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♪ ♪

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TALITHIA WILLIAMS:
We live our lives surrounded
by numbers.

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REPORTER:
Tipping the scales at a
whopping 14 pounds...

4
00:00:14,133 --> 00:00:15,800
The price? $400.

5
00:00:15,800 --> 00:00:17,666
... at 145,000 new infections...

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WILLIAMS:
But they didn't all arrive
at once.

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00:00:20,633 --> 00:00:22,400
Why did it take so long...

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00:00:22,400 --> 00:00:25,466
PATRICK KIMANI:
Its utility in mathematics
is undisputed.

9
00:00:25,466 --> 00:00:27,833
WILLIAMS:
...for one number in
particular...

10
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It's more like a concept
than a number.

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WILLIAMS:
...to gain full "numberhood"?

12
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I'd call it a
very significant number.

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WILLIAMS:
What's so scary...

14
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VINODH CHELLAMUTHU:
You divide a number by it,

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you blow up.

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WILLIAMS:
...about zero?

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In science and mathematics,

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the simplest ideas end up

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the most influential,
the most profound.

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WILLIAMS:
From zero,
where do numbers lead?

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Can we follow them

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all the way to infinity?

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AISHA ARROYO:
Infinity and zero

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are two sides to the same coin.

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WILLIAMS:
Can one infinity be
bigger than another?

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EUGENIA CHENG:
How much there is

27
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to understand, that's where

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all the amazingness
of infinity is!

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WILLIAMS:
What happens when
mathematicians mix

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the clout of zero
with the power of infinity?

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STEVE STROGATZ:
It's all one big principle.

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WILLIAMS:
Nothing less than our
modern world.

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Come join me,
Talithia Williams,

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as we dance with two
of the strangest beasts

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in all of mathematics.

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It's nothing... and everything.

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"Zero to Infinity,"

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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

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to an alien.

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♪ ♪

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Since they are an alien,

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you start with how long it takes
for Earth

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to travel around the sun.

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One year.

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So far, so good.

50
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Then you explain we break
a year

51
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down into 12 months,

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though they don't fit exactly.

53
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And we break months
into four weeks.

54
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Though that's not an
exact fit, either.

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At this point, the alien
might think, "One, 12, four.

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Is there a pattern forming?"

57
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But then you go on and explain

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a week is made up of
seven days.

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And that a day is made
of 24 hours.

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And an hour is made up
of 60 minutes.

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So that's groups of one, 12,
four, seven, 24, and 60.

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That's the "system."

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Even the alien's buddies
can't figure it out.

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Maybe they can wrap their heads
around another number--

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a dance number!

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♪ ♪

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It's easy to imagine

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that the real
universal language

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should be mathematics.

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And maybe it is.

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Though on Earth,

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over the course of our history,

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how we represent numbers
has been anything but universal.

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Over thousands of years,

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we humans have tried out
a lot of systems,

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but there is one that
many of us use today.

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With just ten numerals--

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zero through nine--

79
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we can, in principle,
write out any number we want,

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however large or small.

81
00:04:15,933 --> 00:04:17,266
Though writing out some

82
00:04:17,266 --> 00:04:22,300
may require an eternity--
I'm looking at you, pi!

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♪ ♪

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So where did all these
numbers come from?

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And do they really
go on forever?

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My name is Talithia Williams.

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And when I'm not
on an alien planet,

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you can find me...

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♪ ♪

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...here, at Harvey Mudd College
in Claremont, California,

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where I'm a professor of
mathematics and a statistician.

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(speaking indistinctly)

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WILLIAMS:
Statistics is a
mathematical science

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that looks for patterns in
data...

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So it is really key here
that our data...

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WILLIAMS:
...information that researchers
can gather from anywhere,

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but all of which is ultimately
translated into numbers

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00:05:06,500 --> 00:05:10,200
using the very digits we learn
by counting,

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well, our digits.

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One, two, three, four, five,
and so on.

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They can be arranged as
whole steps

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00:05:20,700 --> 00:05:25,066
on a number line that
extends off into the distance,

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heading toward something
we learned to call "infinity,"

104
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which we shall see can be
a very strange place, indeed.

105
00:05:33,333 --> 00:05:37,900
Though there is one number
that tends to be overlooked--

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at least at first.

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Most of us learn to count
starting with one.

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But is that really
the beginning?

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Or is the start a number that
isn't there at all-- zero?

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♪ ♪

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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,

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two apples, three apples,
and we don't think about,

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well, what about everywhere else
where there are no apples?

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♪ ♪

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Zero is a special number,

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which makes every other
number meaningful.

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

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But as it turns out,
unlike the counting numbers--

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one, two, three, and so on--

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zero was late to the party.

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Maybe that's understandable.

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Numbers help us keep track
of things,

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like the number of sheep
you have,

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or chickens, or cows.

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So why keep track of zero goats?

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LAURIE KEATTS:
Then there would be an
infinite number

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of things that we're not
counting.

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The number zero may seem like
it's been with us forever,

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but ancient civilizations
had numbers and mathematics

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for thousands of years
without it.

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For example,
those of Mesopotamia.

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That's the historical name
for an area that includes

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parts of modern Iraq, Iran,
Syria, and Turkey.

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It was home to some of
the earliest cities

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and the earliest civilizations
in the world,

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as well as an
influential numeral system

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based on the number 60.

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First invented by the Sumerians,

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and later developed
by the Babylonians,

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it survived for
thousands of years,

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and its legacy is with us today

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in the 60 minutes in an hour.

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Nearby, and at about
the same time,

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were the Ancient Egyptians.

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They developed sophisticated
mathematics,

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geometry, and astronomy.

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00:07:58,766 --> 00:08:02,866
They also had their own
hieroglyphic numeral system

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00:08:02,866 --> 00:08:04,533
that evolved over time.

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And just like the Mesopotamians,

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00:08:07,133 --> 00:08:11,600
the Ancient Egyptians
didn't use the number zero.

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Neither did the Greeks

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nor the Romans.

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Now remember, we're talking
about zero as a number.

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For us, zero also acts
as a placeholder,

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a way to distinguish
44 from 404.

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Some ancient numeral systems
had placeholders, as well,

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filling in blank spots.

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But they weren't seen
as a number.

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They were just a way
to keep things organized.

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In fact, as far as
historians can tell,

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using zero as a number
has only turned up twice.

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The Mayans had the idea.

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They represented
the number zero with a shell.

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00:09:00,600 --> 00:09:03,233
But the zero that we
commonly use today

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00:09:03,233 --> 00:09:06,000
came from another part
of the world.

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♪ ♪

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The Indian subcontinent

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has been home to many societies,
cultures, and traditions,

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00:09:17,933 --> 00:09:21,466
some dating back hundreds,
if not thousands, of years.

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For example,
the colorful festival of Holi,

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which celebrates the divine love
of Radha and Krishna.

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00:09:31,066 --> 00:09:36,633
And it was here in India
about 1,700 years ago

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that one of the most powerful
ideas in all of mathematics

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is thought by some
to have taken hold-- zero.

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♪ ♪

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To learn more about
India's critical role

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in zero's history, I've traveled

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to Princeton University
to speak with one of the most

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highly regarded mathematicians
in the world,

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Manjul Bhargava,
also an accomplished player

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00:10:08,333 --> 00:10:11,100
of the primary percussion
instrument

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00:10:11,100 --> 00:10:14,533
in Indian classical music,
the tabla.

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00:10:14,533 --> 00:10:17,300
(playing rapid rhythm)

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00:10:25,866 --> 00:10:28,200
Manjul, we've had number systems
for thousands of years,

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00:10:28,200 --> 00:10:30,733
from the Egyptians
to the Babylonians,

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00:10:30,733 --> 00:10:33,100
uh, but they didn't seem to
have a need for zero.

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00:10:33,100 --> 00:10:36,766
Why do you think it started
in India at this time?

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00:10:36,766 --> 00:10:41,266
The concept of zero started off
in philosophical works.

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00:10:41,266 --> 00:10:44,000
The state of zero-ness.
Mm-hmm.

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00:10:44,000 --> 00:10:47,566
The state that we all try
to achieve when we meditate.

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♪ ♪

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WILLIAMS:
In the Hindu and Buddhist
traditions,

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00:10:52,800 --> 00:10:56,533
both with deep roots
on the Indian subcontinent,

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the concept of emptiness
plays a key role.

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00:11:03,100 --> 00:11:04,900
BHARGAVA:
Emptying the mind

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00:11:04,900 --> 00:11:06,966
of all sensations,
of all temptations,

199
00:11:06,966 --> 00:11:09,233
of ego, of thoughts,
of emotions.

200
00:11:09,233 --> 00:11:11,200
And so that really

201
00:11:11,200 --> 00:11:13,966
put zero in the air
as, as an important concept.

202
00:11:13,966 --> 00:11:16,866
But the first symbolic
representation of a zero

203
00:11:16,866 --> 00:11:18,833
actually happened in the field
of linguistics.

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00:11:18,833 --> 00:11:25,533
WILLIAMS:
In about the fifth century BCE,
an Indian scholar, Panini,

205
00:11:25,533 --> 00:11:28,700
laid out the linguistic rules
of what came to be called

206
00:11:28,700 --> 00:11:31,600
Classical Sanskrit.

207
00:11:31,600 --> 00:11:33,833
BHARGAVA:
Sometimes, when you're
pronouncing things,

208
00:11:33,833 --> 00:11:35,200
you like to leave out

209
00:11:35,200 --> 00:11:37,433
a sound when you're, when you're
pronouncing quickly.

210
00:11:39,233 --> 00:11:42,033
So Panini, who is one of the
great grammarians of India,

211
00:11:42,033 --> 00:11:46,466
had a special symbol
when a sound gets deleted.

212
00:11:46,466 --> 00:11:48,333
That was called a lopa.

213
00:11:48,333 --> 00:11:49,566
And that's like a
linguistic zero.

214
00:11:49,566 --> 00:11:50,966
Very parallel to
the modern

215
00:11:50,966 --> 00:11:52,766
apostrophe in
the English language.
Yeah.

216
00:11:52,766 --> 00:11:57,633
(tabla playing)

217
00:11:57,633 --> 00:12:01,633
WILLIAMS:
Traditional Indian music
of the type Manjul plays

218
00:12:01,633 --> 00:12:06,266
is greatly influenced by the
poetic traditions of Sanskrit.

219
00:12:06,266 --> 00:12:12,133
It too will sometimes
omit sounds.

220
00:12:12,133 --> 00:12:13,700
BHARGAVA:
So, when the lopa came to music,

221
00:12:13,700 --> 00:12:18,866
that void is considered just
as important as an actual sound

222
00:12:18,866 --> 00:12:21,066
and can be
just as powerful.

223
00:12:21,066 --> 00:12:24,100
So, occasionally, to emphasize
the downbeat, you won't play it.

224
00:12:24,100 --> 00:12:25,100
So it'll go...

225
00:12:25,100 --> 00:12:28,066
(vocalizing beats)

226
00:12:38,366 --> 00:12:41,433
And so that's how a
musical zero came about.

227
00:12:41,433 --> 00:12:42,433
And a musical zero can be
very powerful.

228
00:12:42,433 --> 00:12:45,633
A zero is like any other note,

229
00:12:45,633 --> 00:12:47,033
that you can use it
in very important moments

230
00:12:47,033 --> 00:12:48,700
and just put the void there.

231
00:12:48,700 --> 00:12:51,366
♪ ♪

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00:12:51,366 --> 00:12:53,400
WILLIAMS:
The centrality of emptiness

233
00:12:53,400 --> 00:12:56,000
in Indian philosophical
traditions,

234
00:12:56,000 --> 00:12:58,700
and the symbolic
linguistic zero,

235
00:12:58,700 --> 00:13:02,900
may have set the stage
for the number zero.

236
00:13:02,900 --> 00:13:06,300
Many scholars date
its development to sometime

237
00:13:06,300 --> 00:13:09,033
in the first half of
the first millennium,

238
00:13:09,033 --> 00:13:11,400
between the third and fifth
centuries.

239
00:13:11,400 --> 00:13:15,900
But that opinion was originally
based on indirect evidence

240
00:13:15,900 --> 00:13:19,100
because no hard physical proof
had ever been found.

241
00:13:19,100 --> 00:13:21,600
♪ ♪

242
00:13:21,600 --> 00:13:24,733
Some believe that changed
in 2017,

243
00:13:24,733 --> 00:13:27,866
when Oxford University's
Bodleian Libraries

244
00:13:27,866 --> 00:13:33,400
made a surprising announcement
about one of their treasures.

245
00:13:33,400 --> 00:13:35,300
Now scientists from
the University of Oxford

246
00:13:35,300 --> 00:13:38,100
have found a manuscript
that originated in India

247
00:13:38,100 --> 00:13:40,766
and pushes back the discovery
of the concept of zero

248
00:13:40,766 --> 00:13:42,633
by at least 500 years.

249
00:13:42,633 --> 00:13:47,800
WILLIAMS:
The Bakhshali manuscript,
about 70 birch bark pages

250
00:13:47,800 --> 00:13:50,566
of mathematical writings
in Sanskrit,

251
00:13:50,566 --> 00:13:54,133
had been dated to around
800 C.E.

252
00:13:54,133 --> 00:13:57,566
But new carbon dating of
one of its pages

253
00:13:57,566 --> 00:14:01,300
pushed that back about
500 years.

254
00:14:01,300 --> 00:14:03,433
The page shows a dot,
which has been interpreted

255
00:14:03,433 --> 00:14:05,666
to represent zero.

256
00:14:05,666 --> 00:14:06,666
BHARGAVA:
There we see the zero

257
00:14:06,666 --> 00:14:09,200
used in the
Indian number system

258
00:14:09,200 --> 00:14:11,800
just the way that
we write them today.

259
00:14:11,800 --> 00:14:15,666
With one difference, is that
the zero is written as a dot.

260
00:14:15,666 --> 00:14:19,966
WILLIAMS:
If the dating is correct,
the manuscript is now

261
00:14:19,966 --> 00:14:24,700
the earliest evidence
of zero's use as a number.

262
00:14:24,700 --> 00:14:27,566
Not all scholars agree, however,

263
00:14:27,566 --> 00:14:30,600
and the assertion that
the writing is that old

264
00:14:30,600 --> 00:14:33,533
is hotly contested.

265
00:14:33,533 --> 00:14:36,833
However, there's little question

266
00:14:36,833 --> 00:14:39,833
that zero was in use
in mathematics in India

267
00:14:39,833 --> 00:14:41,633
by the seventh century,

268
00:14:41,633 --> 00:14:46,300
in the time of the great
astronomer and mathematician

269
00:14:46,300 --> 00:14:48,333
Brahmagupta.

270
00:14:48,333 --> 00:14:50,100
BHARGAVA:
Brahmagupta came around,

271
00:14:50,100 --> 00:14:52,700
and he said, "Well, zero is a
number just like any other."

272
00:14:52,700 --> 00:14:54,100
So, he actually goes

273
00:14:54,100 --> 00:14:57,300
and writes down rules
for multiplication

274
00:14:57,300 --> 00:14:58,666
and addition
and subtraction of zero.

275
00:14:58,666 --> 00:15:00,300
WILLIAMS:
So he's the first person
to have, like,

276
00:15:00,300 --> 00:15:03,033
thought of how we work
with zero today.

277
00:15:03,033 --> 00:15:04,033
Thought of zero's...
Right, right.
Yeah.

278
00:15:04,033 --> 00:15:07,433
WILLIAMS:
Along with zero,

279
00:15:07,433 --> 00:15:11,466
Brahmagupta also investigated
negative numbers.

280
00:15:11,466 --> 00:15:15,466
Today, when we place zero at the
center of the number line,

281
00:15:15,466 --> 00:15:18,566
between positive
and negative numbers,

282
00:15:18,566 --> 00:15:21,733
that is a legacy of his work.

283
00:15:21,733 --> 00:15:23,100
BHARGAVA:
So, when we talk
about the history of the zero,

284
00:15:23,100 --> 00:15:25,166
from a mathematician's
point of view,

285
00:15:25,166 --> 00:15:26,400
this was the grand moment

286
00:15:26,400 --> 00:15:29,066
where zero became
a full-fledged number

287
00:15:29,066 --> 00:15:31,700
as part of our mathematics,
and that really,

288
00:15:31,700 --> 00:15:33,366
that really changed mathematics.

289
00:15:33,366 --> 00:15:37,966
Do you think it's the, it's the
best idea ever in mathematics?

290
00:15:37,966 --> 00:15:39,500
In science and mathematics,
it's often

291
00:15:39,500 --> 00:15:45,633
the simplest and the most basic
ideas that end up becoming

292
00:15:45,633 --> 00:15:46,833
the most influent...
Revolutionizing the...

293
00:15:46,833 --> 00:15:47,966
Yeah, the most
influential,

294
00:15:47,966 --> 00:15:49,000
the most profound.

295
00:15:49,000 --> 00:15:50,866
Like the wheel.

296
00:15:50,866 --> 00:15:52,700
And it really did

297
00:15:52,700 --> 00:15:55,366
change mathematics and science.
Yeah.

298
00:15:55,366 --> 00:16:01,166
♪ ♪

299
00:16:01,166 --> 00:16:04,766
Before the Indian system became
widely adopted,

300
00:16:04,766 --> 00:16:06,866
the main purpose
of written numerals

301
00:16:06,866 --> 00:16:10,566
was for recording numbers,
not calculating with them.

302
00:16:10,566 --> 00:16:13,766
Instead, calculations
were done with a variety

303
00:16:13,766 --> 00:16:15,166
of techniques and devices--

304
00:16:15,166 --> 00:16:21,733
such as abacuses or counting
boards that used pebbles.

305
00:16:21,733 --> 00:16:25,466
Numerals were only for
storing the results.

306
00:16:25,466 --> 00:16:29,000
But the Indian system
uses the same numerals

307
00:16:29,000 --> 00:16:32,333
for calculation and storage.

308
00:16:32,333 --> 00:16:35,533
Like the number zero, that's
a fundamental breakthrough

309
00:16:35,533 --> 00:16:37,933
we all just take for granted.

310
00:16:37,933 --> 00:16:42,000
The innovative Indian system
would eventually become

311
00:16:42,000 --> 00:16:45,166
the most popular in the world,

312
00:16:45,166 --> 00:16:47,133
but not immediately.

313
00:16:47,133 --> 00:16:50,533
A crucial step in that journey

314
00:16:50,533 --> 00:16:55,000
came out of the remarkable
rise of the Islamic Empire.

315
00:16:55,000 --> 00:16:57,166
Originating in the
Arabian Peninsula

316
00:16:57,166 --> 00:16:59,133
in the seventh century,

317
00:16:59,133 --> 00:17:01,600
after only about a hundred
years,

318
00:17:01,600 --> 00:17:05,900
it had reached India in the east
and Spain in the west.

319
00:17:05,900 --> 00:17:09,733
To learn more about
the key role of Islam

320
00:17:09,733 --> 00:17:12,100
in the spread of Indian numerals
and zero,

321
00:17:12,100 --> 00:17:14,800
I'm visiting the
Hispanic Society of America

322
00:17:14,800 --> 00:17:16,600
in New York City,

323
00:17:16,600 --> 00:17:20,266
which houses perhaps the most
influential work

324
00:17:20,266 --> 00:17:23,633
in that journey.

325
00:17:23,633 --> 00:17:26,933
I'm joined by Waleed el-Ansary.

326
00:17:26,933 --> 00:17:29,800
He's an expert in
Islamic studies

327
00:17:29,800 --> 00:17:33,100
and the intersection
of religion, science,

328
00:17:33,100 --> 00:17:36,166
and economics, and like me,
eager to see

329
00:17:36,166 --> 00:17:39,033
the rare manuscript.

330
00:17:39,033 --> 00:17:40,633
Its roots go back to
what was then

331
00:17:40,633 --> 00:17:42,766
a recently constructed city

332
00:17:42,766 --> 00:17:46,766
and a new political
and cultural center of Islam:

333
00:17:46,766 --> 00:17:49,600
Baghdad.

334
00:17:49,600 --> 00:17:52,466
EL-ANSARY:
So, Baghdad was designed
in a circular shape,

335
00:17:52,466 --> 00:17:55,333
after Euclid's writings.

336
00:17:55,333 --> 00:17:59,200
And the circle is viewed
as the perfect shape,

337
00:17:59,200 --> 00:18:03,233
and therefore it's
a symbol, in a sense, of God.

338
00:18:03,233 --> 00:18:06,833
WILLIAMS:
Strategically located
at the crossroads

339
00:18:06,833 --> 00:18:11,333
of several trade routes,
the city quickly grew.

340
00:18:11,333 --> 00:18:13,000
And it became the
largest city in the world.

341
00:18:13,000 --> 00:18:15,400
It's really quite amazing.

342
00:18:15,400 --> 00:18:20,366
This center for trade
on one hand,

343
00:18:20,366 --> 00:18:22,100
as well as
intellectual trade.
Hm.

344
00:18:22,100 --> 00:18:26,366
The transfer and
transmission of ideas.

345
00:18:26,366 --> 00:18:29,233
WILLIAMS:
Scholars translated texts
that had been gathered

346
00:18:29,233 --> 00:18:34,300
from across the Islamic world
and beyond,

347
00:18:34,300 --> 00:18:37,500
including those about
Indian mathematics.

348
00:18:37,500 --> 00:18:40,500
EL-ANSARY:
They viewed all knowledge
coming from these

349
00:18:40,500 --> 00:18:43,433
other civilizations
that was consistent with

350
00:18:43,433 --> 00:18:45,000
the unity of God

351
00:18:45,000 --> 00:18:47,633
as being Islamic in the
deepest sense of the word.
Mm.

352
00:18:47,633 --> 00:18:51,166
And so it was very easy for the
Muslims to integrate that

353
00:18:51,166 --> 00:18:52,833
into their worldview.

354
00:18:52,833 --> 00:18:57,033
Sounds like they were also
the curators of this knowledge.

355
00:18:57,033 --> 00:18:59,066
And, and once they sort of
brought it together,

356
00:18:59,066 --> 00:19:00,733
they then built on it, as well.

357
00:19:00,733 --> 00:19:04,866
That's right, it wasn't just
Aristotle in Arabic.
That's right.

358
00:19:04,866 --> 00:19:06,933
Yeah.
It was more than that.

359
00:19:06,933 --> 00:19:13,300
♪ ♪

360
00:19:13,300 --> 00:19:15,800
WILLIAMS:
In the early part
of the ninth century,

361
00:19:15,800 --> 00:19:19,300
Muhammad ibn Musa al-Khwarizmi,

362
00:19:19,300 --> 00:19:22,466
a Persian scholar in
a variety of subjects,

363
00:19:22,466 --> 00:19:26,766
wrote several
hugely influential books.

364
00:19:26,766 --> 00:19:32,333
Two had a powerful impact
on mathematics.

365
00:19:32,333 --> 00:19:36,966
In one, he laid out
the foundations of algebra.

366
00:19:36,966 --> 00:19:39,566
In fact, part of the title
of the book would give

367
00:19:39,566 --> 00:19:42,466
the subject its name.

368
00:19:42,466 --> 00:19:45,800
Another of his key works
in mathematics,

369
00:19:45,800 --> 00:19:50,400
which only survives today in a
13th-century Latin translation,

370
00:19:50,400 --> 00:19:53,833
is what's brought us to the
Hispanic Society of America,

371
00:19:53,833 --> 00:19:59,066
home to one of the oldest
and the most complete version.

372
00:19:59,066 --> 00:20:00,733
EL-ANSARY:
This is a gem.

373
00:20:00,733 --> 00:20:04,633
And so you can see here
the Indian Arabic

374
00:20:04,633 --> 00:20:07,133
numeral system.
Yeah.

375
00:20:07,133 --> 00:20:10,600
With zero, one, two,
three, four, five,

376
00:20:10,600 --> 00:20:13,566
six, seven, eight, nine.

377
00:20:13,566 --> 00:20:18,333
And some of them
are shaped very similar

378
00:20:18,333 --> 00:20:20,933
to what we have today,
some of them are not.
WILLIAMS: Mm-hmm.

379
00:20:20,933 --> 00:20:22,433
Mathematics today,
the foundation

380
00:20:22,433 --> 00:20:24,700
is right here in front of us.

381
00:20:24,700 --> 00:20:25,700
That's right.
WILLIAMS:
Yeah.

382
00:20:25,700 --> 00:20:28,200
Which is unbelievable.

383
00:20:28,200 --> 00:20:29,633
(laughs)

384
00:20:29,633 --> 00:20:33,700
WILLIAMS:
The purpose of the book
was to promote

385
00:20:33,700 --> 00:20:38,400
the Indian numeral system and
explain its key innovations,

386
00:20:38,400 --> 00:20:42,900
zero and the use of the
numerals for arithmetic.

387
00:20:42,900 --> 00:20:45,666
♪ ♪

388
00:20:45,666 --> 00:20:49,566
The book also included
procedures for computation

389
00:20:49,566 --> 00:20:53,300
that would come to be known as
algorithms,

390
00:20:53,300 --> 00:20:56,466
a corruption of
al-Khwarizmi's name.

391
00:20:56,466 --> 00:21:01,033
EL-ANSARY:
So it's a little manual
to show people

392
00:21:01,033 --> 00:21:02,966
how to operate with these.

393
00:21:02,966 --> 00:21:05,733
And we learn this as, as kids,
so in some ways,

394
00:21:05,733 --> 00:21:07,433
we take it for granted,
but you're right, it's,

395
00:21:07,433 --> 00:21:09,833
someone had to say,
"This is the process

396
00:21:09,833 --> 00:21:11,333
"that we're going to use

397
00:21:11,333 --> 00:21:13,633
in order to build this
mathematical knowledge."

398
00:21:13,633 --> 00:21:15,100
And here it is.
That's right.

399
00:21:15,100 --> 00:21:17,633
Wow, wow.
That's right, so this
is very foundational.

400
00:21:17,633 --> 00:21:21,400
WILLIAMS:
Al-Khwarizmi's work,

401
00:21:21,400 --> 00:21:24,600
along with that of other
Islamic mathematicians,

402
00:21:24,600 --> 00:21:26,866
helped spread the
Indian numeral system

403
00:21:26,866 --> 00:21:31,100
throughout the Islamic world,
and eventually beyond.

404
00:21:31,100 --> 00:21:34,466
The Islamic promotion
of the Indian numeral system

405
00:21:34,466 --> 00:21:38,400
was so successful, the numbers
would even come to be known

406
00:21:38,400 --> 00:21:43,500
as Arabic numerals, somewhat
obscuring their Indian origins.

407
00:21:43,500 --> 00:21:47,266
So what we're looking at here
is something that is now

408
00:21:47,266 --> 00:21:48,533
not only used

409
00:21:48,533 --> 00:21:51,033
in the Islamic world
and the West,

410
00:21:51,033 --> 00:21:54,066
but really is the most important
numeral system

411
00:21:54,066 --> 00:21:55,833
for the entire world.
Yeah.

412
00:21:55,833 --> 00:21:58,533
And so I can hardly
overemphasize

413
00:21:58,533 --> 00:22:00,966
the significance of this text.

414
00:22:00,966 --> 00:22:05,500
♪ ♪

415
00:22:05,500 --> 00:22:08,266
WILLIAMS:
In Europe, the Indian-Arabic
numeral system,

416
00:22:08,266 --> 00:22:10,933
with its revolutionary zero,

417
00:22:10,933 --> 00:22:14,266
would eventually have
a powerful role

418
00:22:14,266 --> 00:22:17,733
in the advancement of science.

419
00:22:17,733 --> 00:22:21,566
But the earliest users
were Italian merchants

420
00:22:21,566 --> 00:22:24,033
who saw its immediate advantages

421
00:22:24,033 --> 00:22:27,133
for calculations
and business records.

422
00:22:27,133 --> 00:22:32,000
In fact, in 1202,
the son of a merchant,

423
00:22:32,000 --> 00:22:36,733
Leonardo of Pisa-- better known
today as Fibonacci--

424
00:22:36,733 --> 00:22:41,566
wrote "Liber abaci,"
an influential book

425
00:22:41,566 --> 00:22:44,766
about the new numerals
advocating for their use.

426
00:22:44,766 --> 00:22:49,066
Ultimately, it would take
hundreds of years

427
00:22:49,066 --> 00:22:53,266
for the new numerals to displace
both the existing systems

428
00:22:53,266 --> 00:22:55,133
for recording numbers,

429
00:22:55,133 --> 00:23:00,000
such as Roman numerals,
and the various devices

430
00:23:00,000 --> 00:23:03,500
and techniques used
for calculating.

431
00:23:03,500 --> 00:23:06,333
But by the late 16th century,

432
00:23:06,333 --> 00:23:09,266
in part aided by the advent
of the printing press

433
00:23:09,266 --> 00:23:11,466
and growing literacy,

434
00:23:11,466 --> 00:23:14,933
the new system had been
widely adopted in Europe.

435
00:23:14,933 --> 00:23:17,966
♪ ♪

436
00:23:17,966 --> 00:23:19,366
BHARGAVA:
Because of the European
Renaissance,

437
00:23:19,366 --> 00:23:22,833
it started becoming impossible
to really make those

438
00:23:22,833 --> 00:23:26,233
huge scientific leaps without
switching over to zero

439
00:23:26,233 --> 00:23:28,733
and the Indian system
of enumeration,

440
00:23:28,733 --> 00:23:30,466
the system that allowed you
to really

441
00:23:30,466 --> 00:23:33,000
do computations easily.

442
00:23:33,000 --> 00:23:35,866
And so it started becoming
impossible not to use them.

443
00:23:35,866 --> 00:23:39,800
And so by the 17th century, they
started becoming in regular use

444
00:23:39,800 --> 00:23:41,433
in Europe
and then around the world,

445
00:23:41,433 --> 00:23:44,300
and the rest is history.

446
00:23:44,300 --> 00:23:49,333
♪ ♪

447
00:23:54,733 --> 00:23:58,033
Treating zero as a number
transformed mathematics,

448
00:23:58,033 --> 00:24:00,766
but it did take
some getting used to.

449
00:24:00,766 --> 00:24:05,366
Because, in some ways, zero
isn't like any other number.

450
00:24:05,366 --> 00:24:07,400
First of all,
it, it has unique properties.

451
00:24:07,400 --> 00:24:10,466
Zero has some properties
of number,

452
00:24:10,466 --> 00:24:12,200
but also some properties
that make it more

453
00:24:12,200 --> 00:24:13,933
like a concept than a number.

454
00:24:15,433 --> 00:24:18,433
WILLIAMS:
In addition, subtraction,
and multiplication,

455
00:24:18,433 --> 00:24:20,900
zero behaves differently

456
00:24:20,900 --> 00:24:23,400
than every other number.

457
00:24:23,400 --> 00:24:27,933
But where zero really
creates havoc is in division.

458
00:24:27,933 --> 00:24:30,400
You get to division, and all of
a sudden, it's the first time

459
00:24:30,400 --> 00:24:32,533
that you're sort of told, like,
"Well, that's impossible."

460
00:24:32,533 --> 00:24:37,100
WILLIAMS:
You can divide any number
by every other number

461
00:24:37,100 --> 00:24:39,000
except zero.

462
00:24:39,000 --> 00:24:42,800
When you divide a number by
zero, for example, you blow up.

463
00:24:42,800 --> 00:24:45,433
ANNOUNCER:
Three, two, one, zero.

464
00:24:45,433 --> 00:24:49,500
I have no apples, and I share
that among six students,

465
00:24:49,500 --> 00:24:52,133
wouldn't everybody get
zero apples?

466
00:24:52,133 --> 00:24:54,700
There are no apples to share.

467
00:24:54,700 --> 00:24:57,233
But if I have six apples
and they are shared

468
00:24:57,233 --> 00:25:02,166
among zero students, I, the,
the concept becomes messy now.

469
00:25:02,166 --> 00:25:04,533
How do we make sense of that?

470
00:25:04,533 --> 00:25:07,166
The problem is, you can't.

471
00:25:07,166 --> 00:25:08,933
Think of it this way:

472
00:25:08,933 --> 00:25:14,400
dividing six by zero is the
same thing as asking what number

473
00:25:14,400 --> 00:25:19,066
multiplied by zero
will give you six?

474
00:25:19,066 --> 00:25:22,800
Since everything multiplied
by zero always equals zero,

475
00:25:22,800 --> 00:25:24,966
there's no solution.

476
00:25:24,966 --> 00:25:27,766
So mathematicians officially
consider the answer

477
00:25:27,766 --> 00:25:30,700
as undefined.

478
00:25:30,700 --> 00:25:33,900
Now, you might wonder,
is that sort of

479
00:25:33,900 --> 00:25:37,100
hole in the bucket of division
a problem?

480
00:25:37,100 --> 00:25:39,400
Does it get you into trouble?

481
00:25:39,400 --> 00:25:44,533
Turns out it certainly does,
under the right circumstances.

482
00:25:44,533 --> 00:25:47,766
In fact, a Greek philosopher

483
00:25:47,766 --> 00:25:49,533
who lived thousands
of years ago,

484
00:25:49,533 --> 00:25:52,066
before zero even came to be,

485
00:25:52,066 --> 00:25:55,866
invented a paradox that captures
the problem.

486
00:25:55,866 --> 00:25:58,700
His name was Zeno of Elea.

487
00:25:58,700 --> 00:26:02,900
And the paradox was about
an arrow.

488
00:26:02,900 --> 00:26:06,866
♪ ♪

489
00:26:06,866 --> 00:26:10,533
To help me demonstrate
Zeno's Paradox,

490
00:26:10,533 --> 00:26:13,666
I've turned to Eric Bennett
from Surprise, Arizona.

491
00:26:13,666 --> 00:26:17,700
VF is what we're looking for.

492
00:26:17,700 --> 00:26:20,866
WILLIAMS:
He's a physics and engineering
teacher at a local high school.

493
00:26:20,866 --> 00:26:22,633
And he's a Paralympian in
archery,

494
00:26:22,633 --> 00:26:25,600
four times over.

495
00:26:25,600 --> 00:26:27,666
So Eric, what does it feel like

496
00:26:27,666 --> 00:26:29,133
to have participated in
the Paralympics

497
00:26:29,133 --> 00:26:30,966
four times?

498
00:26:30,966 --> 00:26:32,866
Um, it makes me feel old
a little bit.

499
00:26:32,866 --> 00:26:34,366
(both laugh)

500
00:26:34,366 --> 00:26:35,800
But, um, it's, it's amazing.

501
00:26:35,800 --> 00:26:38,266
I've been competing at a really
high level for 15 years.

502
00:26:38,266 --> 00:26:40,166
Wow, wow.

503
00:26:40,166 --> 00:26:42,433
So how far away is
the target here?

504
00:26:42,433 --> 00:26:44,133
The target is the
standard Olympic

505
00:26:44,133 --> 00:26:47,133
competition distance of 70,
meters,

506
00:26:47,133 --> 00:26:49,233
which is about three-quarters
of a football field.

507
00:26:49,233 --> 00:26:50,533
No way!
Yes,

508
00:26:50,533 --> 00:26:52,800
actually, it's pretty far.
(both laugh)

509
00:26:52,800 --> 00:26:54,900
Okay, all right,
I want to see you shoot this.

510
00:26:56,600 --> 00:26:58,100
WILLIAMS:
At 15 years old,

511
00:26:58,100 --> 00:27:01,800
Eric lost an arm in an
automobile accident.

512
00:27:01,800 --> 00:27:05,900
So he draws the bowstring
back with his teeth.

513
00:27:07,533 --> 00:27:08,700
(arrow hits target)

514
00:27:08,700 --> 00:27:11,166
The arrow finds its mark.

515
00:27:11,166 --> 00:27:14,466
(laughs):
Wow, that's awesome.

516
00:27:14,466 --> 00:27:17,433
All right, so you're going to
show me how to use one of these?

517
00:27:17,433 --> 00:27:18,466
Absolutely, yup.

518
00:27:18,466 --> 00:27:20,733
Okay, from, from 70 meters?

519
00:27:20,733 --> 00:27:22,433
No, and that's okay.
I can try!

520
00:27:22,433 --> 00:27:23,933
Are you trying to
say I can't hit it

521
00:27:23,933 --> 00:27:25,400
from this distance?
No, I just want to make sure

522
00:27:25,400 --> 00:27:27,000
that you're super-successful
on your first try.

523
00:27:27,000 --> 00:27:28,700
Okay, I appreciate that--
I appreciate it.
Yeah.

524
00:27:30,333 --> 00:27:33,866
WILLIAMS:
Eric offers me
a try with a beginner's bow

525
00:27:33,866 --> 00:27:37,700
and a target about 20 yards
away.

526
00:27:37,700 --> 00:27:40,533
Let it go and it will go
right into the bullseye.

527
00:27:40,533 --> 00:27:43,000
(both laugh)

528
00:27:44,633 --> 00:27:47,533
So, I channel my inner Katniss
Everdeen

529
00:27:47,533 --> 00:27:49,766
from "The Hunger Games."

530
00:27:49,766 --> 00:27:54,866
♪ ♪

531
00:27:54,866 --> 00:27:57,533
And as a statistician,

532
00:27:57,533 --> 00:28:01,500
"May the odds be ever in my
favor."

533
00:28:04,433 --> 00:28:05,733
(arrow misses)

534
00:28:05,733 --> 00:28:07,466
Whoa!
What, I don't know--
where'd it go?

535
00:28:07,466 --> 00:28:09,133
(laughs)

536
00:28:09,133 --> 00:28:11,700
That is, like, a hundred yards
down the road we'll find it.

537
00:28:11,700 --> 00:28:13,000
(laughs)

538
00:28:13,000 --> 00:28:15,633
Got a lot of work to do,
Eric, come on.
Yeah.

539
00:28:15,633 --> 00:28:18,166
WILLIAMS:
Well, I think it's going to be
a while

540
00:28:18,166 --> 00:28:20,833
before I'm ready to compete.

541
00:28:20,833 --> 00:28:22,600
I had a lot of power
you know?

542
00:28:22,600 --> 00:28:23,833
Yeah!
And so, um...

543
00:28:23,833 --> 00:28:27,500
WILLIAMS:
But back to Zeno and that
paradox.

544
00:28:29,766 --> 00:28:34,000
All of Zeno's original writings
have been lost,

545
00:28:34,000 --> 00:28:36,266
but according to a later Greek
philosopher,

546
00:28:36,266 --> 00:28:38,266
Zeno suggested

547
00:28:38,266 --> 00:28:42,033
that we consider an arrow
in flight

548
00:28:42,033 --> 00:28:44,433
at any instant in time.

549
00:28:44,433 --> 00:28:45,633
And at that instant,

550
00:28:45,633 --> 00:28:49,900
that "now" moment,

551
00:28:49,900 --> 00:28:55,066
the arrow is frozen in space,
motionless.

552
00:28:55,066 --> 00:28:58,466
It's neither arriving
nor leaving.

553
00:28:58,466 --> 00:29:01,400
And if you consider the entire
flight...

554
00:29:03,466 --> 00:29:08,033
...there's an infinity of those
motionless, frozen moments

555
00:29:08,033 --> 00:29:10,200
in time and space.

556
00:29:10,200 --> 00:29:14,100
So, Zeno asked,
is the flight of the arrow,

557
00:29:14,100 --> 00:29:18,266
and all motion,
really just an illusion?

558
00:29:22,566 --> 00:29:25,366
STEVEN STROGATZ:
His radical conclusion is that
motion is impossible.

559
00:29:25,366 --> 00:29:29,933
At a given instant,
that arrow is someplace.

560
00:29:29,933 --> 00:29:33,600
And then click time forward.

561
00:29:33,600 --> 00:29:37,300
(chuckles):
It's at some other place,
but at no moment was it moving.

562
00:29:37,300 --> 00:29:40,533
Okay.
BENNETT:
And when you're ready, let go.

563
00:29:40,533 --> 00:29:42,200
(arrow hits target)

564
00:29:42,200 --> 00:29:44,766
What? Did you hear that?
Did you hear that?

565
00:29:46,000 --> 00:29:50,200
WILLIAMS:
Well, the motion of an arrow
looks real enough for me.

566
00:29:50,200 --> 00:29:52,633
That's right, Katniss--
got nothing on me.

567
00:29:54,100 --> 00:29:55,700
WILLIAMS:
But you can see why Zeno's

568
00:29:55,700 --> 00:29:59,700
timeless frozen moments
are so problematic.

569
00:29:59,700 --> 00:30:04,700
Our whole notion of speed
depends on time.

570
00:30:04,700 --> 00:30:08,000
Here's the formula:

571
00:30:08,000 --> 00:30:10,966
distance traveled divided by
length of time

572
00:30:10,966 --> 00:30:13,100
equals speed.

573
00:30:13,100 --> 00:30:19,300
But Zeno's frozen moment has a
length of time of zero.

574
00:30:19,300 --> 00:30:22,533
That means trying to divide by
zero,

575
00:30:22,533 --> 00:30:24,833
which is against
the rules of division.

576
00:30:26,000 --> 00:30:27,433
But at the same time,

577
00:30:27,433 --> 00:30:29,600
we often want to know the speed
of something

578
00:30:29,600 --> 00:30:32,033
in motion at a particular
instant.

579
00:30:33,333 --> 00:30:37,300
One solution to the problem
of instantaneous speed

580
00:30:37,300 --> 00:30:39,133
is a concept called

581
00:30:39,133 --> 00:30:42,166
a limit.

582
00:30:42,166 --> 00:30:44,466
Let's consider a stick figure

583
00:30:44,466 --> 00:30:48,433
who walks half the distance to
a wall,

584
00:30:48,433 --> 00:30:53,333
and does that again,
and again, and again.

585
00:30:53,333 --> 00:30:56,766
If the stick figure keeps going
half the distance to the wall,

586
00:30:56,766 --> 00:31:00,000
they'll get closer and closer,

587
00:31:00,000 --> 00:31:03,333
but the steps will get smaller
and smaller,

588
00:31:03,333 --> 00:31:06,000
and they'll never reach the
wall.

589
00:31:06,000 --> 00:31:08,766
The wall is an example
of a limit.

590
00:31:08,766 --> 00:31:11,600
As the number of steps
heads to infinity,

591
00:31:11,600 --> 00:31:15,300
the distance to the wall
decreases towards zero,

592
00:31:15,300 --> 00:31:19,433
but the figure
will never reach the wall.

593
00:31:19,433 --> 00:31:21,500
You're getting infinitely close
to a limit,

594
00:31:21,500 --> 00:31:24,633
as far as you're gonna get, but
you never actually get there.

595
00:31:24,633 --> 00:31:26,033
Which, yeah, it's one of those
concepts

596
00:31:26,033 --> 00:31:27,433
that bothers a lot of people.

597
00:31:27,433 --> 00:31:29,700
Even mathematicians it bothers,
I think.

598
00:31:29,700 --> 00:31:31,400
I can never start with a whole
number

599
00:31:31,400 --> 00:31:34,466
and divide it by
something to get zero.

600
00:31:34,466 --> 00:31:37,633
There is nothing-- there is no
way for me to ever get to zero.

601
00:31:37,633 --> 00:31:39,666
Even if you have an itty-bitty
bit

602
00:31:39,666 --> 00:31:43,833
and you divide it in half,
you still don't have zero.

603
00:31:45,400 --> 00:31:48,800
WILLIAMS:
Harnessing the power of
infinity through limits

604
00:31:48,800 --> 00:31:51,300
gives mathematicians a
work-around

605
00:31:51,300 --> 00:31:53,633
to the problem of dividing by
zero,

606
00:31:53,633 --> 00:31:57,900
and in turn opens the door
to a world of solutions

607
00:31:57,900 --> 00:32:01,433
to some extremely difficult
problems.

608
00:32:01,433 --> 00:32:06,600
It helped create a new field
of mathematics: calculus.

609
00:32:06,600 --> 00:32:09,066
And that's really the big idea
at the heart of calculus

610
00:32:09,066 --> 00:32:12,000
as understood in modern terms,
this idea of a limit.

611
00:32:12,000 --> 00:32:13,800
That you're supposed to think,

612
00:32:13,800 --> 00:32:18,366
how far did I go over a
microsecond?

613
00:32:18,366 --> 00:32:21,100
That gives me an approximation
to my instantaneous velocity,

614
00:32:21,100 --> 00:32:23,700
you know, the distance traveled
divided by that duration,

615
00:32:23,700 --> 00:32:25,800
but that's not yet an instant.

616
00:32:25,800 --> 00:32:28,333
So rather than a microsecond,
I think now a nanosecond--

617
00:32:28,333 --> 00:32:31,433
a thousand times shorter--
how far did I travel then?

618
00:32:31,433 --> 00:32:33,800
That gives me a better
approximation.

619
00:32:33,800 --> 00:32:36,266
And then this limit, as the
duration of time goes to zero,

620
00:32:36,266 --> 00:32:38,066
you often find

621
00:32:38,066 --> 00:32:41,633
you'll get a well-defined
limiting answer for the,

622
00:32:41,633 --> 00:32:43,700
for the speed, and that limit
is what's called

623
00:32:43,700 --> 00:32:45,400
the instantaneous velocity.

624
00:32:47,600 --> 00:32:50,000
WILLIAMS:
It sounds like a clever trick,

625
00:32:50,000 --> 00:32:52,433
but does it get the job done?

626
00:32:52,433 --> 00:32:55,533
To find out, I travel to New
York City

627
00:32:55,533 --> 00:33:00,466
to the National Museum of
Mathematics, MoMath.

628
00:33:00,466 --> 00:33:02,200
STROGATZ: May I?
WILLIAMS: Please, thank you.

629
00:33:02,200 --> 00:33:03,833
STROGATZ:
Take your pick.

630
00:33:03,833 --> 00:33:06,766
WILLIAMS:
Here, Cornell University
mathematician

631
00:33:06,766 --> 00:33:09,366
Steve Strogatz is
enjoying a year

632
00:33:09,366 --> 00:33:11,700
as a distinguished visiting
professor.

633
00:33:11,700 --> 00:33:14,133
13 points,
thank you very much!
(laughs)

634
00:33:14,133 --> 00:33:15,966
WILLIAMS:
He shows me around.

635
00:33:15,966 --> 00:33:17,533
Ooh.

636
00:33:17,533 --> 00:33:19,600
WILLIAMS:
But I'm here for a
specific reason.

637
00:33:19,600 --> 00:33:21,566
Steve is going to demonstrate

638
00:33:21,566 --> 00:33:25,233
the problem-solving power
of limits and infinity,

639
00:33:25,233 --> 00:33:26,933
though, as it turns out...

640
00:33:26,933 --> 00:33:28,233
Whoa!

641
00:33:28,233 --> 00:33:30,233
WILLIAMS:
...we're missing the key
component.

642
00:33:30,233 --> 00:33:31,266
(squeals, laughs)

643
00:33:31,266 --> 00:33:33,666
(crew exclaiming)

644
00:33:33,666 --> 00:33:36,333
If you want to understand what
infinity can do,

645
00:33:36,333 --> 00:33:38,900
we're gonna need pizza.

646
00:33:38,900 --> 00:33:40,300
Pizza?

647
00:33:40,300 --> 00:33:41,800
WILLIAMS:
Yes!

648
00:33:41,800 --> 00:33:44,033
There's a science
to making pizza.

649
00:33:44,033 --> 00:33:45,133
WILLIAMS:
We don't typically
associate

650
00:33:45,133 --> 00:33:47,800
pizza with infinity.

651
00:33:47,800 --> 00:33:48,900
Ay-yi-yi!

652
00:33:48,900 --> 00:33:51,033
WILLIAMS:
So how can New York City's

653
00:33:51,033 --> 00:33:52,300
most famous food...

654
00:33:52,300 --> 00:33:53,433
(Strogatz chortles)

655
00:33:53,433 --> 00:33:54,833
WILLIAMS:
...help solve one of

656
00:33:54,833 --> 00:33:57,500
the most elusive mysteries of
early mathematics?

657
00:33:59,300 --> 00:34:00,933
(both laugh)

658
00:34:05,033 --> 00:34:06,833
WILLIAMS:
So Steve,

659
00:34:06,833 --> 00:34:10,400
how is this pizza going to help
us understand infinity?

660
00:34:10,400 --> 00:34:12,366
Huh, I would say it
the other way.

661
00:34:12,366 --> 00:34:15,733
Infinity and the pizza are
gonna help us understand

662
00:34:15,733 --> 00:34:18,233
one of the oldest problems
in math.

663
00:34:18,233 --> 00:34:20,300
Mm-hmm?
What's the area of a circle?

664
00:34:20,300 --> 00:34:22,300
Which is not intuitive.
No!

665
00:34:22,300 --> 00:34:23,766
You know, what's hard about it,

666
00:34:23,766 --> 00:34:25,900
you might think a circle is a
beautiful, simple shape.

667
00:34:25,900 --> 00:34:28,066
But actually, it's got
this nasty property

668
00:34:28,066 --> 00:34:30,566
that it doesn't have any
straight lines in it.
Right.

669
00:34:30,566 --> 00:34:32,900
Ancient civilizations
didn't know how to find

670
00:34:32,900 --> 00:34:36,833
the area of a shape like that.

671
00:34:37,833 --> 00:34:42,900
WILLIAMS:
How to find the exact area of a
circle isn't obvious.

672
00:34:42,900 --> 00:34:44,900
For a square or rectangle,

673
00:34:44,900 --> 00:34:48,066
you just multiply the sides.

674
00:34:48,066 --> 00:34:50,700
But what do you do with a
circle?

675
00:34:50,700 --> 00:34:52,200
So what did they do?

676
00:34:52,200 --> 00:34:55,066
Well, they came up with an
argument that you can convert

677
00:34:55,066 --> 00:34:58,300
a round shape into a rectangle
if you use infinity.

678
00:34:58,300 --> 00:35:00,666
So we're basically gonna
kind of deconstruct this pizza,

679
00:35:00,666 --> 00:35:02,100
make it into a rectangle...
Beautiful.

680
00:35:02,100 --> 00:35:03,766
And then we're gonna
know the area.
That's it.

681
00:35:03,766 --> 00:35:06,400
So I'm gonna start
with four pieces.
Okay.

682
00:35:06,400 --> 00:35:10,966
STROGATZ:
To do that, I'm gonna go one
point up and one point down.

683
00:35:10,966 --> 00:35:12,400
WILLIAMS:
Mm-hmm.

684
00:35:12,400 --> 00:35:16,066
And then one point up and one
point down, and...

685
00:35:16,066 --> 00:35:17,566
Yeah, like that.

686
00:35:17,566 --> 00:35:19,100
Uh, how'd you do in geometry?

687
00:35:19,100 --> 00:35:22,200
STROGATZ (laughs):
You don't think that
looks like a rectangle?

688
00:35:22,200 --> 00:35:23,700
That is not close
to a rectangle.

689
00:35:23,700 --> 00:35:24,900
No, no.
No, it's not, it's not.

690
00:35:24,900 --> 00:35:27,033
But come on,
I'm only using four pieces.

691
00:35:27,033 --> 00:35:29,466
If I use more,
I can get closer.
Okay, all right.

692
00:35:29,466 --> 00:35:31,700
So we gotta cut
these babies in half.
Let's cut 'em.

693
00:35:34,200 --> 00:35:36,333
Let's rearrange them,
same trick.

694
00:35:36,333 --> 00:35:38,566
Alternating point up
and point down.

695
00:35:40,466 --> 00:35:42,333
One up and one down.

696
00:35:43,633 --> 00:35:45,633
And one up and one down.

697
00:35:45,633 --> 00:35:47,066
Now we are ready!

698
00:35:47,066 --> 00:35:49,633
That is looking a lot better!
Aw!

699
00:35:49,633 --> 00:35:51,233
What do you think,
is that a rectangle?

700
00:35:51,233 --> 00:35:54,066
Um, it's, it's not quite a
rectangle,

701
00:35:54,066 --> 00:35:55,466
but it's getting closer.
It is, right?

702
00:35:55,466 --> 00:35:57,100
Yeah!

703
00:35:57,100 --> 00:35:59,833
WILLIAMS:
In both the four-piece
and eight-piece versions,

704
00:35:59,833 --> 00:36:04,633
half the crust sits at the top
and half at the bottom.

705
00:36:04,633 --> 00:36:08,500
But with eight pieces, the edge
becomes less scalloped,

706
00:36:08,500 --> 00:36:10,700
closer to a straight line.

707
00:36:10,700 --> 00:36:12,500
So we need to go at
least a step further.

708
00:36:12,500 --> 00:36:14,133
STROGATZ:
Let's go more--
we gotta do 16.

709
00:36:16,700 --> 00:36:18,866
So we have to just change

710
00:36:18,866 --> 00:36:20,600
every other one--
am I going to mess this up?

711
00:36:20,600 --> 00:36:22,266
I mean, that's...
Wow.

712
00:36:22,266 --> 00:36:23,866
That's a parallelogram

713
00:36:23,866 --> 00:36:26,700
that's aspiring
to be a rectangle.
(laughs)

714
00:36:26,700 --> 00:36:27,700
That's got aspirations!
Yeah, it's got high hopes.

715
00:36:27,700 --> 00:36:29,700
It's got high hopes, I tell you.

716
00:36:29,700 --> 00:36:33,333
WILLIAMS:
From four slices,

717
00:36:33,333 --> 00:36:37,366
to eight slices,

718
00:36:37,366 --> 00:36:40,300
to 16 slices,

719
00:36:40,300 --> 00:36:42,766
and even 32 slices,

720
00:36:42,766 --> 00:36:46,233
there's a clear progression
towards a rectangle.

721
00:36:46,233 --> 00:36:50,566
With one piece out of 32 cut in
half to create vertical sides,

722
00:36:50,566 --> 00:36:53,466
the rectangle is almost
complete,

723
00:36:53,466 --> 00:36:56,633
except for the wavy top
and bottom.

724
00:36:56,633 --> 00:36:59,833
But as the number of slices
increases,

725
00:36:59,833 --> 00:37:03,966
the straighter and straighter
those edges would become.

726
00:37:03,966 --> 00:37:06,933
And the argument here is that if
we could keep doing this

727
00:37:06,933 --> 00:37:08,566
all the way out to infinity...
Mm-hmm.

728
00:37:08,566 --> 00:37:10,800
...so that this would be
infinitely many slices,

729
00:37:10,800 --> 00:37:12,400
infinitesimally thin,

730
00:37:12,400 --> 00:37:14,633
this really would become
a rectangle.
Yeah.

731
00:37:14,633 --> 00:37:17,133
STROGATZ:
And we can read off the area.

732
00:37:17,133 --> 00:37:18,600
WILLIAMS: That's right.
STROGATZ: It's this radius,

733
00:37:18,600 --> 00:37:21,266
that's the distance from the
center out to the crust...

734
00:37:21,266 --> 00:37:24,100
WILLIAMS: Mm-hmm...
STROGATZ:
...times half the circumference,

735
00:37:24,100 --> 00:37:26,800
which is half the crust,
half the curvy stuff.

736
00:37:26,800 --> 00:37:28,666
And that's a famous formula.

737
00:37:28,666 --> 00:37:30,400
Half the crust
times the radius.
Yeah!

738
00:37:30,400 --> 00:37:31,633
(One-half C)R.

739
00:37:31,633 --> 00:37:33,233
That's what the C is for?

740
00:37:33,233 --> 00:37:35,666
Usually, C for circumference,
but you could see it's crust.

741
00:37:35,666 --> 00:37:38,600
So, at the limit, once we
got all the way out there,

742
00:37:38,600 --> 00:37:40,000
it's going to look like
a rectangle.

743
00:37:40,000 --> 00:37:40,966
It would be a rectangle,

744
00:37:40,966 --> 00:37:42,233
and that is actually

745
00:37:42,233 --> 00:37:44,333
the first calculus argument
in history...

746
00:37:44,333 --> 00:37:46,166
Yeah?
...like, 250 B.C.,

747
00:37:46,166 --> 00:37:48,033
to find the area of a circle.

748
00:37:48,033 --> 00:37:49,833
Who knew you could learn
so much from pizza?

749
00:37:49,833 --> 00:37:51,133
(laughs)

750
00:37:51,133 --> 00:37:53,133
Infinity is your friend in math.

751
00:37:53,133 --> 00:37:56,033
And that's the great insight of
calculus, that you can,

752
00:37:56,033 --> 00:37:58,633
you can rebuild the world out of
much simpler objects,

753
00:37:58,633 --> 00:38:01,966
as long as you're willing to use
infinitely many of them.

754
00:38:01,966 --> 00:38:07,000
♪ ♪

755
00:38:08,333 --> 00:38:11,133
WILLIAMS:
By embracing infinity through
calculus,

756
00:38:11,133 --> 00:38:17,166
mathematicians created one of
their most powerful tools.

757
00:38:19,466 --> 00:38:22,466
For this professor of applied
mathematics,

758
00:38:22,466 --> 00:38:24,833
it is part of how he sees the
world.

759
00:38:27,733 --> 00:38:29,533
STROGATZ:
Do you remember that
movie "The Sixth Sense,"

760
00:38:29,533 --> 00:38:30,633
where the kid says...

761
00:38:30,633 --> 00:38:33,233
I want to tell
you my secret now.

762
00:38:33,233 --> 00:38:34,466
Okay.

763
00:38:34,466 --> 00:38:37,066
STROGATZ:
..."I see dead people"?

764
00:38:38,700 --> 00:38:41,966
That's sort of what I feel
like, except I see math.

765
00:38:46,633 --> 00:38:50,766
When I go out and see the
New York skyline,

766
00:38:50,766 --> 00:38:54,633
I see all the rectangles and
pyramids in the skyscrapers.

767
00:38:57,166 --> 00:38:59,833
I see the patterns of geometry,

768
00:38:59,833 --> 00:39:03,366
I see hidden algebraic
relationships.

769
00:39:03,366 --> 00:39:07,566
There's traffic flow, and the
cars look like corpuscles,

770
00:39:07,566 --> 00:39:09,666
which makes me think about
blood flow in arteries,

771
00:39:09,666 --> 00:39:13,666
laws of fluid dynamics
and aerodynamics.

772
00:39:17,066 --> 00:39:19,333
Patterns of cylinders, and the

773
00:39:19,333 --> 00:39:22,300
rings on the cylinders are
spaced unevenly

774
00:39:22,300 --> 00:39:25,566
because of the way hydrostatic
pressure works.

775
00:39:27,400 --> 00:39:29,033
There's so much math
in the real world,

776
00:39:29,033 --> 00:39:31,000
and it's all one big principle.

777
00:39:31,000 --> 00:39:32,900
♪ ♪

778
00:39:32,900 --> 00:39:35,000
The whole world runs on
calculus,

779
00:39:35,000 --> 00:39:37,766
and math is everywhere-- I
just can't help but notice it.

780
00:39:41,800 --> 00:39:43,666
I see math.

781
00:39:43,666 --> 00:39:45,366
Actually, I see dead people,
too.

782
00:39:45,366 --> 00:39:48,833
(laughs)

783
00:39:50,300 --> 00:39:53,600
WILLIAMS:
Calculus is applied everywhere.

784
00:39:53,600 --> 00:39:55,966
And if you're looking for how
infinity

785
00:39:55,966 --> 00:39:58,300
comes into play in the modern
world,

786
00:39:58,300 --> 00:40:01,700
you need search no further.

787
00:40:01,700 --> 00:40:04,166
But even with the advent of
calculus,

788
00:40:04,166 --> 00:40:09,333
infinity itself in mathematics
remained poorly understood.

789
00:40:09,333 --> 00:40:12,233
It was only in the late 19th
century

790
00:40:12,233 --> 00:40:14,900
that new mind-bending ideas

791
00:40:14,900 --> 00:40:19,466
helped tame that strange beast
infinity.

792
00:40:19,466 --> 00:40:21,800
♪ ♪

793
00:40:21,800 --> 00:40:27,500
When I asked my friend, author
and mathematician Eugenia Cheng,

794
00:40:27,500 --> 00:40:30,200
to discuss her thoughts on
infinity,

795
00:40:30,200 --> 00:40:34,500
she suggested that we visit the
imaginary Hilbert's Hotel,

796
00:40:34,500 --> 00:40:37,766
a thought experiment first
proposed

797
00:40:37,766 --> 00:40:40,833
by mathematician David Hilbert
in the 1920s...

798
00:40:40,833 --> 00:40:42,700
♪ ♪

799
00:40:42,700 --> 00:40:47,866
...to demonstrate some of the
odd properties of infinity.

800
00:40:47,866 --> 00:40:51,566
And this hotel is definitely
an odd property.

801
00:40:51,566 --> 00:40:55,133
♪ ♪

802
00:40:55,133 --> 00:40:57,166
Well, the Hilbert Hotel
is a pretty amazing hotel.

803
00:40:57,166 --> 00:41:00,400
CHENG:
It has an infinite number of
rooms.

804
00:41:00,400 --> 00:41:02,000
Wouldn't that be great?

805
00:41:02,000 --> 00:41:04,200
You might think that you could
always fit more people in.

806
00:41:04,200 --> 00:41:06,666
But what if an infinite number
of people showed up?

807
00:41:06,666 --> 00:41:09,366
WILLIAMS: Mm.
CHENG: And then the hotel would
be full.

808
00:41:09,366 --> 00:41:10,800
Oh, dear!

809
00:41:10,800 --> 00:41:12,066
Then, if another person came
along,

810
00:41:12,066 --> 00:41:13,600
what would you do?

811
00:41:13,600 --> 00:41:15,233
Well, if you weren't very
astute,

812
00:41:15,233 --> 00:41:16,866
then you might just
say, "Sorry, we're full."

813
00:41:18,066 --> 00:41:20,266
WILLIAMS:
That's one solution.

814
00:41:20,266 --> 00:41:21,900
Or you might think,

815
00:41:21,900 --> 00:41:25,966
given there are an infinite
number of rooms,

816
00:41:25,966 --> 00:41:28,333
you can just assign the late
guest

817
00:41:28,333 --> 00:41:30,866
the room that comes after
the one given

818
00:41:30,866 --> 00:41:32,733
to the last guest that checked
in,

819
00:41:32,733 --> 00:41:35,466
you know, just farther down
the hall.

820
00:41:35,466 --> 00:41:37,466
Just put this person
at the end of the line.

821
00:41:37,466 --> 00:41:38,533
Why can't we do that?

822
00:41:38,533 --> 00:41:40,766
Where is the end of the line?

823
00:41:40,766 --> 00:41:43,366
Sounds like a philosophical
question, but the thing is,

824
00:41:43,366 --> 00:41:45,166
you can't just tell them to go
to the end.

825
00:41:45,166 --> 00:41:46,166
You have to give them a room
number.

826
00:41:46,166 --> 00:41:47,400
And all the rooms are full.

827
00:41:48,833 --> 00:41:52,500
WILLIAMS:
Hm, seems unsolvable.

828
00:41:52,500 --> 00:41:55,933
But luckily, any manager of a
hotel

829
00:41:55,933 --> 00:41:58,533
with an infinite number of
rooms,

830
00:41:58,533 --> 00:42:01,600
and an infinite number of
guests,

831
00:42:01,600 --> 00:42:05,700
has to have an infinite number
of tricks up their sleeve.

832
00:42:05,700 --> 00:42:10,666
CHENG:
Okay, how about the person in
room one moves into room two,

833
00:42:10,666 --> 00:42:13,466
and the person in room two moves
into room three,

834
00:42:13,466 --> 00:42:17,566
and the person in room three
moves into room four, and so on?

835
00:42:18,666 --> 00:42:21,133
Everybody has another room they
can move into,

836
00:42:21,133 --> 00:42:23,566
because everyone just
adds one to their room number.

837
00:42:23,566 --> 00:42:25,433
And that will leave
room one empty.

838
00:42:25,433 --> 00:42:26,866
WILLIAMS:
So, a new person comes.
CHENG: Mm-hmm.

839
00:42:26,866 --> 00:42:28,266
Welcome-- you know what?

840
00:42:28,266 --> 00:42:30,333
We're just going to have
everybody scoot over for you.

841
00:42:30,333 --> 00:42:32,366
Just scoot, goes in room one.
Mm-hmm.

842
00:42:32,366 --> 00:42:34,500
And then what
if two people showed up?

843
00:42:34,500 --> 00:42:35,533
Mm.
That's fine.

844
00:42:35,533 --> 00:42:37,833
Everyone moves up two rooms.

845
00:42:39,700 --> 00:42:41,500
What if five people show up?
That's fine.

846
00:42:43,100 --> 00:42:45,633
WILLIAMS:
But what if an infinite
number showed up?

847
00:42:45,633 --> 00:42:47,266
(bell ringing)

848
00:42:47,266 --> 00:42:49,533
Say, because of a fire

849
00:42:49,533 --> 00:42:53,766
at a second, nearby, completely
full Hilbert's Hotel?

850
00:42:56,233 --> 00:43:00,266
Is there room for a second
infinity of guests?

851
00:43:02,733 --> 00:43:05,566
You've now got an infinite
number of people.

852
00:43:05,566 --> 00:43:07,266
You can't just get everyone to
move up

853
00:43:07,266 --> 00:43:09,500
an infinite number of rooms,
because where would they go?

854
00:43:09,500 --> 00:43:12,566
WILLIAMS:
There is a solution:

855
00:43:12,566 --> 00:43:15,633
the manager asks each person
checked into a room

856
00:43:15,633 --> 00:43:19,966
to multiply their room number by
two, and move there.

857
00:43:19,966 --> 00:43:22,866
So one goes to two, two goes to
four,

858
00:43:22,866 --> 00:43:27,466
three goes to six, and so on.

859
00:43:27,466 --> 00:43:29,466
Which means they will all move
into an even-numbered room,

860
00:43:29,466 --> 00:43:31,733
and that will leave all the
odd-numbered rooms,

861
00:43:31,733 --> 00:43:33,533
and that's an infinite number of
rooms.

862
00:43:33,533 --> 00:43:35,633
And so all the new infinite

863
00:43:35,633 --> 00:43:37,766
number of people can move into
the odd-numbered rooms.

864
00:43:39,833 --> 00:43:41,433
WILLIAMS:
So then it feels like we've got

865
00:43:41,433 --> 00:43:43,033
twice the number of rooms,

866
00:43:43,033 --> 00:43:44,100
although we're still at
infinity.

867
00:43:44,100 --> 00:43:45,700
Mm-hmm!

868
00:43:45,700 --> 00:43:50,500
WILLIAMS:
In fact, the hotel can
accommodate all the guests

869
00:43:50,500 --> 00:43:54,033
from an infinite number of
infinite hotels.

870
00:43:54,033 --> 00:43:58,533
But you'll have to stop
in to learn how.

871
00:43:58,533 --> 00:44:02,966
I guess here at Hilbert's Hotel,
there's always room

872
00:44:02,966 --> 00:44:05,566
for one more!

873
00:44:06,566 --> 00:44:08,933
While Hilbert's Hotel is named

874
00:44:08,933 --> 00:44:10,900
for the person who conceived
of it,

875
00:44:10,900 --> 00:44:15,333
the ideas it plays with came
from Georg Cantor,

876
00:44:15,333 --> 00:44:19,766
a German mathematician who,
in the late 19th century,

877
00:44:19,766 --> 00:44:24,800
introduced a radically new
understanding of infinity.

878
00:44:24,800 --> 00:44:26,566
He built that understanding

879
00:44:26,566 --> 00:44:29,766
based on another area of
mathematics he created:

880
00:44:29,766 --> 00:44:31,733
set theory.

881
00:44:31,733 --> 00:44:35,166
A set is a well-defined
collection of things,

882
00:44:35,166 --> 00:44:38,466
like all the bright red shoes
you own,

883
00:44:38,466 --> 00:44:40,566
or all the possible outcomes

884
00:44:40,566 --> 00:44:44,166
from rolling a typical six-sided
die.

885
00:44:44,166 --> 00:44:48,533
Cantor used sets as a way
of comparing quantity.

886
00:44:48,533 --> 00:44:51,333
If you can match up the die
roll possibilities

887
00:44:51,333 --> 00:44:54,300
in a one-to-one correspondence
with your shoes,

888
00:44:54,300 --> 00:44:56,633
with none left over in either
set,

889
00:44:56,633 --> 00:44:59,600
then you know they have the
same quantity.

890
00:44:59,600 --> 00:45:02,400
All of this may seem elementary,

891
00:45:02,400 --> 00:45:04,366
like counting with your
fingers,

892
00:45:04,366 --> 00:45:06,366
but they are ideas

893
00:45:06,366 --> 00:45:10,433
that will carry you to some
strange places.

894
00:45:10,433 --> 00:45:12,233
Counting in pure math is
very profound,

895
00:45:12,233 --> 00:45:13,700
and it doesn't just mean

896
00:45:13,700 --> 00:45:15,966
that, list everything and label
them one, two, three.

897
00:45:15,966 --> 00:45:18,433
It often means, find some

898
00:45:18,433 --> 00:45:20,900
perfect correspondence...
Mm-hmm.

899
00:45:20,900 --> 00:45:22,066
...in the ideas

900
00:45:22,066 --> 00:45:24,300
so that you don't have to list
them all,

901
00:45:24,300 --> 00:45:27,966
but you can know that they match
up perfectly

902
00:45:27,966 --> 00:45:30,433
without listing them all,
and so, there are some

903
00:45:30,433 --> 00:45:32,933
really counterintuitive things
we can do.

904
00:45:32,933 --> 00:45:37,866
WILLIAMS:
Consider this: which
infinity is bigger,

905
00:45:37,866 --> 00:45:40,400
the set of counting numbers--

906
00:45:40,400 --> 00:45:42,866
one, two, three, four,
et cetera--

907
00:45:42,866 --> 00:45:46,266
or the set of just the even
numbers--

908
00:45:46,266 --> 00:45:49,633
two, four, six, and so on?

909
00:45:49,633 --> 00:45:51,766
And intuitively we might go,
"Well, that's half of them."

910
00:45:51,766 --> 00:45:54,233
That's half, right, yeah.
Right?

911
00:45:54,233 --> 00:45:56,500
But we could still perfectly
match them up

912
00:45:56,500 --> 00:45:59,933
with all the numbers, because
all we have to do is

913
00:45:59,933 --> 00:46:04,066
multiply each of the ordinary
numbers by two.

914
00:46:04,066 --> 00:46:07,866
And that will make a perfect
correspondence.

915
00:46:07,866 --> 00:46:10,066
WILLIAMS:
So, the set of counting
numbers

916
00:46:10,066 --> 00:46:13,466
and the set of even numbers are
both infinite

917
00:46:13,466 --> 00:46:17,600
and both the same size.

918
00:46:17,600 --> 00:46:20,000
Cantor called these kinds of
infinities,

919
00:46:20,000 --> 00:46:23,566
with a one-to-one correspondence
to the counting numbers,

920
00:46:23,566 --> 00:46:26,100
countable.

921
00:46:26,100 --> 00:46:28,966
And he investigated other kinds
of infinities,

922
00:46:28,966 --> 00:46:32,433
like that of the
prime numbers,

923
00:46:32,433 --> 00:46:34,000
whole numbers greater than one

924
00:46:34,000 --> 00:46:38,666
that can only be evenly divided
by themselves or one.

925
00:46:38,666 --> 00:46:40,966
Cantor found the infinity of
the prime numbers

926
00:46:40,966 --> 00:46:44,800
was also countable.

927
00:46:44,800 --> 00:46:48,100
And even the infinity of the
rational numbers--

928
00:46:48,100 --> 00:46:51,000
all the negative and all
the positive integers,

929
00:46:51,000 --> 00:46:54,333
plus all the fractions that can
be made up from them--

930
00:46:54,333 --> 00:46:57,200
even that infinity was countable

931
00:46:57,200 --> 00:47:00,733
and the same size as the
others.

932
00:47:04,833 --> 00:47:10,233
♪ ♪

933
00:47:13,000 --> 00:47:15,466
But now for the ultimate
challenge.

934
00:47:18,466 --> 00:47:21,966
If you take all the rational
numbers

935
00:47:21,966 --> 00:47:24,966
and add in the irrational
numbers,

936
00:47:24,966 --> 00:47:29,400
like pi or the square
root of 2--

937
00:47:29,400 --> 00:47:33,166
numbers you can't represent as
fractions using integers.

938
00:47:34,166 --> 00:47:36,366
You know, the ones that have
decimals

939
00:47:36,366 --> 00:47:39,466
that go on forever without
repeating.

940
00:47:39,466 --> 00:47:44,966
Then you have the real numbers,
the complete number line.

941
00:47:46,533 --> 00:47:51,566
Every possible number in decimal
notation.

942
00:47:51,566 --> 00:47:54,933
So is the infinity of the real
numbers,

943
00:47:54,933 --> 00:47:59,000
just like the others,
countable?

944
00:47:59,000 --> 00:48:00,800
Well, since the other sets of
numbers are,

945
00:48:00,800 --> 00:48:03,133
this one has to be, too, right?

946
00:48:04,466 --> 00:48:07,766
In Cantor's work, for an
infinity to be countable,

947
00:48:07,766 --> 00:48:10,600
it has to have a one-to-one
correspondence

948
00:48:10,600 --> 00:48:12,566
with the counting numbers,

949
00:48:12,566 --> 00:48:16,433
like we saw with the infinity
of the even numbers.

950
00:48:16,433 --> 00:48:19,133
So to do that, you need to
be able

951
00:48:19,133 --> 00:48:22,966
to list the infinity's members--
not literally.

952
00:48:22,966 --> 00:48:24,400
It's infinite and would take
forever.

953
00:48:24,400 --> 00:48:28,700
But just the way the list of all
the counting numbers

954
00:48:28,700 --> 00:48:32,800
marches off toward infinity,
adding one with each step,

955
00:48:32,800 --> 00:48:35,433
is there a way to list all the
real numbers

956
00:48:35,433 --> 00:48:37,633
to prove that they're countable?

957
00:48:37,633 --> 00:48:41,933
Cantor demonstrated the answer
is no

958
00:48:41,933 --> 00:48:45,433
with an ingenious argument.

959
00:48:45,433 --> 00:48:48,500
Imagine you presented Cantor
with what you think

960
00:48:48,500 --> 00:48:52,433
is complete list of all the
real numbers.

961
00:48:52,433 --> 00:48:55,100
To keep it simple, we will only
do the ones

962
00:48:55,100 --> 00:48:56,966
between zero and one.

963
00:48:56,966 --> 00:49:00,800
And for consistency, a number
that terminates exactly,

964
00:49:00,800 --> 00:49:04,800
like .5, will receive an endless
series of zeroes

965
00:49:04,800 --> 00:49:08,133
after the last digit.

966
00:49:08,133 --> 00:49:11,466
The list, of course, goes down
the page infinitely,

967
00:49:11,466 --> 00:49:13,833
and off the page to the right,

968
00:49:13,833 --> 00:49:16,300
because the numbers are
infinitely long.

969
00:49:16,300 --> 00:49:17,933
Cantor looks at your list,

970
00:49:17,933 --> 00:49:20,833
and starts to construct a new
number.

971
00:49:20,833 --> 00:49:24,233
He takes the first digit of the
number in the first row,

972
00:49:24,233 --> 00:49:25,966
and adds one to it.

973
00:49:25,966 --> 00:49:29,133
If it's a nine, it becomes a
zero.

974
00:49:29,133 --> 00:49:31,833
Now he knows his new number
won't match

975
00:49:31,833 --> 00:49:34,233
the one in the first row.

976
00:49:34,233 --> 00:49:38,133
Next, he takes the second digit
of the second row's number,

977
00:49:38,133 --> 00:49:40,366
and does the same.

978
00:49:40,366 --> 00:49:42,066
Now he knows his new number

979
00:49:42,066 --> 00:49:46,566
won't match the one in the
second row.

980
00:49:46,566 --> 00:49:51,166
And he does the same thing with
the third row's number.

981
00:49:51,166 --> 00:49:54,733
He continues down the list,
moving diagonally,

982
00:49:54,733 --> 00:49:56,933
building the new number,

983
00:49:56,933 --> 00:49:59,833
making sure that in at least
one position,

984
00:49:59,833 --> 00:50:02,566
a digit will be different when
compared

985
00:50:02,566 --> 00:50:05,466
to any other number on the list.

986
00:50:05,466 --> 00:50:09,166
This famous diagonal proof
shows that any attempt

987
00:50:09,166 --> 00:50:14,300
to list all the real numbers
will always be incomplete.

988
00:50:14,300 --> 00:50:17,533
And if you can't create a
complete list

989
00:50:17,533 --> 00:50:21,566
of the real numbers,
they can't be counted.

990
00:50:22,866 --> 00:50:28,033
Cantor called the infinity of
the real numbers uncountable,

991
00:50:28,033 --> 00:50:32,833
a bigger-size infinity than all
those countable infinities.

992
00:50:33,833 --> 00:50:36,566
Well, the idea of infinity had
been around for a long time,

993
00:50:36,566 --> 00:50:38,300
but the idea

994
00:50:38,300 --> 00:50:41,200
that some infinities could
bigger than others,

995
00:50:41,200 --> 00:50:43,366
that's what Cantor's
diagonalization argument

996
00:50:43,366 --> 00:50:46,533
demonstrated, and his argument
is so simple.

997
00:50:46,533 --> 00:50:47,933
It's one, again, one of those
simple ideas

998
00:50:47,933 --> 00:50:50,733
that is just so profound.

999
00:50:50,733 --> 00:50:52,333
It's one of the most
ingenious,

1000
00:50:52,333 --> 00:50:56,400
innovative ideas ever inserted
into the study of numbers.

1001
00:50:56,400 --> 00:50:58,600
And our understanding of
infinity is forever changed

1002
00:50:58,600 --> 00:51:01,300
because of Cantor's incredible
work.

1003
00:51:04,366 --> 00:51:08,133
WILLIAMS:
For humankind, the
journey from zero to infinity

1004
00:51:08,133 --> 00:51:10,500
has been extraordinary.

1005
00:51:10,500 --> 00:51:12,900
Zero, introduced thousands of
years

1006
00:51:12,900 --> 00:51:14,766
after the birth of mathematics,

1007
00:51:14,766 --> 00:51:16,500
revolutionized it,

1008
00:51:16,500 --> 00:51:19,866
enabling a new means of
calculation

1009
00:51:19,866 --> 00:51:23,200
that helped the advancement of
science.

1010
00:51:23,200 --> 00:51:27,433
Harnessing the power of zero
and infinity

1011
00:51:27,433 --> 00:51:28,733
together through calculus

1012
00:51:28,733 --> 00:51:32,000
made many of the technological
breakthroughs

1013
00:51:32,000 --> 00:51:34,600
that we take for
granted possible.

1014
00:51:34,600 --> 00:51:37,500
And Cantor's work on infinity?

1015
00:51:37,500 --> 00:51:42,266
He unveiled a new strange
vision of it for all to see.

1016
00:51:42,266 --> 00:51:45,766
His ideas and methods laid a
foundation

1017
00:51:45,766 --> 00:51:47,366
for the development
of mathematics

1018
00:51:47,366 --> 00:51:50,033
in the 20th and the 21st
centuries.

1019
00:51:50,033 --> 00:51:52,233
But for me personally,

1020
00:51:52,233 --> 00:51:55,800
I think his imagination helps us
appreciate

1021
00:51:55,800 --> 00:51:59,800
that we live in a universe of
infinite possibilities.

1022
00:51:59,800 --> 00:52:03,100
No doubt new wonders still
await us

1023
00:52:03,100 --> 00:52:06,833
on the road from zero to
infinity.

1024
00:52:32,800 --> 00:52:40,333
♪ ♪

1025
00:52:47,566 --> 00:52:52,433
ANNOUNCER:
To order this program on DVD,
visit ShopPBS.

1026
00:52:52,433 --> 00:52:55,166
Or call 1-800-PLAY-PBS.

1027
00:52:55,166 --> 00:52:58,033
Episodes of "NOVA"
are available with Passport.

1028
00:52:58,033 --> 00:53:01,833
"NOVA" is also available
on Amazon Prime Video.

1029
00:53:01,833 --> 00:53:07,033
♪ ♪

1030
00:53:15,833 --> 00:53:23,000
♪ ♪