi!

83
00:04:22,305 --> 00:04:24,205
♪ ♪

84
00:04:24,205 --> 00:04:27,571
So where did all these
numbers come from?

85
00:04:27,571 --> 00:04:30,638
And do they really
go on forever?

86
00:04:30,638 --> 00:04:33,171
My name is Talithia Williams.

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00:04:33,171 --> 00:04:36,138
And when I'm not
on an alien planet,

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you can find me...

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♪ ♪

90
00:04:40,438 --> 00:04:45,371
...here, at Harvey Mudd College
in Claremont, California,

91
00:04:45,371 --> 00:04:50,171
where I'm a professor of
mathematics and a statistician.

92
00:04:50,171 --> 00:04:51,971
(speaking indistinctly)

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00:04:51,971 --> 00:04:54,771
WILLIAMS:
Statistics is a
mathematical science

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00:04:54,771 --> 00:04:57,305
that looks for patterns in
data...

95
00:04:57,305 --> 00:04:59,138
So it is really key here
that our data...

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00:04:59,138 --> 00:05:01,805
WILLIAMS:
...information that researchers
can gather from anywhere,

97
00:05:01,805 --> 00:05:06,505
but all of which is ultimately
translated into numbers

98
00:05:06,505 --> 00:05:10,205
using the very digits we learn
by counting,

99
00:05:10,205 --> 00:05:12,605
well, our digits.

100
00:05:14,271 --> 00:05:18,671
One, two, three, four, five,
and so on.

101
00:05:18,671 --> 00:05:20,705
They can be arranged as
whole steps

102
00:05:20,705 --> 00:05:25,071
on a number line that
extends off into the distance,

103
00:05:25,071 --> 00:05:28,838
heading toward something
we learned to call "infinity,"

104
00:05:28,838 --> 00:05:33,338
which we shall see can be
a very strange place, indeed.

105
00:05:33,338 --> 00:05:37,905
Though there is one number
that tends to be overlooked--

106
00:05:37,905 --> 00:05:39,238
at least at first.

107
00:05:39,238 --> 00:05:43,571
Most of us learn to count
starting with one.

108
00:05:43,571 --> 00:05:46,471
But is that really
the beginning?

109
00:05:46,471 --> 00:05:53,338
Or is the start a number that
isn't there at all-- zero?

110
00:05:53,338 --> 00:05:56,938
♪ ♪

111
00:05:56,938 --> 00:06:00,505
When we talk about zero,

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we're talking about nothing.

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So we start, you know, teaching
children, here's one apple,

114
00:06:07,405 --> 00:06:09,538
two apples, three apples,
and we don't think about,

115
00:06:09,538 --> 00:06:12,538
well, what about everywhere else
where there are no apples?

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00:06:12,538 --> 00:06:14,638
♪ ♪

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00:06:14,638 --> 00:06:16,471
Zero is a special number,

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00:06:16,471 --> 00:06:20,705
which makes every other
number meaningful.

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WILLIAMS:
These days, most of us
take zero for granted.

120
00:06:25,605 --> 00:06:29,871
But as it turns out,
unlike the counting numbers--

121
00:06:29,871 --> 00:06:32,238
one, two, three, and so on--

122
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zero was late to the party.

123
00:06:35,105 --> 00:06:38,271
Maybe that's understandable.

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00:06:38,271 --> 00:06:40,971
Numbers help us keep track
of things,

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00:06:40,971 --> 00:06:43,505
like the number of sheep
you have,

126
00:06:43,505 --> 00:06:46,738
or chickens, or cows.

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00:06:46,738 --> 00:06:51,205
So why keep track of zero goats?

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00:06:51,205 --> 00:06:52,871
LAURIE KEATTS:
Then there would be an
infinite number

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00:06:52,871 --> 00:06:55,938
of things that we're not
counting.

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00:06:55,938 --> 00:07:00,738
The number zero may seem like
it's been with us forever,

131
00:07:00,738 --> 00:07:04,538
but ancient civilizations
had numbers and mathematics

132
00:07:04,538 --> 00:07:07,938
for thousands of years
without it.

133
00:07:07,938 --> 00:07:10,838
For example,
those of Mesopotamia.

134
00:07:10,838 --> 00:07:15,038
That's the historical name
for an area that includes

135
00:07:15,038 --> 00:07:20,271
parts of modern Iraq, Iran,
Syria, and Turkey.

136
00:07:20,271 --> 00:07:23,138
It was home to some of
the earliest cities

137
00:07:23,138 --> 00:07:25,771
and the earliest civilizations
in the world,

138
00:07:25,771 --> 00:07:28,238
as well as an
influential numeral system

139
00:07:28,238 --> 00:07:30,871
based on the number 60.

140
00:07:30,871 --> 00:07:33,305
First invented by the Sumerians,

141
00:07:33,305 --> 00:07:36,105
and later developed
by the Babylonians,

142
00:07:36,105 --> 00:07:40,238
it survived for
thousands of years,

143
00:07:40,238 --> 00:07:42,438
and its legacy is with us today

144
00:07:42,438 --> 00:07:45,638
in the 60 minutes in an hour.

145
00:07:46,771 --> 00:07:49,638
Nearby, and at about
the same time,

146
00:07:49,638 --> 00:07:51,471
were the Ancient Egyptians.

147
00:07:51,471 --> 00:07:55,605
They developed sophisticated
mathematics,

148
00:07:55,605 --> 00:07:58,771
geometry, and astronomy.

149
00:07:58,771 --> 00:08:02,871
They also had their own
hieroglyphic numeral system

150
00:08:02,871 --> 00:08:04,538
that evolved over time.

151
00:08:04,538 --> 00:08:07,138
And just like the Mesopotamians,

152
00:08:07,138 --> 00:08:11,605
the Ancient Egyptians
didn't use the number zero.

153
00:08:11,605 --> 00:08:14,471
Neither did the Greeks

154
00:08:14,471 --> 00:08:17,071
nor the Romans.

155
00:08:17,071 --> 00:08:21,938
Now remember, we're talking
about zero as a number.

156
00:08:21,938 --> 00:08:25,671
For us, zero also acts
as a placeholder,

157
00:08:25,671 --> 00:08:32,038
a way to distinguish
44 from 404.

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00:08:32,038 --> 00:08:36,571
Some ancient numeral systems
had placeholders, as well,

159
00:08:36,571 --> 00:08:38,305
filling in blank spots.

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00:08:38,305 --> 00:08:41,438
But they weren't seen
as a number.

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00:08:41,438 --> 00:08:44,071
They were just a way
to keep things organized.

162
00:08:44,071 --> 00:08:49,338
In fact, as far as
historians can tell,

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00:08:49,338 --> 00:08:54,138
using zero as a number
has only turned up twice.

164
00:08:54,138 --> 00:08:55,905
The Mayans had the idea.

165
00:08:55,905 --> 00:09:00,605
They represented
the number zero with a shell.

166
00:09:00,605 --> 00:09:03,238
But the zero that we
commonly use today

167
00:09:03,238 --> 00:09:06,005
came from another part
of the world.

168
00:09:06,005 --> 00:09:11,038
♪ ♪

169
00:09:11,038 --> 00:09:12,438
The Indian subcontinent

170
00:09:12,438 --> 00:09:17,938
has been home to many societies,
cultures, and traditions,

171
00:09:17,938 --> 00:09:21,471
some dating back hundreds,
if not thousands, of years.

172
00:09:21,471 --> 00:09:26,671
For example,
the colorful festival of Holi,

173
00:09:26,671 --> 00:09:31,071
which celebrates the divine love
of Radha and Krishna.

174
00:09:31,071 --> 00:09:36,638
And it was here in India
about 1,700 years ago

175
00:09:36,638 --> 00:09:40,405
that one of the most powerful
ideas in all of mathematics

176
00:09:40,405 --> 00:09:46,038
is thought by some
to have taken hold-- zero.

177
00:09:46,038 --> 00:09:51,305
♪ ♪

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00:09:51,305 --> 00:09:54,071
To learn more about
India's critical role

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00:09:54,071 --> 00:09:56,971
in zero's history, I've traveled

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00:09:56,971 --> 00:10:00,405
to Princeton University
to speak with one of the most

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00:10:00,405 --> 00:10:04,238
highly regarded mathematicians
in the world,

182
00:10:04,238 --> 00:10:08,338
Manjul Bhargava,
also an accomplished player

183
00:10:08,338 --> 00:10:11,105
of the primary percussion
instrument

184
00:10:11,105 --> 00:10:14,538
in Indian classical music,
the tabla.

185
00:10:14,538 --> 00:10:17,305
(playing rapid rhythm)

186
00:10:25,871 --> 00:10:28,205
Manjul, we've had number systems
for thousands of years,

187
00:10:28,205 --> 00:10:30,738
from the Egyptians
to the Babylonians,

188
00:10:30,738 --> 00:10:33,105
uh, but they didn't seem to
have a need for zero.

189
00:10:33,105 --> 00:10:36,771
Why do you think it started
in India at this time?

190
00:10:36,771 --> 00:10:41,271
The concept of zero started off
in philosophical works.

191
00:10:41,271 --> 00:10:44,005
The state of zero-ness.
Mm-hmm.

192
00:10:44,005 --> 00:10:47,571
The state that we all try
to achieve when we meditate.

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00:10:47,571 --> 00:10:50,205
♪ ♪

194
00:10:50,205 --> 00:10:52,805
WILLIAMS:
In the Hindu and Buddhist
traditions,

195
00:10:52,805 --> 00:10:56,538
both with deep roots
on the Indian subcontinent,

196
00:10:56,538 --> 00:11:03,105
the concept of emptiness
plays a key role.

197
00:11:03,105 --> 00:11:04,905
BHARGAVA:
Emptying the mind

198
00:11:04,905 --> 00:11:06,971
of all sensations,
of all temptations,

199
00:11:06,971 --> 00:11:09,238
of ego, of thoughts,
of emotions.

200
00:11:09,238 --> 00:11:11,205
And so that really

201
00:11:11,205 --> 00:11:13,971
put zero in the air
as, as an important concept.

202
00:11:13,971 --> 00:11:16,871
But the first symbolic
representation of a zero

203
00:11:16,871 --> 00:11:18,838
actually happened in the field
of linguistics.

204
00:11:18,838 --> 00:11:25,538
WILLIAMS:
In about the fifth century BCE,
an Indian scholar, Panini,

205
00:11:25,538 --> 00:11:28,705
laid out the linguistic rules
of what came to be called

206
00:11:28,705 --> 00:11:31,605
Classical Sanskrit.

207
00:11:31,605 --> 00:11:33,838
BHARGAVA:
Sometimes, when you're
pronouncing things,

208
00:11:33,838 --> 00:11:35,205
you like to leave out

209
00:11:35,205 --> 00:11:37,438
a sound when you're, when you're
pronouncing quickly.

210
00:11:39,238 --> 00:11:42,038
So Panini, who is one of the
great grammarians of India,

211
00:11:42,038 --> 00:11:46,471
had a special symbol
when a sound gets deleted.

212
00:11:46,471 --> 00:11:48,338
That was called a lopa.

213
00:11:48,338 --> 00:11:49,571
And that's like a
linguistic zero.

214
00:11:49,571 --> 00:11:50,971
Very parallel to
the modern

215
00:11:50,971 --> 00:11:52,771
apostrophe in
the English language.
Yeah.

216
00:11:52,771 --> 00:11:57,638
(tabla playing)

217
00:11:57,638 --> 00:12:01,638
WILLIAMS:
Traditional Indian music
of the type Manjul plays

218
00:12:01,638 --> 00:12:06,271
is greatly influenced by the
poetic traditions of Sanskrit.

219
00:12:06,271 --> 00:12:12,138
It too will sometimes
omit sounds.

220
00:12:12,138 --> 00:12:13,705
BHARGAVA:
So, when the lopa came to music,

221
00:12:13,705 --> 00:12:18,871
that void is considered just
as important as an actual sound

222
00:12:18,871 --> 00:12:21,071
and can be
just as powerful.

223
00:12:21,071 --> 00:12:24,105
So, occasionally, to emphasize
the downbeat, you won't play it.

224
00:12:24,105 --> 00:12:25,105
So it'll go...

225
00:12:25,105 --> 00:12:28,071
(vocalizing beats)

226
00:12:38,371 --> 00:12:41,438
And so that's how a
musical zero came about.

227
00:12:41,438 --> 00:12:42,438
And a musical zero can be
very powerful.

228
00:12:42,438 --> 00:12:45,638
A zero is like any other note,

229
00:12:45,638 --> 00:12:47,038
that you can use it
in very important moments

230
00:12:47,038 --> 00:12:48,705
and just put the void there.

231
00:12:48,705 --> 00:12:51,371
♪ ♪

232
00:12:51,371 --> 00:12:53,405
WILLIAMS:
The centrality of emptiness

233
00:12:53,405 --> 00:12:56,005
in Indian philosophical
traditions,

234
00:12:56,005 --> 00:12:58,705
and the symbolic
linguistic zero,

235
00:12:58,705 --> 00:13:02,905
may have set the stage
for the number zero.

236
00:13:02,905 --> 00:13:06,305
Many scholars date
its development to sometime

237
00:13:06,305 --> 00:13:09,038
in the first half of
the first millennium,

238
00:13:09,038 --> 00:13:11,405
between the third and fifth
centuries.

239
00:13:11,405 --> 00:13:15,905
But that opinion was originally
based on indirect evidence

240
00:13:15,905 --> 00:13:19,105
because no hard physical proof
had ever been found.

241
00:13:19,105 --> 00:13:21,605
♪ ♪

242
00:13:21,605 --> 00:13:24,738
Some believe that changed
in 2017,

243
00:13:24,738 --> 00:13:27,871
when Oxford University's
Bodleian Libraries

244
00:13:27,871 --> 00:13:33,405
made a surprising announcement
about one of their treasures.

245
00:13:33,405 --> 00:13:35,305
Now scientists from
the University of Oxford

246
00:13:35,305 --> 00:13:38,105
have found a manuscript
that originated in India

247
00:13:38,105 --> 00:13:40,771
and pushes back the discovery
of the concept of zero

248
00:13:40,771 --> 00:13:42,638
by at least 500 years.

249
00:13:42,638 --> 00:13:47,805
WILLIAMS:
The Bakhshali manuscript,
about 70 birch bark pages

250
00:13:47,805 --> 00:13:50,571
of mathematical writings
in Sanskrit,

251
00:13:50,571 --> 00:13:54,138
had been dated to around
800 C.E.

252
00:13:54,138 --> 00:13:57,571
But new carbon dating of
one of its pages

253
00:13:57,571 --> 00:14:01,305
pushed that back about
500 years.

254
00:14:01,305 --> 00:14:03,438
The page shows a dot,
which has been interpreted

255
00:14:03,438 --> 00:14:05,671
to represent zero.

256
00:14:05,671 --> 00:14:06,671
BHARGAVA:
There we see the zero

257
00:14:06,671 --> 00:14:09,205
used in the
Indian number system

258
00:14:09,205 --> 00:14:11,805
just the way that
we write them today.

259
00:14:11,805 --> 00:14:15,671
With one difference, is that
the zero is written as a dot.

260
00:14:15,671 --> 00:14:19,971
WILLIAMS:
If the dating is correct,
the manuscript is now

261
00:14:19,971 --> 00:14:24,705
the earliest evidence
of zero's use as a number.

262
00:14:24,705 --> 00:14:27,571
Not all scholars agree, however,

263
00:14:27,571 --> 00:14:30,605
and the assertion that
the writing is that old

264
00:14:30,605 --> 00:14:33,538
is hotly contested.

265
00:14:33,538 --> 00:14:36,838
However, there's little question

266
00:14:36,838 --> 00:14:39,838
that zero was in use
in mathematics in India

267
00:14:39,838 --> 00:14:41,638
by the seventh century,

268
00:14:41,638 --> 00:14:46,305
in the time of the great
astronomer and mathematician

269
00:14:46,305 --> 00:14:48,338
Brahmagupta.

270
00:14:48,338 --> 00:14:50,105
BHARGAVA:
Brahmagupta came around,

271
00:14:50,105 --> 00:14:52,705
and he said, "Well, zero is a
number just like any other."

272
00:14:52,705 --> 00:14:54,105
So, he actually goes

273
00:14:54,105 --> 00:14:57,305
and writes down rules
for multiplication

274
00:14:57,305 --> 00:14:58,671
and addition
and subtraction of zero.

275
00:14:58,671 --> 00:15:00,305
WILLIAMS:
So he's the first person
to have, like,

276
00:15:00,305 --> 00:15:03,038
thought of how we work
with zero today.

277
00:15:03,038 --> 00:15:04,038
Thought of zero's...
Right, right.
Yeah.

278
00:15:04,038 --> 00:15:07,438
WILLIAMS:
Along with zero,

279
00:15:07,438 --> 00:15:11,471
Brahmagupta also investigated
negative numbers.

280
00:15:11,471 --> 00:15:15,471
Today, when we place zero at the
center of the number line,

281
00:15:15,471 --> 00:15:18,571
between positive
and negative numbers,

282
00:15:18,571 --> 00:15:21,738
that is a legacy of his work.

283
00:15:21,738 --> 00:15:23,105
BHARGAVA:
So, when we talk
about the history of the zero,

284
00:15:23,105 --> 00:15:25,171
from a mathematician's
point of view,

285
00:15:25,171 --> 00:15:26,405
this was the grand moment

286
00:15:26,405 --> 00:15:29,071
where zero became
a full-fledged number

287
00:15:29,071 --> 00:15:31,705
as part of our mathematics,
and that really,

288
00:15:31,705 --> 00:15:33,371
that really changed mathematics.

289
00:15:33,371 --> 00:15:37,971
Do you think it's the, it's the
best idea ever in mathematics?

290
00:15:37,971 --> 00:15:39,505
In science and mathematics,
it's often

291
00:15:39,505 --> 00:15:45,638
the simplest and the most basic
ideas that end up becoming

292
00:15:45,638 --> 00:15:46,838
the most influent...
Revolutionizing the...

293
00:15:46,838 --> 00:15:47,971
Yeah, the most
influential,

294
00:15:47,971 --> 00:15:49,005
the most profound.

295
00:15:49,005 --> 00:15:50,871
Like the wheel.

296
00:15:50,871 --> 00:15:52,705
And it really did

297
00:15:52,705 --> 00:15:55,371
change mathematics and science.
Yeah.

298
00:15:55,371 --> 00:16:01,171
♪ ♪

299
00:16:01,171 --> 00:16:04,771
Before the Indian system became
widely adopted,

300
00:16:04,771 --> 00:16:06,871
the main purpose
of written numerals

301
00:16:06,871 --> 00:16:10,571
was for recording numbers,
not calculating with them.

302
00:16:10,571 --> 00:16:13,771
Instead, calculations
were done with a variety

303
00:16:13,771 --> 00:16:15,171
of techniques and devices--

304
00:16:15,171 --> 00:16:21,738
such as abacuses or counting
boards that used pebbles.

305
00:16:21,738 --> 00:16:25,471
Numerals were only for
storing the results.

306
00:16:25,471 --> 00:16:29,005
But the Indian system
uses the same numerals

307
00:16:29,005 --> 00:16:32,338
for calculation and storage.

308
00:16:32,338 --> 00:16:35,538
Like the number zero, that's
a fundamental breakthrough

309
00:16:35,538 --> 00:16:37,938
we all just take for granted.

310
00:16:37,938 --> 00:16:42,005
The innovative Indian system
would eventually become

311
00:16:42,005 --> 00:16:45,171
the most popular in the world,

312
00:16:45,171 --> 00:16:47,138
but not immediately.

313
00:16:47,138 --> 00:16:50,538
A crucial step in that journey

314
00:16:50,538 --> 00:16:55,005
came out of the remarkable
rise of the Islamic Empire.

315
00:16:55,005 --> 00:16:57,171
Originating in the
Arabian Peninsula

316
00:16:57,171 --> 00:16:59,138
in the seventh century,

317
00:16:59,138 --> 00:17:01,605
after only about a hundred
years,

318
00:17:01,605 --> 00:17:05,905
it had reached India in the east
and Spain in the west.

319
00:17:05,905 --> 00:17:09,738
To learn more about
the key role of Islam

320
00:17:09,738 --> 00:17:12,105
in the spread of Indian numerals
and zero,

321
00:17:12,105 --> 00:17:14,805
I'm visiting the
Hispanic Society of America

322
00:17:14,805 --> 00:17:16,605
in New York City,

323
00:17:16,605 --> 00:17:20,271
which houses perhaps the most
influential work

324
00:17:20,271 --> 00:17:23,638
in that journey.

325
00:17:23,638 --> 00:17:26,938
I'm joined by Waleed el-Ansary.

326
00:17:26,938 --> 00:17:29,805
He's an expert in
Islamic studies

327
00:17:29,805 --> 00:17:33,105
and the intersection
of religion, science,

328
00:17:33,105 --> 00:17:36,171
and economics, and like me,
eager to see

329
00:17:36,171 --> 00:17:39,038
the rare manuscript.

330
00:17:39,038 --> 00:17:40,638
Its roots go back to
what was then

331
00:17:40,638 --> 00:17:42,771
a recently constructed city

332
00:17:42,771 --> 00:17:46,771
and a new political
and cultural center of Islam:

333
00:17:46,771 --> 00:17:49,605
Baghdad.

334
00:17:49,605 --> 00:17:52,471
EL-ANSARY:
So, Baghdad was designed
in a circular shape,

335
00:17:52,471 --> 00:17:55,338
after Euclid's writings.

336
00:17:55,338 --> 00:17:59,205
And the circle is viewed
as the perfect shape,

337
00:17:59,205 --> 00:18:03,238
and therefore it's
a symbol, in a sense, of God.

338
00:18:03,238 --> 00:18:06,838
WILLIAMS:
Strategically located
at the crossroads

339
00:18:06,838 --> 00:18:11,338
of several trade routes,
the city quickly grew.

340
00:18:11,338 --> 00:18:13,005
And it became the
largest city in the world.

341
00:18:13,005 --> 00:18:15,405
It's really quite amazing.

342
00:18:15,405 --> 00:18:20,371
This center for trade
on one hand,

343
00:18:20,371 --> 00:18:22,105
as well as
intellectual trade.
Hm.

344
00:18:22,105 --> 00:18:26,371
The transfer and
transmission of ideas.

345
00:18:26,371 --> 00:18:29,238
WILLIAMS:
Scholars translated texts
that had been gathered

346
00:18:29,238 --> 00:18:34,305
from across the Islamic world
and beyond,

347
00:18:34,305 --> 00:18:37,505
including those about
Indian mathematics.

348
00:18:37,505 --> 00:18:40,505
EL-ANSARY:
They viewed all knowledge
coming from these

349
00:18:40,505 --> 00:18:43,438
other civilizations
that was consistent with

350
00:18:43,438 --> 00:18:45,005
the unity of God

351
00:18:45,005 --> 00:18:47,638
as being Islamic in the
deepest sense of the word.
Mm.

352
00:18:47,638 --> 00:18:51,171
And so it was very easy for the
Muslims to integrate that

353
00:18:51,171 --> 00:18:52,838
into their worldview.

354
00:18:52,838 --> 00:18:57,038
Sounds like they were also
the curators of this knowledge.

355
00:18:57,038 --> 00:18:59,071
And, and once they sort of
brought it together,

356
00:18:59,071 --> 00:19:00,738
they then built on it, as well.

357
00:19:00,738 --> 00:19:04,871
That's right, it wasn't just
Aristotle in Arabic.
That's right.

358
00:19:04,871 --> 00:19:06,938
Yeah.
It was more than that.

359
00:19:06,938 --> 00:19:13,305
♪ ♪

360
00:19:13,305 --> 00:19:15,805
WILLIAMS:
In the early part
of the ninth century,

361
00:19:15,805 --> 00:19:19,305
Muhammad ibn Musa al-Khwarizmi,

362
00:19:19,305 --> 00:19:22,471
a Persian scholar in
a variety of subjects,

363
00:19:22,471 --> 00:19:26,771
wrote several
hugely influential books.

364
00:19:26,771 --> 00:19:32,338
Two had a powerful impact
on mathematics.

365
00:19:32,338 --> 00:19:36,971
In one, he laid out
the foundations of algebra.

366
00:19:36,971 --> 00:19:39,571
In fact, part of the title
of the book would give

367
00:19:39,571 --> 00:19:42,471
the subject its name.

368
00:19:42,471 --> 00:19:45,805
Another of his key works
in mathematics,

369
00:19:45,805 --> 00:19:50,405
which only survives today in a
13th-century Latin translation,

370
00:19:50,405 --> 00:19:53,838
is what's brought us to the
Hispanic Society of America,

371
00:19:53,838 --> 00:19:59,071
home to one of the oldest
and the most complete version.

372
00:19:59,071 --> 00:20:00,738
EL-ANSARY:
This is a gem.

373
00:20:00,738 --> 00:20:04,638
And so you can see here
the Indian Arabic

374
00:20:04,638 --> 00:20:07,138
numeral system.
Yeah.

375
00:20:07,138 --> 00:20:10,605
With zero, one, two,
three, four, five,

376
00:20:10,605 --> 00:20:13,571
six, seven, eight, nine.

377
00:20:13,571 --> 00:20:18,338
And some of them
are shaped very similar

378
00:20:18,338 --> 00:20:20,938
to what we have today,
some of them are not.
WILLIAMS: Mm-hmm.

379
00:20:20,938 --> 00:20:22,438
Mathematics today,
the foundation

380
00:20:22,438 --> 00:20:24,705
is right here in front of us.

381
00:20:24,705 --> 00:20:25,705
That's right.
WILLIAMS:
Yeah.

382
00:20:25,705 --> 00:20:28,205
Which is unbelievable.

383
00:20:28,205 --> 00:20:29,638
(laughs)

384
00:20:29,638 --> 00:20:33,705
WILLIAMS:
The purpose of the book
was to promote

385
00:20:33,705 --> 00:20:38,405
the Indian numeral system and
explain its key innovations,

386
00:20:38,405 --> 00:20:42,905
zero and the use of the
numerals for arithmetic.

387
00:20:42,905 --> 00:20:45,671
♪ ♪

388
00:20:45,671 --> 00:20:49,571
The book also included
procedures for computation

389
00:20:49,571 --> 00:20:53,305
that would come to be known as
algorithms,

390
00:20:53,305 --> 00:20:56,471
a corruption of
al-Khwarizmi's name.

391
00:20:56,471 --> 00:21:01,038
EL-ANSARY:
So it's a little manual
to show people

392
00:21:01,038 --> 00:21:02,971
how to operate with these.

393
00:21:02,971 --> 00:21:05,738
And we learn this as, as kids,
so in some ways,

394
00:21:05,738 --> 00:21:07,438
we take it for granted,
but you're right, it's,

395
00:21:07,438 --> 00:21:09,838
someone had to say,
"This is the process

396
00:21:09,838 --> 00:21:11,338
"that we're going to use

397
00:21:11,338 --> 00:21:13,638
in order to build this
mathematical knowledge."

398
00:21:13,638 --> 00:21:15,105
And here it is.
That's right.

399
00:21:15,105 --> 00:21:17,638
Wow, wow.
That's right, so this
is very foundational.

400
00:21:17,638 --> 00:21:21,405
WILLIAMS:
Al-Khwarizmi's work,

401
00:21:21,405 --> 00:21:24,605
along with that of other
Islamic mathematicians,

402
00:21:24,605 --> 00:21:26,871
helped spread the
Indian numeral system

403
00:21:26,871 --> 00:21:31,105
throughout the Islamic world,
and eventually beyond.

404
00:21:31,105 --> 00:21:34,471
The Islamic promotion
of the Indian numeral system

405
00:21:34,471 --> 00:21:38,405
was so successful, the numbers
would even come to be known

406
00:21:38,405 --> 00:21:43,505
as Arabic numerals, somewhat
obscuring their Indian origins.

407
00:21:43,505 --> 00:21:47,271
So what we're looking at here
is something that is now

408
00:21:47,271 --> 00:21:48,538
not only used

409
00:21:48,538 --> 00:21:51,038
in the Islamic world
and the West,

410
00:21:51,038 --> 00:21:54,071
but really is the most important
numeral system

411
00:21:54,071 --> 00:21:55,838
for the entire world.
Yeah.

412
00:21:55,838 --> 00:21:58,538
And so I can hardly
overemphasize

413
00:21:58,538 --> 00:22:00,971
the significance of this text.

414
00:22:00,971 --> 00:22:05,505
♪ ♪

415
00:22:05,505 --> 00:22:08,271
WILLIAMS:
In Europe, the Indian-Arabic
numeral system,

416
00:22:08,271 --> 00:22:10,938
with its revolutionary zero,

417
00:22:10,938 --> 00:22:14,271
would eventually have
a powerful role

418
00:22:14,271 --> 00:22:17,738
in the advancement of science.

419
00:22:17,738 --> 00:22:21,571
But the earliest users
were Italian merchants

420
00:22:21,571 --> 00:22:24,038
who saw its immediate advantages

421
00:22:24,038 --> 00:22:27,138
for calculations
and business records.

422
00:22:27,138 --> 00:22:32,005
In fact, in 1202,
the son of a merchant,

423
00:22:32,005 --> 00:22:36,738
Leonardo of Pisa-- better known
today as Fibonacci--

424
00:22:36,738 --> 00:22:41,571
wrote "Liber abaci,"
an influential book

425
00:22:41,571 --> 00:22:44,771
about the new numerals
advocating for their use.

426
00:22:44,771 --> 00:22:49,071
Ultimately, it would take
hundreds of years

427
00:22:49,071 --> 00:22:53,271
for the new numerals to displace
both the existing systems

428
00:22:53,271 --> 00:22:55,138
for recording numbers,

429
00:22:55,138 --> 00:23:00,005
such as Roman numerals,
and the various devices

430
00:23:00,005 --> 00:23:03,505
and techniques used
for calculating.

431
00:23:03,505 --> 00:23:06,338
But by the late 16th century,

432
00:23:06,338 --> 00:23:09,271
in part aided by the advent
of the printing press

433
00:23:09,271 --> 00:23:11,471
and growing literacy,

434
00:23:11,471 --> 00:23:14,938
the new system had been
widely adopted in Europe.

435
00:23:14,938 --> 00:23:17,971
♪ ♪

436
00:23:17,971 --> 00:23:19,371
BHARGAVA:
Because of the European
Renaissance,

437
00:23:19,371 --> 00:23:22,838
it started becoming impossible
to really make those

438
00:23:22,838 --> 00:23:26,238
huge scientific leaps without
switching over to zero

439
00:23:26,238 --> 00:23:28,738
and the Indian system
of enumeration,

440
00:23:28,738 --> 00:23:30,471
the system that allowed you
to really

441
00:23:30,471 --> 00:23:33,005
do computations easily.

442
00:23:33,005 --> 00:23:35,871
And so it started becoming
impossible not to use them.

443
00:23:35,871 --> 00:23:39,805
And so by the 17th century, they
started becoming in regular use

444
00:23:39,805 --> 00:23:41,438
in Europe
and then around the world,

445
00:23:41,438 --> 00:23:44,305
and the rest is history.

446
00:23:44,305 --> 00:23:49,338
♪ ♪

447
00:23:54,738 --> 00:23:58,038
Treating zero as a number
transformed mathematics,

448
00:23:58,038 --> 00:24:00,771
but it did take
some getting used to.

449
00:24:00,771 --> 00:24:05,371
Because, in some ways, zero
isn't like any other number.

450
00:24:05,371 --> 00:24:07,405
First of all,
it, it has unique properties.

451
00:24:07,405 --> 00:24:10,471
Zero has some properties
of number,

452
00:24:10,471 --> 00:24:12,205
but also some properties
that make it more

453
00:24:12,205 --> 00:24:13,938
like a concept than a number.

454
00:24:15,438 --> 00:24:18,438
WILLIAMS:
In addition, subtraction,
and multiplication,

455
00:24:18,438 --> 00:24:20,905
zero behaves differently

456
00:24:20,905 --> 00:24:23,405
than every other number.

457
00:24:23,405 --> 00:24:27,938
But where zero really
creates havoc is in division.

458
00:24:27,938 --> 00:24:30,405
You get to division, and all of
a sudden, it's the first time

459
00:24:30,405 --> 00:24:32,538
that you're sort of told, like,
"Well, that's impossible."

460
00:24:32,538 --> 00:24:37,105
WILLIAMS:
You can divide any number
by every other number

461
00:24:37,105 --> 00:24:39,005
except zero.

462
00:24:39,005 --> 00:24:42,805
When you divide a number by
zero, for example, you blow up.

463
00:24:42,805 --> 00:24:45,438
ANNOUNCER:
Three, two, one, zero.

464
00:24:45,438 --> 00:24:49,505
I have no apples, and I share
that among six students,

465
00:24:49,505 --> 00:24:52,138
wouldn't everybody get
zero apples?

466
00:24:52,138 --> 00:24:54,705
There are no apples to share.

467
00:24:54,705 --> 00:24:57,238
But if I have six apples
and they are shared

468
00:24:57,238 --> 00:25:02,171
among zero students, I, the,
the concept becomes messy now.

469
00:25:02,171 --> 00:25:04,538
How do we make sense of that?

470
00:25:04,538 --> 00:25:07,171
The problem is, you can't.

471
00:25:07,171 --> 00:25:08,938
Think of it this way:

472
00:25:08,938 --> 00:25:14,405
dividing six by zero is the
same thing as asking what number

473
00:25:14,405 --> 00:25:19,071
multiplied by zero
will give you six?

474
00:25:19,071 --> 00:25:22,805
Since everything multiplied
by zero always equals zero,

475
00:25:22,805 --> 00:25:24,971
there's no solution.

476
00:25:24,971 --> 00:25:27,771
So mathematicians officially
consider the answer

477
00:25:27,771 --> 00:25:30,705
as undefined.

478
00:25:30,705 --> 00:25:33,905
Now, you might wonder,
is that sort of

479
00:25:33,905 --> 00:25:37,105
hole in the bucket of division
a problem?

480
00:25:37,105 --> 00:25:39,405
Does it get you into trouble?

481
00:25:39,405 --> 00:25:44,538
Turns out it certainly does,
under the right circumstances.

482
00:25:44,538 --> 00:25:47,771
In fact, a Greek philosopher

483
00:25:47,771 --> 00:25:49,538
who lived thousands
of years ago,

484
00:25:49,538 --> 00:25:52,071
before zero even came to be,

485
00:25:52,071 --> 00:25:55,871
invented a paradox that captures
the problem.

486
00:25:55,871 --> 00:25:58,705
His name was Zeno of Elea.

487
00:25:58,705 --> 00:26:02,905
And the paradox was about
an arrow.

488
00:26:02,905 --> 00:26:06,871
♪ ♪

489
00:26:06,871 --> 00:26:10,538
To help me demonstrate
Zeno's Paradox,

490
00:26:10,538 --> 00:26:13,671
I've turned to Eric Bennett
from Surprise, Arizona.

491
00:26:13,671 --> 00:26:17,705
VF is what we're looking for.

492
00:26:17,705 --> 00:26:20,871
WILLIAMS:
He's a physics and engineering
teacher at a local high school.

493
00:26:20,871 --> 00:26:22,638
And he's a Paralympian in
archery,

494
00:26:22,638 --> 00:26:25,605
four times over.

495
00:26:25,605 --> 00:26:27,671
So Eric, what does it feel like

496
00:26:27,671 --> 00:26:29,138
to have participated in
the Paralympics

497
00:26:29,138 --> 00:26:30,971
four times?

498
00:26:30,971 --> 00:26:32,871
Um, it makes me feel old
a little bit.

499
00:26:32,871 --> 00:26:34,371
(both laugh)

500
00:26:34,371 --> 00:26:35,805
But, um, it's, it's amazing.

501
00:26:35,805 --> 00:26:38,271
I've been competing at a really
high level for 15 years.

502
00:26:38,271 --> 00:26:40,171
Wow, wow.

503
00:26:40,171 --> 00:26:42,438
So how far away is
the target here?

504
00:26:42,438 --> 00:26:44,138
The target is the
standard Olympic

505
00:26:44,138 --> 00:26:47,138
competition distance of 70,
meters,

506
00:26:47,138 --> 00:26:49,238
which is about three-quarters
of a football field.

507
00:26:49,238 --> 00:26:50,538
No way!
Yes,

508
00:26:50,538 --> 00:26:52,805
actually, it's pretty far.
(both laugh)

509
00:26:52,805 --> 00:26:54,905
Okay, all right,
I want to see you shoot this.

510
00:26:56,605 --> 00:26:58,105
WILLIAMS:
At 15 years old,

511
00:26:58,105 --> 00:27:01,805
Eric lost an arm in an
automobile accident.

512
00:27:01,805 --> 00:27:05,905
So he draws the bowstring
back with his teeth.

513
00:27:07,538 --> 00:27:08,705
(arrow hits target)

514
00:27:08,705 --> 00:27:11,171
The arrow finds its mark.

515
00:27:11,171 --> 00:27:14,471
(laughs):
Wow, that's awesome.

516
00:27:14,471 --> 00:27:17,438
All right, so you're going to
show me how to use one of these?

517
00:27:17,438 --> 00:27:18,471
Absolutely, yup.

518
00:27:18,471 --> 00:27:20,738
Okay, from, from 70 meters?

519
00:27:20,738 --> 00:27:22,438
No, and that's okay.
I can try!

520
00:27:22,438 --> 00:27:23,938
Are you trying to
say I can't hit it

521
00:27:23,938 --> 00:27:25,405
from this distance?
No, I just want to make sure

522
00:27:25,405 --> 00:27:27,005
that you're super-successful
on your first try.

523
00:27:27,005 --> 00:27:28,705
Okay, I appreciate that--
I appreciate it.
Yeah.

524
00:27:30,338 --> 00:27:33,871
WILLIAMS:
Eric offers me
a try with a beginner's bow

525
00:27:33,871 --> 00:27:37,705
and a target about 20 yards
away.

526
00:27:37,705 --> 00:27:40,538
Let it go and it will go
right into the bullseye.

527
00:27:40,538 --> 00:27:43,005
(both laugh)

528
00:27:44,638 --> 00:27:47,538
So, I channel my inner Katniss
Everdeen

529
00:27:47,538 --> 00:27:49,771
from "The Hunger Games."

530
00:27:49,771 --> 00:27:54,871
♪ ♪

531
00:27:54,871 --> 00:27:57,538
And as a statistician,

532
00:27:57,538 --> 00:28:01,505
"May the odds be ever in my
favor."

533
00:28:04,438 --> 00:28:05,738
(arrow misses)

534
00:28:05,738 --> 00:28:07,471
Whoa!
What, I don't know--
where'd it go?

535
00:28:07,471 --> 00:28:09,138
(laughs)

536
00:28:09,138 --> 00:28:11,705
That is, like, a hundred yards
down the road we'll find it.

537
00:28:11,705 --> 00:28:13,005
(laughs)

538
00:28:13,005 --> 00:28:15,638
Got a lot of work to do,
Eric, come on.
Yeah.

539
00:28:15,638 --> 00:28:18,171
WILLIAMS:
Well, I think it's going to be
a while

540
00:28:18,171 --> 00:28:20,838
before I'm ready to compete.

541
00:28:20,838 --> 00:28:22,605
I had a lot of power
you know?

542
00:28:22,605 --> 00:28:23,838
Yeah!
And so, um...

543
00:28:23,838 --> 00:28:27,505
WILLIAMS:
But back to Zeno and that
paradox.

544
00:28:29,771 --> 00:28:34,005
All of Zeno's original writings
have been lost,

545
00:28:34,005 --> 00:28:36,271
but according to a later Greek
philosopher,

546
00:28:36,271 --> 00:28:38,271
Zeno suggested

547
00:28:38,271 --> 00:28:42,038
that we consider an arrow
in flight

548
00:28:42,038 --> 00:28:44,438
at any instant in time.

549
00:28:44,438 --> 00:28:45,638
And at that instant,

550
00:28:45,638 --> 00:28:49,905
that "now" moment,

551
00:28:49,905 --> 00:28:55,071
the arrow is frozen in space,
motionless.

552
00:28:55,071 --> 00:28:58,471
It's neither arriving
nor leaving.

553
00:28:58,471 --> 00:29:01,405
And if you consider the entire
flight...

554
00:29:03,471 --> 00:29:08,038
...there's an infinity of those
motionless, frozen moments

555
00:29:08,038 --> 00:29:10,205
in time and space.

556
00:29:10,205 --> 00:29:14,105
So, Zeno asked,
is the flight of the arrow,

557
00:29:14,105 --> 00:29:18,271
and all motion,
really just an illusion?

558
00:29:22,571 --> 00:29:25,371
STEVEN STROGATZ:
His radical conclusion is that
motion is impossible.

559
00:29:25,371 --> 00:29:29,938
At a given instant,
that arrow is someplace.

560
00:29:29,938 --> 00:29:33,605
And then click time forward.

561
00:29:33,605 --> 00:29:37,305
(chuckles):
It's at some other place,
but at no moment was it moving.

562
00:29:37,305 --> 00:29:40,538
Okay.
BENNETT:
And when you're ready, let go.

563
00:29:40,538 --> 00:29:42,205
(arrow hits target)

564
00:29:42,205 --> 00:29:44,771
What? Did you hear that?
Did you hear that?

565
00:29:46,005 --> 00:29:50,205
WILLIAMS:
Well, the motion of an arrow
looks real enough for me.

566
00:29:50,205 --> 00:29:52,638
That's right, Katniss--
got nothing on me.

567
00:29:54,105 --> 00:29:55,705
WILLIAMS:
But you can see why Zeno's

568
00:29:55,705 --> 00:29:59,705
timeless frozen moments
are so problematic.

569
00:29:59,705 --> 00:30:04,705
Our whole notion of speed
depends on time.

570
00:30:04,705 --> 00:30:08,005
Here's the formula:

571
00:30:08,005 --> 00:30:10,971
distance traveled divided by
length of time

572
00:30:10,971 --> 00:30:13,105
equals speed.

573
00:30:13,105 --> 00:30:19,305
But Zeno's frozen moment has a
length of time of zero.

574
00:30:19,305 --> 00:30:22,538
That means trying to divide by
zero,

575
00:30:22,538 --> 00:30:24,838
which is against
the rules of division.

576
00:30:26,005 --> 00:30:27,438
But at the same time,

577
00:30:27,438 --> 00:30:29,605
we often want to know the speed
of something

578
00:30:29,605 --> 00:30:32,038
in motion at a particular
instant.

579
00:30:33,338 --> 00:30:37,305
One solution to the problem
of instantaneous speed

580
00:30:37,305 --> 00:30:39,138
is a concept called

581
00:30:39,138 --> 00:30:42,171
a limit.

582
00:30:42,171 --> 00:30:44,471
Let's consider a stick figure

583
00:30:44,471 --> 00:30:48,438
who walks half the distance to
a wall,

584
00:30:48,438 --> 00:30:53,338
and does that again,
and again, and again.

585
00:30:53,338 --> 00:30:56,771
If the stick figure keeps going
half the distance to the wall,

586
00:30:56,771 --> 00:31:00,005
they'll get closer and closer,

587
00:31:00,005 --> 00:31:03,338
but the steps will get smaller
and smaller,

588
00:31:03,338 --> 00:31:06,005
and they'll never reach the
wall.

589
00:31:06,005 --> 00:31:08,771
The wall is an example
of a limit.

590
00:31:08,771 --> 00:31:11,605
As the number of steps
heads to infinity,

591
00:31:11,605 --> 00:31:15,305
the distance to the wall
decreases towards zero,

592
00:31:15,305 --> 00:31:19,438
but the figure
will never reach the wall.

593
00:31:19,438 --> 00:31:21,505
You're getting infinitely close
to a limit,

594
00:31:21,505 --> 00:31:24,638
as far as you're gonna get, but
you never actually get there.

595
00:31:24,638 --> 00:31:26,038
Which, yeah, it's one of those
concepts

596
00:31:26,038 --> 00:31:27,438
that bothers a lot of people.

597
00:31:27,438 --> 00:31:29,705
Even mathematicians it bothers,
I think.

598
00:31:29,705 --> 00:31:31,405
I can never start with a whole
number

599
00:31:31,405 --> 00:31:34,471
and divide it by
something to get zero.

600
00:31:34,471 --> 00:31:37,638
There is nothing-- there is no
way for me to ever get to zero.

601
00:31:37,638 --> 00:31:39,671
Even if you have an itty-bitty
bit

602
00:31:39,671 --> 00:31:43,838
and you divide it in half,
you still don't have zero.

603
00:31:45,405 --> 00:31:48,805
WILLIAMS:
Harnessing the power of
infinity through limits

604
00:31:48,805 --> 00:31:51,305
gives mathematicians a
work-around

605
00:31:51,305 --> 00:31:53,638
to the problem of dividing by
zero,

606
00:31:53,638 --> 00:31:57,905
and in turn opens the door
to a world of solutions

607
00:31:57,905 --> 00:32:01,438
to some extremely difficult
problems.

608
00:32:01,438 --> 00:32:06,605
It helped create a new field
of mathematics: calculus.

609
00:32:06,605 --> 00:32:09,071
And that's really the big idea
at the heart of calculus

610
00:32:09,071 --> 00:32:12,005
as understood in modern terms,
this idea of a limit.

611
00:32:12,005 --> 00:32:13,805
That you're supposed to think,

612
00:32:13,805 --> 00:32:18,371
how far did I go over a
microsecond?

613
00:32:18,371 --> 00:32:21,105
That gives me an approximation
to my instantaneous velocity,

614
00:32:21,105 --> 00:32:23,705
you know, the distance traveled
divided by that duration,

615
00:32:23,705 --> 00:32:25,805
but that's not yet an instant.

616
00:32:25,805 --> 00:32:28,338
So rather than a microsecond,
I think now a nanosecond--

617
00:32:28,338 --> 00:32:31,438
a thousand times shorter--
how far did I travel then?

618
00:32:31,438 --> 00:32:33,805
That gives me a better
approximation.

619
00:32:33,805 --> 00:32:36,271
And then this limit, as the
duration of time goes to zero,

620
00:32:36,271 --> 00:32:38,071
you often find

621
00:32:38,071 --> 00:32:41,638
you'll get a well-defined
limiting answer for the,

622
00:32:41,638 --> 00:32:43,705
for the speed, and that limit
is what's called

623
00:32:43,705 --> 00:32:45,405
the instantaneous velocity.

624
00:32:47,605 --> 00:32:50,005
WILLIAMS:
It sounds like a clever trick,

625
00:32:50,005 --> 00:32:52,438
but does it get the job done?

626
00:32:52,438 --> 00:32:55,538
To find out, I travel to New
York City

627
00:32:55,538 --> 00:33:00,471
to the National Museum of
Mathematics, MoMath.

628
00:33:00,471 --> 00:33:02,205
STROGATZ: May I?
WILLIAMS: Please, thank you.

629
00:33:02,205 --> 00:33:03,838
STROGATZ:
Take your pick.

630
00:33:03,838 --> 00:33:06,771
WILLIAMS:
Here, Cornell University
mathematician

631
00:33:06,771 --> 00:33:09,371
Steve Strogatz is
enjoying a year

632
00:33:09,371 --> 00:33:11,705
as a distinguished visiting
professor.

633
00:33:11,705 --> 00:33:14,138
13 points,
thank you very much!
(laughs)

634
00:33:14,138 --> 00:33:15,971
WILLIAMS:
He shows me around.

635
00:33:15,971 --> 00:33:17,538
Ooh.

636
00:33:17,538 --> 00:33:19,605
WILLIAMS:
But I'm here for a
specific reason.

637
00:33:19,605 --> 00:33:21,571
Steve is going to demonstrate

638
00:33:21,571 --> 00:33:25,238
the problem-solving power
of limits and infinity,

639
00:33:25,238 --> 00:33:26,938
though, as it turns out...

640
00:33:26,938 --> 00:33:28,238
Whoa!

641
00:33:28,238 --> 00:33:30,238
WILLIAMS:
...we're missing the key
component.

642
00:33:30,238 --> 00:33:31,271
(squeals, laughs)

643
00:33:31,271 --> 00:33:33,671
(crew exclaiming)

644
00:33:33,671 --> 00:33:36,338
If you want to understand what
infinity can do,

645
00:33:36,338 --> 00:33:38,905
we're gonna need pizza.

646
00:33:38,905 --> 00:33:40,305
Pizza?

647
00:33:40,305 --> 00:33:41,805
WILLIAMS:
Yes!

648
00:33:41,805 --> 00:33:44,038
There's a science
to making pizza.

649
00:33:44,038 --> 00:33:45,138
WILLIAMS:
We don't typically
associate

650
00:33:45,138 --> 00:33:47,805
pizza with infinity.

651
00:33:47,805 --> 00:33:48,905
Ay-yi-yi!

652
00:33:48,905 --> 00:33:51,038
WILLIAMS:
So how can New York City's

653
00:33:51,038 --> 00:33:52,305
most famous food...

654
00:33:52,305 --> 00:33:53,438
(Strogatz chortles)

655
00:33:53,438 --> 00:33:54,838
WILLIAMS:
...help solve one of

656
00:33:54,838 --> 00:33:57,505
the most elusive mysteries of
early mathematics?

657
00:33:59,305 --> 00:34:00,938
(both laugh)

658
00:34:05,038 --> 00:34:06,838
WILLIAMS:
So Steve,

659
00:34:06,838 --> 00:34:10,405
how is this pizza going to help
us understand infinity?

660
00:34:10,405 --> 00:34:12,371
Huh, I would say it
the other way.

661
00:34:12,371 --> 00:34:15,738
Infinity and the pizza are
gonna help us understand

662
00:34:15,738 --> 00:34:18,238
one of the oldest problems
in math.

663
00:34:18,238 --> 00:34:20,305
Mm-hmm?
What's the area of a circle?

664
00:34:20,305 --> 00:34:22,305
Which is not intuitive.
No!

665
00:34:22,305 --> 00:34:23,771
You know, what's hard about it,

666
00:34:23,771 --> 00:34:25,905
you might think a circle is a
beautiful, simple shape.

667
00:34:25,905 --> 00:34:28,071
But actually, it's got
this nasty property

668
00:34:28,071 --> 00:34:30,571
that it doesn't have any
straight lines in it.
Right.

669
00:34:30,571 --> 00:34:32,905
Ancient civilizations
didn't know how to find

670
00:34:32,905 --> 00:34:36,838
the area of a shape like that.

671
00:34:37,838 --> 00:34:42,905
WILLIAMS:
How to find the exact area of a
circle isn't obvious.

672
00:34:42,905 --> 00:34:44,905
For a square or rectangle,

673
00:34:44,905 --> 00:34:48,071
you just multiply the sides.

674
00:34:48,071 --> 00:34:50,705
But what do you do with a
circle?

675
00:34:50,705 --> 00:34:52,205
So what did they do?

676
00:34:52,205 --> 00:34:55,071
Well, they came up with an
argument that you can convert

677
00:34:55,071 --> 00:34:58,305
a round shape into a rectangle
if you use infinity.

678
00:34:58,305 --> 00:35:00,671
So we're basically gonna
kind of deconstruct this pizza,

679
00:35:00,671 --> 00:35:02,105
make it into a rectangle...
Beautiful.

680
00:35:02,105 --> 00:35:03,771
And then we're gonna
know the area.
That's it.

681
00:35:03,771 --> 00:35:06,405
So I'm gonna start
with four pieces.
Okay.

682
00:35:06,405 --> 00:35:10,971
STROGATZ:
To do that, I'm gonna go one
point up and one point down.

683
00:35:10,971 --> 00:35:12,405
WILLIAMS:
Mm-hmm.

684
00:35:12,405 --> 00:35:16,071
And then one point up and one
point down, and...

685
00:35:16,071 --> 00:35:17,571
Yeah, like that.

686
00:35:17,571 --> 00:35:19,105
Uh, how'd you do in geometry?

687
00:35:19,105 --> 00:35:22,205
STROGATZ (laughs):
You don't think that
looks like a rectangle?

688
00:35:22,205 --> 00:35:23,705
That is not close
to a rectangle.

689
00:35:23,705 --> 00:35:24,905
No, no.
No, it's not, it's not.

690
00:35:24,905 --> 00:35:27,038
But come on,
I'm only using four pieces.

691
00:35:27,038 --> 00:35:29,471
If I use more,
I can get closer.
Okay, all right.

692
00:35:29,471 --> 00:35:31,705
So we gotta cut
these babies in half.
Let's cut 'em.

693
00:35:34,205 --> 00:35:36,338
Let's rearrange them,
same trick.

694
00:35:36,338 --> 00:35:38,571
Alternating point up
and point down.

695
00:35:40,471 --> 00:35:42,338
One up and one down.

696
00:35:43,638 --> 00:35:45,638
And one up and one down.

697
00:35:45,638 --> 00:35:47,071
Now we are ready!

698
00:35:47,071 --> 00:35:49,638
That is looking a lot better!
Aw!

699
00:35:49,638 --> 00:35:51,238
What do you think,
is that a rectangle?

700
00:35:51,238 --> 00:35:54,071
Um, it's, it's not quite a
rectangle,

701
00:35:54,071 --> 00:35:55,471
but it's getting closer.
It is, right?

702
00:35:55,471 --> 00:35:57,105
Yeah!

703
00:35:57,105 --> 00:35:59,838
WILLIAMS:
In both the four-piece
and eight-piece versions,

704
00:35:59,838 --> 00:36:04,638
half the crust sits at the top
and half at the bottom.

705
00:36:04,638 --> 00:36:08,505
But with eight pieces, the edge
becomes less scalloped,

706
00:36:08,505 --> 00:36:10,705
closer to a straight line.

707
00:36:10,705 --> 00:36:12,505
So we need to go at
least a step further.

708
00:36:12,505 --> 00:36:14,138
STROGATZ:
Let's go more--
we gotta do 16.

709
00:36:16,705 --> 00:36:18,871
So we have to just change

710
00:36:18,871 --> 00:36:20,605
every other one--
am I going to mess this up?

711
00:36:20,605 --> 00:36:22,271
I mean, that's...
Wow.

712
00:36:22,271 --> 00:36:23,871
That's a parallelogram

713
00:36:23,871 --> 00:36:26,705
that's aspiring
to be a rectangle.
(laughs)

714
00:36:26,705 --> 00:36:27,705
That's got aspirations!
Yeah, it's got high hopes.

715
00:36:27,705 --> 00:36:29,705
It's got high hopes, I tell you.

716
00:36:29,705 --> 00:36:33,338
WILLIAMS:
From four slices,

717
00:36:33,338 --> 00:36:37,371
to eight slices,

718
00:36:37,371 --> 00:36:40,305
to 16 slices,

719
00:36:40,305 --> 00:36:42,771
and even 32 slices,

720
00:36:42,771 --> 00:36:46,238
there's a clear progression
towards a rectangle.

721
00:36:46,238 --> 00:36:50,571
With one piece out of 32 cut in
half to create vertical sides,

722
00:36:50,571 --> 00:36:53,471
the rectangle is almost
complete,

723
00:36:53,471 --> 00:36:56,638
except for the wavy top
and bottom.

724
00:36:56,638 --> 00:36:59,838
But as the number of slices
increases,

725
00:36:59,838 --> 00:37:03,971
the straighter and straighter
those edges would become.

726
00:37:03,971 --> 00:37:06,938
And the argument here is that if
we could keep doing this

727
00:37:06,938 --> 00:37:08,571
all the way out to infinity...
Mm-hmm.

728
00:37:08,571 --> 00:37:10,805
...so that this would be
infinitely many slices,

729
00:37:10,805 --> 00:37:12,405
infinitesimally thin,

730
00:37:12,405 --> 00:37:14,638
this really would become
a rectangle.
Yeah.

731
00:37:14,638 --> 00:37:17,138
STROGATZ:
And we can read off the area.

732
00:37:17,138 --> 00:37:18,605
WILLIAMS: That's right.
STROGATZ: It's this radius,

733
00:37:18,605 --> 00:37:21,271
that's the distance from the
center out to the crust...

734
00:37:21,271 --> 00:37:24,105
WILLIAMS: Mm-hmm...
STROGATZ:
...times half the circumference,

735
00:37:24,105 --> 00:37:26,805
which is half the crust,
half the curvy stuff.

736
00:37:26,805 --> 00:37:28,671
And that's a famous formula.

737
00:37:28,671 --> 00:37:30,405
Half the crust
times the radius.
Yeah!

738
00:37:30,405 --> 00:37:31,638
(One-half C)R.

739
00:37:31,638 --> 00:37:33,238
That's what the C is for?

740
00:37:33,238 --> 00:37:35,671
Usually, C for circumference,
but you could see it's crust.

741
00:37:35,671 --> 00:37:38,605
So, at the limit, once we
got all the way out there,

742
00:37:38,605 --> 00:37:40,005
it's going to look like
a rectangle.

743
00:37:40,005 --> 00:37:40,971
It would be a rectangle,

744
00:37:40,971 --> 00:37:42,238
and that is actually

745
00:37:42,238 --> 00:37:44,338
the first calculus argument
in history...

746
00:37:44,338 --> 00:37:46,171
Yeah?
...like, 250 B.C.,

747
00:37:46,171 --> 00:37:48,038
to find the area of a circle.

748
00:37:48,038 --> 00:37:49,838
Who knew you could learn
so much from pizza?

749
00:37:49,838 --> 00:37:51,138
(laughs)

750
00:37:51,138 --> 00:37:53,138
Infinity is your friend in math.

751
00:37:53,138 --> 00:37:56,038
And that's the great insight of
calculus, that you can,

752
00:37:56,038 --> 00:37:58,638
you can rebuild the world out of
much simpler objects,

753
00:37:58,638 --> 00:38:01,971
as long as you're willing to use
infinitely many of them.

754
00:38:01,971 --> 00:38:07,005
♪ ♪

755
00:38:08,338 --> 00:38:11,138
WILLIAMS:
By embracing infinity through
calculus,

756
00:38:11,138 --> 00:38:17,171
mathematicians created one of
their most powerful tools.

757
00:38:19,471 --> 00:38:22,471
For this professor of applied
mathematics,

758
00:38:22,471 --> 00:38:24,838
it is part of how he sees the
world.

759
00:38:27,738 --> 00:38:29,538
STROGATZ:
Do you remember that
movie "The Sixth Sense,"

760
00:38:29,538 --> 00:38:30,638
where the kid says...

761
00:38:30,638 --> 00:38:33,238
I want to tell
you my secret now.

762
00:38:33,238 --> 00:38:34,471
Okay.

763
00:38:34,471 --> 00:38:37,071
STROGATZ:
..."I see dead people"?

764
00:38:38,705 --> 00:38:41,971
That's sort of what I feel
like, except I see math.

765
00:38:46,638 --> 00:38:50,771
When I go out and see the
New York skyline,

766
00:38:50,771 --> 00:38:54,638
I see all the rectangles and
pyramids in the skyscrapers.

767
00:38:57,171 --> 00:38:59,838
I see the patterns of geometry,

768
00:38:59,838 --> 00:39:03,371
I see hidden algebraic
relationships.

769
00:39:03,371 --> 00:39:07,571
There's traffic flow, and the
cars look like corpuscles,

770
00:39:07,571 --> 00:39:09,671
which makes me think about
blood flow in arteries,

771
00:39:09,671 --> 00:39:13,671
laws of fluid dynamics
and aerodynamics.

772
00:39:17,071 --> 00:39:19,338
Patterns of cylinders, and the

773
00:39:19,338 --> 00:39:22,305
rings on the cylinders are
spaced unevenly

774
00:39:22,305 --> 00:39:25,571
because of the way hydrostatic
pressure works.

775
00:39:27,405 --> 00:39:29,038
There's so much math
in the real world,

776
00:39:29,038 --> 00:39:31,005
and it's all one big principle.

777
00:39:31,005 --> 00:39:32,905
♪ ♪

778
00:39:32,905 --> 00:39:35,005
The whole world runs on
calculus,

779
00:39:35,005 --> 00:39:37,771
and math is everywhere-- I
just can't help but notice it.

780
00:39:41,805 --> 00:39:43,671
I see math.

781
00:39:43,671 --> 00:39:45,371
Actually, I see dead people,
too.

782
00:39:45,371 --> 00:39:48,838
(laughs)

783
00:39:50,305 --> 00:39:53,605
WILLIAMS:
Calculus is applied everywhere.

784
00:39:53,605 --> 00:39:55,971
And if you're looking for how
infinity

785
00:39:55,971 --> 00:39:58,305
comes into play in the modern
world,

786
00:39:58,305 --> 00:40:01,705
you need search no further.

787
00:40:01,705 --> 00:40:04,171
But even with the advent of
calculus,

788
00:40:04,171 --> 00:40:09,338
infinity itself in mathematics
remained poorly understood.

789
00:40:09,338 --> 00:40:12,238
It was only in the late 19th
century

790
00:40:12,238 --> 00:40:14,905
that new mind-bending ideas

791
00:40:14,905 --> 00:40:19,471
helped tame that strange beast
infinity.

792
00:40:19,471 --> 00:40:21,805
♪ ♪

793
00:40:21,805 --> 00:40:27,505
When I asked my friend, author
and mathematician Eugenia Cheng,

794
00:40:27,505 --> 00:40:30,205
to discuss her thoughts on
infinity,

795
00:40:30,205 --> 00:40:34,505
she suggested that we visit the
imaginary Hilbert's Hotel,

796
00:40:34,505 --> 00:40:37,771
a thought experiment first
proposed

797
00:40:37,771 --> 00:40:40,838
by mathematician David Hilbert
in the 1920s...

798
00:40:40,838 --> 00:40:42,705
♪ ♪

799
00:40:42,705 --> 00:40:47,871
...to demonstrate some of the
odd properties of infinity.

800
00:40:47,871 --> 00:40:51,571
And this hotel is definitely
an odd property.

801
00:40:51,571 --> 00:40:55,138
♪ ♪

802
00:40:55,138 --> 00:40:57,171
Well, the Hilbert Hotel
is a pretty amazing hotel.

803
00:40:57,171 --> 00:41:00,405
CHENG:
It has an infinite number of
rooms.

804
00:41:00,405 --> 00:41:02,005
Wouldn't that be great?

805
00:41:02,005 --> 00:41:04,205
You might think that you could
always fit more people in.

806
00:41:04,205 --> 00:41:06,671
But what if an infinite number
of people showed up?

807
00:41:06,671 --> 00:41:09,371
WILLIAMS: Mm.
CHENG: And then the hotel would
be full.

808
00:41:09,371 --> 00:41:10,805
Oh, dear!

809
00:41:10,805 --> 00:41:12,071
Then, if another person came
along,

810
00:41:12,071 --> 00:41:13,605
what would you do?

811
00:41:13,605 --> 00:41:15,238
Well, if you weren't very
astute,

812
00:41:15,238 --> 00:41:16,871
then you might just
say, "Sorry, we're full."

813
00:41:18,071 --> 00:41:20,271
WILLIAMS:
That's one solution.

814
00:41:20,271 --> 00:41:21,905
Or you might think,

815
00:41:21,905 --> 00:41:25,971
given there are an infinite
number of rooms,

816
00:41:25,971 --> 00:41:28,338
you can just assign the late
guest

817
00:41:28,338 --> 00:41:30,871
the room that comes after
the one given

818
00:41:30,871 --> 00:41:32,738
to the last guest that checked
in,

819
00:41:32,738 --> 00:41:35,471
you know, just farther down
the hall.

820
00:41:35,471 --> 00:41:37,471
Just put this person
at the end of the line.

821
00:41:37,471 --> 00:41:38,538
Why can't we do that?

822
00:41:38,538 --> 00:41:40,771
Where is the end of the line?

823
00:41:40,771 --> 00:41:43,371
Sounds like a philosophical
question, but the thing is,

824
00:41:43,371 --> 00:41:45,171
you can't just tell them to go
to the end.

825
00:41:45,171 --> 00:41:46,171
You have to give them a room
number.

826
00:41:46,171 --> 00:41:47,405
And all the rooms are full.

827
00:41:48,838 --> 00:41:52,505
WILLIAMS:
Hm, seems unsolvable.

828
00:41:52,505 --> 00:41:55,938
But luckily, any manager of a
hotel

829
00:41:55,938 --> 00:41:58,538
with an infinite number of
rooms,

830
00:41:58,538 --> 00:42:01,605
and an infinite number of
guests,

831
00:42:01,605 --> 00:42:05,705
has to have an infinite number
of tricks up their sleeve.

832
00:42:05,705 --> 00:42:10,671
CHENG:
Okay, how about the person in
room one moves into room two,

833
00:42:10,671 --> 00:42:13,471
and the person in room two moves
into room three,

834
00:42:13,471 --> 00:42:17,571
and the person in room three
moves into room four, and so on?

835
00:42:18,671 --> 00:42:21,138
Everybody has another room they
can move into,

836
00:42:21,138 --> 00:42:23,571
because everyone just
adds one to their room number.

837
00:42:23,571 --> 00:42:25,438
And that will leave
room one empty.

838
00:42:25,438 --> 00:42:26,871
WILLIAMS:
So, a new person comes.
CHENG: Mm-hmm.

839
00:42:26,871 --> 00:42:28,271
Welcome-- you know what?

840
00:42:28,271 --> 00:42:30,338
We're just going to have
everybody scoot over for you.

841
00:42:30,338 --> 00:42:32,371
Just scoot, goes in room one.
Mm-hmm.

842
00:42:32,371 --> 00:42:34,505
And then what
if two people showed up?

843
00:42:34,505 --> 00:42:35,538
Mm.
That's fine.

844
00:42:35,538 --> 00:42:37,838
Everyone moves up two rooms.

845
00:42:39,705 --> 00:42:41,505
What if five people show up?
That's fine.

846
00:42:43,105 --> 00:42:45,638
WILLIAMS:
But what if an infinite
number showed up?

847
00:42:45,638 --> 00:42:47,271
(bell ringing)

848
00:42:47,271 --> 00:42:49,538
Say, because of a fire

849
00:42:49,538 --> 00:42:53,771
at a second, nearby, completely
full Hilbert's Hotel?

850
00:42:56,238 --> 00:43:00,271
Is there room for a second
infinity of guests?

851
00:43:02,738 --> 00:43:05,571
You've now got an infinite
number of people.

852
00:43:05,571 --> 00:43:07,271
You can't just get everyone to
move up

853
00:43:07,271 --> 00:43:09,505
an infinite number of rooms,
because where would they go?

854
00:43:09,505 --> 00:43:12,571
WILLIAMS:
There is a solution:

855
00:43:12,571 --> 00:43:15,638
the manager asks each person
checked into a room

856
00:43:15,638 --> 00:43:19,971
to multiply their room number by
two, and move there.

857
00:43:19,971 --> 00:43:22,871
So one goes to two, two goes to
four,

858
00:43:22,871 --> 00:43:27,471
three goes to six, and so on.

859
00:43:27,471 --> 00:43:29,471
Which means they will all move
into an even-numbered room,

860
00:43:29,471 --> 00:43:31,738
and that will leave all the
odd-numbered rooms,

861
00:43:31,738 --> 00:43:33,538
and that's an infinite number of
rooms.

862
00:43:33,538 --> 00:43:35,638
And so all the new infinite

863
00:43:35,638 --> 00:43:37,771
number of people can move into
the odd-numbered rooms.

864
00:43:39,838 --> 00:43:41,438
WILLIAMS:
So then it feels like we've got

865
00:43:41,438 --> 00:43:43,038
twice the number of rooms,

866
00:43:43,038 --> 00:43:44,105
although we're still at
infinity.

867
00:43:44,105 --> 00:43:45,705
Mm-hmm!

868
00:43:45,705 --> 00:43:50,505
WILLIAMS:
In fact, the hotel can
accommodate all the guests

869
00:43:50,505 --> 00:43:54,038
from an infinite number of
infinite hotels.

870
00:43:54,038 --> 00:43:58,538
But you'll have to stop
in to learn how.

871
00:43:58,538 --> 00:44:02,971
I guess here at Hilbert's Hotel,
there's always room

872
00:44:02,971 --> 00:44:05,571
for one more!

873
00:44:06,571 --> 00:44:08,938
While Hilbert's Hotel is named

874
00:44:08,938 --> 00:44:10,905
for the person who conceived
of it,

875
00:44:10,905 --> 00:44:15,338
the ideas it plays with came
from Georg Cantor,

876
00:44:15,338 --> 00:44:19,771
a German mathematician who,
in the late 19th century,

877
00:44:19,771 --> 00:44:24,805
introduced a radically new
understanding of infinity.

878
00:44:24,805 --> 00:44:26,571
He built that understanding

879
00:44:26,571 --> 00:44:29,771
based on another area of
mathematics he created:

880
00:44:29,771 --> 00:44:31,738
set theory.

881
00:44:31,738 --> 00:44:35,171
A set is a well-defined
collection of things,

882
00:44:35,171 --> 00:44:38,471
like all the bright red shoes
you own,

883
00:44:38,471 --> 00:44:40,571
or all the possible outcomes

884
00:44:40,571 --> 00:44:44,171
from rolling a typical six-sided
die.

885
00:44:44,171 --> 00:44:48,538
Cantor used sets as a way
of comparing quantity.

886
00:44:48,538 --> 00:44:51,338
If you can match up the die
roll possibilities

887
00:44:51,338 --> 00:44:54,305
in a one-to-one correspondence
with your shoes,

888
00:44:54,305 --> 00:44:56,638
with none left over in either
set,

889
00:44:56,638 --> 00:44:59,605
then you know they have the
same quantity.

890
00:44:59,605 --> 00:45:02,405
All of this may seem elementary,

891
00:45:02,405 --> 00:45:04,371
like counting with your
fingers,

892
00:45:04,371 --> 00:45:06,371
but they are ideas

893
00:45:06,371 --> 00:45:10,438
that will carry you to some
strange places.

894
00:45:10,438 --> 00:45:12,238
Counting in pure math is
very profound,

895
00:45:12,238 --> 00:45:13,705
and it doesn't just mean

896
00:45:13,705 --> 00:45:15,971
that, list everything and label
them one, two, three.

897
00:45:15,971 --> 00:45:18,438
It often means, find some

898
00:45:18,438 --> 00:45:20,905
perfect correspondence...
Mm-hmm.

899
00:45:20,905 --> 00:45:22,071
...in the ideas

900
00:45:22,071 --> 00:45:24,305
so that you don't have to list
them all,

901
00:45:24,305 --> 00:45:27,971
but you can know that they match
up perfectly

902
00:45:27,971 --> 00:45:30,438
without listing them all,
and so, there are some

903
00:45:30,438 --> 00:45:32,938
really counterintuitive things
we can do.

904
00:45:32,938 --> 00:45:37,871
WILLIAMS:
Consider this: which
infinity is bigger,

905
00:45:37,871 --> 00:45:40,405
the set of counting numbers--

906
00:45:40,405 --> 00:45:42,871
one, two, three, four,
et cetera--

907
00:45:42,871 --> 00:45:46,271
or the set of just the even
numbers--

908
00:45:46,271 --> 00:45:49,638
two, four, six, and so on?

909
00:45:49,638 --> 00:45:51,771
And intuitively we might go,
"Well, that's half of them."

910
00:45:51,771 --> 00:45:54,238
That's half, right, yeah.
Right?

911
00:45:54,238 --> 00:45:56,505
But we could still perfectly
match them up

912
00:45:56,505 --> 00:45:59,938
with all the numbers, because
all we have to do is

913
00:45:59,938 --> 00:46:04,071
multiply each of the ordinary
numbers by two.

914
00:46:04,071 --> 00:46:07,871
And that will make a perfect
correspondence.

915
00:46:07,871 --> 00:46:10,071
WILLIAMS:
So, the set of counting
numbers

916
00:46:10,071 --> 00:46:13,471
and the set of even numbers are
both infinite

917
00:46:13,471 --> 00:46:17,605
and both the same size.

918
00:46:17,605 --> 00:46:20,005
Cantor called these kinds of
infinities,

919
00:46:20,005 --> 00:46:23,571
with a one-to-one correspondence
to the counting numbers,

920
00:46:23,571 --> 00:46:26,105
countable.

921
00:46:26,105 --> 00:46:28,971
And he investigated other kinds
of infinities,

922
00:46:28,971 --> 00:46:32,438
like that of the
prime numbers,

923
00:46:32,438 --> 00:46:34,005
whole numbers greater than one

924
00:46:34,005 --> 00:46:38,671
that can only be evenly divided
by themselves or one.

925
00:46:38,671 --> 00:46:40,971
Cantor found the infinity of
the prime numbers

926
00:46:40,971 --> 00:46:44,805
was also countable.

927
00:46:44,805 --> 00:46:48,105
And even the infinity of the
rational numbers--

928
00:46:48,105 --> 00:46:51,005
all the negative and all
the positive integers,

929
00:46:51,005 --> 00:46:54,338
plus all the fractions that can
be made up from them--

930
00:46:54,338 --> 00:46:57,205
even that infinity was countable

931
00:46:57,205 --> 00:47:00,738
and the same size as the
others.

932
00:47:04,838 --> 00:47:10,238
♪ ♪

933
00:47:13,005 --> 00:47:15,471
But now for the ultimate
challenge.

934
00:47:18,471 --> 00:47:21,971
If you take all the rational
numbers

935
00:47:21,971 --> 00:47:24,971
and add in the irrational
numbers,

936
00:47:24,971 --> 00:47:29,405
like pi or the square
root of 2--

937
00:47:29,405 --> 00:47:33,171
numbers you can't represent as
fractions using integers.

938
00:47:34,171 --> 00:47:36,371
You know, the ones that have
decimals

939
00:47:36,371 --> 00:47:39,471
that go on forever without
repeating.

940
00:47:39,471 --> 00:47:44,971
Then you have the real numbers,
the complete number line.

941
00:47:46,538 --> 00:47:51,571
Every possible number in decimal
notation.

942
00:47:51,571 --> 00:47:54,938
So is the infinity of the real
numbers,

943
00:47:54,938 --> 00:47:59,005
just like the others,
countable?

944
00:47:59,005 --> 00:48:00,805
Well, since the other sets of
numbers are,

945
00:48:00,805 --> 00:48:03,138
this one has to be, too, right?

946
00:48:04,471 --> 00:48:07,771
In Cantor's work, for an
infinity to be countable,

947
00:48:07,771 --> 00:48:10,605
it has to have a one-to-one
correspondence

948
00:48:10,605 --> 00:48:12,571
with the counting numbers,

949
00:48:12,571 --> 00:48:16,438
like we saw with the infinity
of the even numbers.

950
00:48:16,438 --> 00:48:19,138
So to do that, you need to
be able

951
00:48:19,138 --> 00:48:22,971
to list the infinity's members--
not literally.

952
00:48:22,971 --> 00:48:24,405
It's infinite and would take
forever.

953
00:48:24,405 --> 00:48:28,705
But just the way the list of all
the counting numbers

954
00:48:28,705 --> 00:48:32,805
marches off toward infinity,
adding one with each step,

955
00:48:32,805 --> 00:48:35,438
is there a way to list all the
real numbers

956
00:48:35,438 --> 00:48:37,638
to prove that they're countable?

957
00:48:37,638 --> 00:48:41,938
Cantor demonstrated the answer
is no

958
00:48:41,938 --> 00:48:45,438
with an ingenious argument.

959
00:48:45,438 --> 00:48:48,505
Imagine you presented Cantor
with what you think

960
00:48:48,505 --> 00:48:52,438
is complete list of all the
real numbers.

961
00:48:52,438 --> 00:48:55,105
To keep it simple, we will only
do the ones

962
00:48:55,105 --> 00:48:56,971
between zero and one.

963
00:48:56,971 --> 00:49:00,805
And for consistency, a number
that terminates exactly,

964
00:49:00,805 --> 00:49:04,805
like .5, will receive an endless
series of zeroes

965
00:49:04,805 --> 00:49:08,138
after the last digit.

966
00:49:08,138 --> 00:49:11,471
The list, of course, goes down
the page infinitely,

967
00:49:11,471 --> 00:49:13,838
and off the page to the right,

968
00:49:13,838 --> 00:49:16,305
because the numbers are
infinitely long.

969
00:49:16,305 --> 00:49:17,938
Cantor looks at your list,

970
00:49:17,938 --> 00:49:20,838
and starts to construct a new
number.

971
00:49:20,838 --> 00:49:24,238
He takes the first digit of the
number in the first row,

972
00:49:24,238 --> 00:49:25,971
and adds one to it.

973
00:49:25,971 --> 00:49:29,138
If it's a nine, it becomes a
zero.

974
00:49:29,138 --> 00:49:31,838
Now he knows his new number
won't match

975
00:49:31,838 --> 00:49:34,238
the one in the first row.

976
00:49:34,238 --> 00:49:38,138
Next, he takes the second digit
of the second row's number,

977
00:49:38,138 --> 00:49:40,371
and does the same.

978
00:49:40,371 --> 00:49:42,071
Now he knows his new number

979
00:49:42,071 --> 00:49:46,571
won't match the one in the
second row.

980
00:49:46,571 --> 00:49:51,171
And he does the same thing with
the third row's number.

981
00:49:51,171 --> 00:49:54,738
He continues down the list,
moving diagonally,

982
00:49:54,738 --> 00:49:56,938
building the new number,

983
00:49:56,938 --> 00:49:59,838
making sure that in at least
one position,

984
00:49:59,838 --> 00:50:02,571
a digit will be different when
compared

985
00:50:02,571 --> 00:50:05,471
to any other number on the list.

986
00:50:05,471 --> 00:50:09,171
This famous diagonal proof
shows that any attempt

987
00:50:09,171 --> 00:50:14,305
to list all the real numbers
will always be incomplete.

988
00:50:14,305 --> 00:50:17,538
And if you can't create a
complete list

989
00:50:17,538 --> 00:50:21,571
of the real numbers,
they can't be counted.

990
00:50:22,871 --> 00:50:28,038
Cantor called the infinity of
the real numbers uncountable,

991
00:50:28,038 --> 00:50:32,838
a bigger-size infinity than all
those countable infinities.

992
00:50:33,838 --> 00:50:36,571
Well, the idea of infinity had
been around for a long time,

993
00:50:36,571 --> 00:50:38,305
but the idea

994
00:50:38,305 --> 00:50:41,205
that some infinities could
bigger than others,

995
00:50:41,205 --> 00:50:43,371
that's what Cantor's
diagonalization argument

996
00:50:43,371 --> 00:50:46,538
demonstrated, and his argument
is so simple.

997
00:50:46,538 --> 00:50:47,938
It's one, again, one of those
simple ideas

998
00:50:47,938 --> 00:50:50,738
that is just so profound.

999
00:50:50,738 --> 00:50:52,338
It's one of the most
ingenious,

1000
00:50:52,338 --> 00:50:56,405
innovative ideas ever inserted
into the study of numbers.

1001
00:50:56,405 --> 00:50:58,605
And our understanding of
infinity is forever changed

1002
00:50:58,605 --> 00:51:01,305
because of Cantor's incredible
work.

1003
00:51:04,371 --> 00:51:08,138
WILLIAMS:
For humankind, the
journey from zero to infinity

1004
00:51:08,138 --> 00:51:10,505
has been extraordinary.

1005
00:51:10,505 --> 00:51:12,905
Zero, introduced thousands of
years

1006
00:51:12,905 --> 00:51:14,771
after the birth of mathematics,

1007
00:51:14,771 --> 00:51:16,505
revolutionized it,

1008
00:51:16,505 --> 00:51:19,871
enabling a new means of
calculation

1009
00:51:19,871 --> 00:51:23,205
that helped the advancement of
science.

1010
00:51:23,205 --> 00:51:27,438
Harnessing the power of zero
and infinity

1011
00:51:27,438 --> 00:51:28,738
together through calculus

1012
00:51:28,738 --> 00:51:32,005
made many of the technological
breakthroughs

1013
00:51:32,005 --> 00:51:34,605
that we take for
granted possible.

1014
00:51:34,605 --> 00:51:37,505
And Cantor's work on infinity?

1015
00:51:37,505 --> 00:51:42,271
He unveiled a new strange
vision of it for all to see.

1016
00:51:42,271 --> 00:51:45,771
His ideas and methods laid a
foundation

1017
00:51:45,771 --> 00:51:47,371
for the development
of mathematics

1018
00:51:47,371 --> 00:51:50,038
in the 20th and the 21st
centuries.

1019
00:51:50,038 --> 00:51:52,238
But for me personally,

1020
00:51:52,238 --> 00:51:55,805
I think his imagination helps us
appreciate

1021
00:51:55,805 --> 00:51:59,805
that we live in a universe of
infinite possibilities.

1022
00:51:59,805 --> 00:52:03,105
No doubt new wonders still
await us

1023
00:52:03,105 --> 00:52:06,838
on the road from zero to
infinity.

1024
00:52:32,805 --> 00:52:40,338
♪ ♪

1025
00:52:47,571 --> 00:52:52,438
ANNOUNCER:
To order this program on DVD,
visit ShopPBS.

1026
00:52:52,438 --> 00:52:55,171
Or call 1-800-PLAY-PBS.

1027
00:52:55,171 --> 00:52:58,038
Episodes of "NOVA"
are available with Passport.

1028
00:52:58,038 --> 00:53:01,838
"NOVA" is also available
on Amazon Prime Video.

1029
00:53:01,838 --> 00:53:07,038
♪ ♪

1030
00:53:15,838 --> 00:53:23,005
♪ ♪