All language subtitles for NOVA S49E18 Zero to Infinity 1080p WEB h264-BAE (1)_track3_[eng]

♪ ♪

TALITHIA WILLIAMS:
We live our lives surrounded
by numbers.

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

The price? $400.

... at 145,000 new infections...

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

Why did it take so long...

PATRICK KIMANI:
Its utility in mathematics
is undisputed.

WILLIAMS:
...for one number in
particular...

It's more like a concept
than a number.

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

I'd call it a
very significant number.

WILLIAMS:
What's so scary...

VINODH CHELLAMUTHU:
You divide a number by it,

you blow up.

WILLIAMS:
...about zero?

In science and mathematics,

the simplest ideas end up

the most influential,
the most profound.

WILLIAMS:
From zero,
where do numbers lead?

Can we follow them

all the way to infinity?

AISHA ARROYO:
Infinity and zero

are two sides to the same coin.

WILLIAMS:
Can one infinity be
bigger than another?

EUGENIA CHENG:
How much there is

to understand, that's where

all the amazingness
of infinity is!

WILLIAMS:
What happens when
mathematicians mix

the clout of zero
with the power of infinity?

STEVE STROGATZ:
It's all one big principle.

WILLIAMS:
Nothing less than our
modern world.

Come join me,
Talithia Williams,

as we dance with two
of the strangest beasts

in all of mathematics.

It's nothing... and everything.

"Zero to Infinity,"

right now, on "NOVA."

♪ ♪

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

♪ ♪

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

to an alien.

♪ ♪

Since they are an alien,

you start with how long it takes
for Earth

to travel around the sun.

One year.

So far, so good.

Then you explain we break
a year

down into 12 months,

though they don't fit exactly.

And we break months
into four weeks.

Though that's not an
exact fit, either.

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

Is there a pattern forming?"

But then you go on and explain

a week is made up of
seven days.

And that a day is made
of 24 hours.

And an hour is made up
of 60 minutes.

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

That's the "system."

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

Maybe they can wrap their heads
around another number--

a dance number!

♪ ♪

It's easy to imagine

that the real
universal language

should be mathematics.

And maybe it is.

Though on Earth,

over the course of our history,

how we represent numbers
has been anything but universal.

Over thousands of years,

we humans have tried out
a lot of systems,

but there is one that
many of us use today.

With just ten numerals--

zero through nine--

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

however large or small.

Though writing out some

may require an eternity--
I'm looking at you, pi!

♪ ♪

So where did all these
numbers come from?

And do they really
go on forever?

My name is Talithia Williams.

And when I'm not
on an alien planet,

you can find me...

♪ ♪

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

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

(speaking indistinctly)

WILLIAMS:
Statistics is a
mathematical science

that looks for patterns in
data...

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

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

but all of which is ultimately
translated into numbers

using the very digits we learn
by counting,

well, our digits.

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

They can be arranged as
whole steps

on a number line that
extends off into the distance,

heading toward something
we learned to call "infinity,"

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

Though there is one number
that tends to be overlooked--

at least at first.

Most of us learn to count
starting with one.

But is that really
the beginning?

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

♪ ♪

When we talk about zero,

we're talking about nothing.

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

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

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

♪ ♪

Zero is a special number,

which makes every other
number meaningful.

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

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

one, two, three, and so on--

zero was late to the party.

Maybe that's understandable.

Numbers help us keep track
of things,

like the number of sheep
you have,

or chickens, or cows.

So why keep track of zero goats?

LAURIE KEATTS:
Then there would be an
infinite number

of things that we're not
counting.

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

but ancient civilizations
had numbers and mathematics

for thousands of years
without it.

For example,
those of Mesopotamia.

That's the historical name
for an area that includes

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

It was home to some of
the earliest cities

and the earliest civilizations
in the world,

as well as an
influential numeral system

based on the number 60.

First invented by the Sumerians,

and later developed
by the Babylonians,

it survived for
thousands of years,

and its legacy is with us today

in the 60 minutes in an hour.

Nearby, and at about
the same time,

were the Ancient Egyptians.

They developed sophisticated
mathematics,

geometry, and astronomy.

They also had their own
hieroglyphic numeral system

that evolved over time.

And just like the Mesopotamians,

the Ancient Egyptians
didn't use the number zero.

Neither did the Greeks

nor the Romans.

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

For us, zero also acts
as a placeholder,

a way to distinguish
44 from 404.

Some ancient numeral systems
had placeholders, as well,

filling in blank spots.

But they weren't seen
as a number.

They were just a way
to keep things organized.

In fact, as far as
historians can tell,

using zero as a number
has only turned up twice.

The Mayans had the idea.

They represented
the number zero with a shell.

But the zero that we
commonly use today

came from another part
of the world.

♪ ♪

The Indian subcontinent

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

some dating back hundreds,
if not thousands, of years.

For example,
the colorful festival of Holi,

which celebrates the divine love
of Radha and Krishna.

And it was here in India
about 1,700 years ago

that one of the most powerful
ideas in all of mathematics

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

♪ ♪

To learn more about
India's critical role

in zero's history, I've traveled

to Princeton University
to speak with one of the most

highly regarded mathematicians
in the world,

Manjul Bhargava,
also an accomplished player

of the primary percussion
instrument

in Indian classical music,
the tabla.

(playing rapid rhythm)

Manjul, we've had number systems
for thousands of years,

from the Egyptians
to the Babylonians,

uh, but they didn't seem to
have a need for zero.

Why do you think it started
in India at this time?

The concept of zero started off
in philosophical works.

The state of zero-ness.
Mm-hmm.

The state that we all try
to achieve when we meditate.

♪ ♪

WILLIAMS:
In the Hindu and Buddhist
traditions,

both with deep roots
on the Indian subcontinent,

the concept of emptiness
plays a key role.

BHARGAVA:
Emptying the mind

of all sensations,
of all temptations,

of ego, of thoughts,
of emotions.

And so that really

put zero in the air
as, as an important concept.

But the first symbolic
representation of a zero

actually happened in the field
of linguistics.

WILLIAMS:
In about the fifth century BCE,
an Indian scholar, Panini,

laid out the linguistic rules
of what came to be called

Classical Sanskrit.

BHARGAVA:
Sometimes, when you're
pronouncing things,

you like to leave out

a sound when you're, when you're
pronouncing quickly.

So Panini, who is one of the
great grammarians of India,

had a special symbol
when a sound gets deleted.

That was called a lopa.

And that's like a
linguistic zero.

Very parallel to
the modern

apostrophe in
the English language.
Yeah.

(tabla playing)

WILLIAMS:
Traditional Indian music
of the type Manjul plays

is greatly influenced by the
poetic traditions of Sanskrit.

It too will sometimes
omit sounds.

BHARGAVA:
So, when the lopa came to music,

that void is considered just
as important as an actual sound

and can be
just as powerful.

So, occasionally, to emphasize
the downbeat, you won't play it.

So it'll go...

(vocalizing beats)

And so that's how a
musical zero came about.

And a musical zero can be
very powerful.

A zero is like any other note,

that you can use it
in very important moments

and just put the void there.

♪ ♪

WILLIAMS:
The centrality of emptiness

in Indian philosophical
traditions,

and the symbolic
linguistic zero,

may have set the stage
for the number zero.

Many scholars date
its development to sometime

in the first half of
the first millennium,

between the third and fifth
centuries.

But that opinion was originally
based on indirect evidence

because no hard physical proof
had ever been found.

♪ ♪

Some believe that changed
in 2017,

when Oxford University's
Bodleian Libraries

made a surprising announcement
about one of their treasures.

Now scientists from
the University of Oxford

have found a manuscript
that originated in India

and pushes back the discovery
of the concept of zero

by at least 500 years.

WILLIAMS:
The Bakhshali manuscript,
about 70 birch bark pages

of mathematical writings
in Sanskrit,

had been dated to around
800 C.E.

But new carbon dating of
one of its pages

pushed that back about
500 years.

The page shows a dot,
which has been interpreted

to represent zero.

BHARGAVA:
There we see the zero

used in the
Indian number system

just the way that
we write them today.

With one difference, is that
the zero is written as a dot.

WILLIAMS:
If the dating is correct,
the manuscript is now

the earliest evidence
of zero's use as a number.

Not all scholars agree, however,

and the assertion that
the writing is that old

is hotly contested.

However, there's little question

that zero was in use
in mathematics in India

by the seventh century,

in the time of the great
astronomer and mathematician

Brahmagupta.

BHARGAVA:
Brahmagupta came around,

and he said, "Well, zero is a
number just like any other."

So, he actually goes

and writes down rules
for multiplication

and addition
and subtraction of zero.

WILLIAMS:
So he's the first person
to have, like,

thought of how we work
with zero today.

Thought of zero's...
Right, right.
Yeah.

WILLIAMS:
Along with zero,

Brahmagupta also investigated
negative numbers.

Today, when we place zero at the
center of the number line,

between positive
and negative numbers,

that is a legacy of his work.

BHARGAVA:
So, when we talk
about the history of the zero,

from a mathematician's
point of view,

this was the grand moment

where zero became
a full-fledged number

as part of our mathematics,
and that really,

that really changed mathematics.

Do you think it's the, it's the
best idea ever in mathematics?

In science and mathematics,
it's often

the simplest and the most basic
ideas that end up becoming

the most influent...
Revolutionizing the...

Yeah, the most
influential,

the most profound.

Like the wheel.

And it really did

change mathematics and science.
Yeah.

♪ ♪

Before the Indian system became
widely adopted,

the main purpose
of written numerals

was for recording numbers,
not calculating with them.

Instead, calculations
were done with a variety

of techniques and devices--

such as abacuses or counting
boards that used pebbles.

Numerals were only for
storing the results.

But the Indian system
uses the same numerals

for calculation and storage.

Like the number zero, that's
a fundamental breakthrough

we all just take for granted.

The innovative Indian system
would eventually become

the most popular in the world,

but not immediately.

A crucial step in that journey

came out of the remarkable
rise of the Islamic Empire.

Originating in the
Arabian Peninsula

in the seventh century,

after only about a hundred
years,

it had reached India in the east
and Spain in the west.

To learn more about
the key role of Islam

in the spread of Indian numerals
and zero,

I'm visiting the
Hispanic Society of America

in New York City,

which houses perhaps the most
influential work

in that journey.

I'm joined by Waleed el-Ansary.

He's an expert in
Islamic studies

and the intersection
of religion, science,

and economics, and like me,
eager to see

the rare manuscript.

Its roots go back to
what was then

a recently constructed city

and a new political
and cultural center of Islam:

Baghdad.

EL-ANSARY:
So, Baghdad was designed
in a circular shape,

after Euclid's writings.

And the circle is viewed
as the perfect shape,

and therefore it's
a symbol, in a sense, of God.

WILLIAMS:
Strategically located
at the crossroads

of several trade routes,
the city quickly grew.

And it became the
largest city in the world.

It's really quite amazing.

This center for trade
on one hand,

as well as
intellectual trade.
Hm.

The transfer and
transmission of ideas.

WILLIAMS:
Scholars translated texts
that had been gathered

from across the Islamic world
and beyond,

including those about
Indian mathematics.

EL-ANSARY:
They viewed all knowledge
coming from these

other civilizations
that was consistent with

the unity of God

as being Islamic in the
deepest sense of the word.
Mm.

And so it was very easy for the
Muslims to integrate that

into their worldview.

Sounds like they were also
the curators of this knowledge.

And, and once they sort of
brought it together,

they then built on it, as well.

That's right, it wasn't just
Aristotle in Arabic.
That's right.

Yeah.
It was more than that.

♪ ♪

WILLIAMS:
In the early part
of the ninth century,

Muhammad ibn Musa al-Khwarizmi,

a Persian scholar in
a variety of subjects,

wrote several
hugely influential books.

Two had a powerful impact
on mathematics.

In one, he laid out
the foundations of algebra.

In fact, part of the title
of the book would give

the subject its name.

Another of his key works
in mathematics,

which only survives today in a
13th-century Latin translation,

is what's brought us to the
Hispanic Society of America,

home to one of the oldest
and the most complete version.

EL-ANSARY:
This is a gem.

And so you can see here
the Indian Arabic

numeral system.
Yeah.

With zero, one, two,
three, four, five,

six, seven, eight, nine.

And some of them
are shaped very similar

to what we have today,
some of them are not.
WILLIAMS: Mm-hmm.

Mathematics today,
the foundation

is right here in front of us.

That's right.
WILLIAMS:
Yeah.

Which is unbelievable.

(laughs)

WILLIAMS:
The purpose of the book
was to promote

the Indian numeral system and
explain its key innovations,

zero and the use of the
numerals for arithmetic.

♪ ♪

The book also included
procedures for computation

that would come to be known as
algorithms,

a corruption of
al-Khwarizmi's name.

EL-ANSARY:
So it's a little manual
to show people

how to operate with these.

And we learn this as, as kids,
so in some ways,

we take it for granted,
but you're right, it's,

someone had to say,
"This is the process

"that we're going to use

in order to build this
mathematical knowledge."

And here it is.
That's right.

Wow, wow.
That's right, so this
is very foundational.

WILLIAMS:
Al-Khwarizmi's work,

along with that of other
Islamic mathematicians,

helped spread the
Indian numeral system

throughout the Islamic world,
and eventually beyond.

The Islamic promotion
of the Indian numeral system

was so successful, the numbers
would even come to be known

as Arabic numerals, somewhat
obscuring their Indian origins.

So what we're looking at here
is something that is now

not only used

in the Islamic world
and the West,

but really is the most important
numeral system

for the entire world.
Yeah.

And so I can hardly
overemphasize

the significance of this text.

♪ ♪

WILLIAMS:
In Europe, the Indian-Arabic
numeral system,

with its revolutionary zero,

would eventually have
a powerful role

in the advancement of science.

But the earliest users
were Italian merchants

who saw its immediate advantages

for calculations
and business records.

In fact, in 1202,
the son of a merchant,

Leonardo of Pisa-- better known
today as Fibonacci--

wrote "Liber abaci,"
an influential book

about the new numerals
advocating for their use.

Ultimately, it would take
hundreds of years

for the new numerals to displace
both the existing systems

for recording numbers,

such as Roman numerals,
and the various devices

and techniques used
for calculating.

But by the late 16th century,

in part aided by the advent
of the printing press

and growing literacy,

the new system had been
widely adopted in Europe.

♪ ♪

BHARGAVA:
Because of the European
Renaissance,

it started becoming impossible
to really make those

huge scientific leaps without
switching over to zero

and the Indian system
of enumeration,

the system that allowed you
to really

do computations easily.

And so it started becoming
impossible not to use them.

And so by the 17th century, they
started becoming in regular use

in Europe
and then around the world,

and the rest is history.

♪ ♪

Treating zero as a number
transformed mathematics,

but it did take
some getting used to.

Because, in some ways, zero
isn't like any other number.

First of all,
it, it has unique properties.

Zero has some properties
of number,

but also some properties
that make it more

like a concept than a number.

WILLIAMS:
In addition, subtraction,
and multiplication,

zero behaves differently

than every other number.

But where zero really
creates havoc is in division.

You get to division, and all of
a sudden, it's the first time

that you're sort of told, like,
"Well, that's impossible."

WILLIAMS:
You can divide any number
by every other number

except zero.

When you divide a number by
zero, for example, you blow up.

ANNOUNCER:
Three, two, one, zero.

I have no apples, and I share
that among six students,

wouldn't everybody get
zero apples?

There are no apples to share.

But if I have six apples
and they are shared

among zero students, I, the,
the concept becomes messy now.

How do we make sense of that?

The problem is, you can't.

Think of it this way:

dividing six by zero is the
same thing as asking what number

multiplied by zero
will give you six?

Since everything multiplied
by zero always equals zero,

there's no solution.

So mathematicians officially
consider the answer

as undefined.

Now, you might wonder,
is that sort of

hole in the bucket of division
a problem?

Does it get you into trouble?

Turns out it certainly does,
under the right circumstances.

In fact, a Greek philosopher

who lived thousands
of years ago,

before zero even came to be,

invented a paradox that captures
the problem.

His name was Zeno of Elea.

And the paradox was about
an arrow.

♪ ♪

To help me demonstrate
Zeno's Paradox,

I've turned to Eric Bennett
from Surprise, Arizona.

VF is what we're looking for.

WILLIAMS:
He's a physics and engineering
teacher at a local high school.

And he's a Paralympian in
archery,

four times over.

So Eric, what does it feel like

to have participated in
the Paralympics

four times?

Um, it makes me feel old
a little bit.

(both laugh)

But, um, it's, it's amazing.

I've been competing at a really
high level for 15 years.

Wow, wow.

So how far away is
the target here?

The target is the
standard Olympic

competition distance of 70,
meters,

which is about three-quarters
of a football field.

No way!
Yes,

actually, it's pretty far.
(both laugh)

Okay, all right,
I want to see you shoot this.

WILLIAMS:
At 15 years old,

Eric lost an arm in an
automobile accident.

So he draws the bowstring
back with his teeth.

(arrow hits target)

The arrow finds its mark.

(laughs):
Wow, that's awesome.

All right, so you're going to
show me how to use one of these?

Absolutely, yup.

Okay, from, from 70 meters?

No, and that's okay.
I can try!

Are you trying to
say I can't hit it

from this distance?
No, I just want to make sure

that you're super-successful
on your first try.

Okay, I appreciate that--
I appreciate it.
Yeah.

WILLIAMS:
Eric offers me
a try with a beginner's bow

and a target about 20 yards
away.

Let it go and it will go
right into the bullseye.

(both laugh)

So, I channel my inner Katniss
Everdeen

from "The Hunger Games."

♪ ♪

And as a statistician,

"May the odds be ever in my
favor."

(arrow misses)

Whoa!
What, I don't know--
where'd it go?

(laughs)

That is, like, a hundred yards
down the road we'll find it.

(laughs)

Got a lot of work to do,
Eric, come on.
Yeah.

WILLIAMS:
Well, I think it's going to be
a while

before I'm ready to compete.

I had a lot of power
you know?

Yeah!
And so, um...

WILLIAMS:
But back to Zeno and that
paradox.

All of Zeno's original writings
have been lost,

but according to a later Greek
philosopher,

Zeno suggested

that we consider an arrow
in flight

at any instant in time.

And at that instant,

that "now" moment,

the arrow is frozen in space,
motionless.

It's neither arriving
nor leaving.

And if you consider the entire
flight...

...there's an infinity of those
motionless, frozen moments

in time and space.

So, Zeno asked,
is the flight of the arrow,

and all motion,
really just an illusion?

STEVEN STROGATZ:
His radical conclusion is that
motion is impossible.

At a given instant,
that arrow is someplace.

And then click time forward.

(chuckles):
It's at some other place,
but at no moment was it moving.

Okay.
BENNETT:
And when you're ready, let go.

(arrow hits target)

What? Did you hear that?
Did you hear that?

WILLIAMS:
Well, the motion of an arrow
looks real enough for me.

That's right, Katniss--
got nothing on me.

WILLIAMS:
But you can see why Zeno's

timeless frozen moments
are so problematic.

Our whole notion of speed
depends on time.

Here's the formula:

distance traveled divided by
length of time

equals speed.

But Zeno's frozen moment has a
length of time of zero.

That means trying to divide by
zero,

which is against
the rules of division.

But at the same time,

we often want to know the speed
of something

in motion at a particular
instant.

One solution to the problem
of instantaneous speed

is a concept called

a limit.

Let's consider a stick figure

who walks half the distance to
a wall,

and does that again,
and again, and again.

If the stick figure keeps going
half the distance to the wall,

they'll get closer and closer,

but the steps will get smaller
and smaller,

and they'll never reach the
wall.

The wall is an example
of a limit.

As the number of steps
heads to infinity,

the distance to the wall
decreases towards zero,

but the figure
will never reach the wall.

You're getting infinitely close
to a limit,

as far as you're gonna get, but
you never actually get there.

Which, yeah, it's one of those
concepts

that bothers a lot of people.

Even mathematicians it bothers,
I think.

I can never start with a whole
number

and divide it by
something to get zero.

There is nothing-- there is no
way for me to ever get to zero.

Even if you have an itty-bitty
bit

and you divide it in half,
you still don't have zero.

WILLIAMS:
Harnessing the power of
infinity through limits

gives mathematicians a
work-around

to the problem of dividing by
zero,

and in turn opens the door
to a world of solutions

to some extremely difficult
problems.

It helped create a new field
of mathematics: calculus.

And that's really the big idea
at the heart of calculus

as understood in modern terms,
this idea of a limit.

That you're supposed to think,

how far did I go over a
microsecond?

That gives me an approximation
to my instantaneous velocity,

you know, the distance traveled
divided by that duration,

but that's not yet an instant.

So rather than a microsecond,
I think now a nanosecond--

a thousand times shorter--
how far did I travel then?

That gives me a better
approximation.

And then this limit, as the
duration of time goes to zero,

you often find

you'll get a well-defined
limiting answer for the,

for the speed, and that limit
is what's called

the instantaneous velocity.

WILLIAMS:
It sounds like a clever trick,

but does it get the job done?

To find out, I travel to New
York City

to the National Museum of
Mathematics, MoMath.

STROGATZ: May I?
WILLIAMS: Please, thank you.

STROGATZ:
Take your pick.

WILLIAMS:
Here, Cornell University
mathematician

Steve Strogatz is
enjoying a year

as a distinguished visiting
professor.

13 points,
thank you very much!
(laughs)

WILLIAMS:
He shows me around.

Ooh.

WILLIAMS:
But I'm here for a
specific reason.

Steve is going to demonstrate

the problem-solving power
of limits and infinity,

though, as it turns out...

Whoa!

WILLIAMS:
...we're missing the key
component.

(squeals, laughs)

(crew exclaiming)

If you want to understand what
infinity can do,

we're gonna need pizza.

Pizza?

WILLIAMS:
Yes!

There's a science
to making pizza.

WILLIAMS:
We don't typically
associate

pizza with infinity.

Ay-yi-yi!

WILLIAMS:
So how can New York City's

most famous food...

(Strogatz chortles)

WILLIAMS:
...help solve one of

the most elusive mysteries of
early mathematics?

(both laugh)

WILLIAMS:
So Steve,

how is this pizza going to help
us understand infinity?

Huh, I would say it
the other way.

Infinity and the pizza are
gonna help us understand

one of the oldest problems
in math.

Mm-hmm?
What's the area of a circle?

Which is not intuitive.
No!

You know, what's hard about it,

you might think a circle is a
beautiful, simple shape.

But actually, it's got
this nasty property

that it doesn't have any
straight lines in it.
Right.

Ancient civilizations
didn't know how to find

the area of a shape like that.

WILLIAMS:
How to find the exact area of a
circle isn't obvious.

For a square or rectangle,

you just multiply the sides.

But what do you do with a
circle?

So what did they do?

Well, they came up with an
argument that you can convert

a round shape into a rectangle
if you use infinity.

So we're basically gonna
kind of deconstruct this pizza,

make it into a rectangle...
Beautiful.

And then we're gonna
know the area.
That's it.

So I'm gonna start
with four pieces.
Okay.

STROGATZ:
To do that, I'm gonna go one
point up and one point down.

WILLIAMS:
Mm-hmm.

And then one point up and one
point down, and...

Yeah, like that.

Uh, how'd you do in geometry?

STROGATZ (laughs):
You don't think that
looks like a rectangle?

That is not close
to a rectangle.

No, no.
No, it's not, it's not.

But come on,
I'm only using four pieces.

If I use more,
I can get closer.
Okay, all right.

So we gotta cut
these babies in half.
Let's cut 'em.

Let's rearrange them,
same trick.

Alternating point up
and point down.

One up and one down.

And one up and one down.

Now we are ready!

That is looking a lot better!
Aw!

What do you think,
is that a rectangle?

Um, it's, it's not quite a
rectangle,

but it's getting closer.
It is, right?

Yeah!

WILLIAMS:
In both the four-piece
and eight-piece versions,

half the crust sits at the top
and half at the bottom.

But with eight pieces, the edge
becomes less scalloped,

closer to a straight line.

So we need to go at
least a step further.

STROGATZ:
Let's go more--
we gotta do 16.

So we have to just change

every other one--
am I going to mess this up?

I mean, that's...
Wow.

That's a parallelogram

that's aspiring
to be a rectangle.
(laughs)

That's got aspirations!
Yeah, it's got high hopes.

It's got high hopes, I tell you.

WILLIAMS:
From four slices,

to eight slices,

to 16 slices,

and even 32 slices,

there's a clear progression
towards a rectangle.

With one piece out of 32 cut in
half to create vertical sides,

the rectangle is almost
complete,

except for the wavy top
and bottom.

But as the number of slices
increases,

the straighter and straighter
those edges would become.

And the argument here is that if
we could keep doing this

all the way out to infinity...
Mm-hmm.

...so that this would be
infinitely many slices,

infinitesimally thin,

this really would become
a rectangle.
Yeah.

STROGATZ:
And we can read off the area.

WILLIAMS: That's right.
STROGATZ: It's this radius,

that's the distance from the
center out to the crust...

WILLIAMS: Mm-hmm...
STROGATZ:
...times half the circumference,

which is half the crust,
half the curvy stuff.

And that's a famous formula.

Half the crust
times the radius.
Yeah!

(One-half C)R.

That's what the C is for?

Usually, C for circumference,
but you could see it's crust.

So, at the limit, once we
got all the way out there,

it's going to look like
a rectangle.

It would be a rectangle,

and that is actually

the first calculus argument
in history...

Yeah?
...like, 250 B.C.,

to find the area of a circle.

Who knew you could learn
so much from pizza?

(laughs)

Infinity is your friend in math.

And that's the great insight of
calculus, that you can,

you can rebuild the world out of
much simpler objects,

as long as you're willing to use
infinitely many of them.

♪ ♪

WILLIAMS:
By embracing infinity through
calculus,

mathematicians created one of
their most powerful tools.

For this professor of applied
mathematics,

it is part of how he sees the
world.

STROGATZ:
Do you remember that
movie "The Sixth Sense,"

where the kid says...

I want to tell
you my secret now.

Okay.

STROGATZ:
..."I see dead people"?

That's sort of what I feel
like, except I see math.

When I go out and see the
New York skyline,

I see all the rectangles and
pyramids in the skyscrapers.

I see the patterns of geometry,

I see hidden algebraic
relationships.

There's traffic flow, and the
cars look like corpuscles,

which makes me think about
blood flow in arteries,

laws of fluid dynamics
and aerodynamics.

Patterns of cylinders, and the

rings on the cylinders are
spaced unevenly

because of the way hydrostatic
pressure works.

There's so much math
in the real world,

and it's all one big principle.

♪ ♪

The whole world runs on
calculus,

and math is everywhere-- I
just can't help but notice it.

I see math.

Actually, I see dead people,
too.

(laughs)

WILLIAMS:
Calculus is applied everywhere.

And if you're looking for how
infinity

comes into play in the modern
world,

you need search no further.

But even with the advent of
calculus,

infinity itself in mathematics
remained poorly understood.

It was only in the late 19th
century

that new mind-bending ideas

helped tame that strange beast
infinity.

♪ ♪

When I asked my friend, author
and mathematician Eugenia Cheng,

to discuss her thoughts on
infinity,

she suggested that we visit the
imaginary Hilbert's Hotel,

a thought experiment first
proposed

by mathematician David Hilbert
in the 1920s...

♪ ♪

...to demonstrate some of the
odd properties of infinity.

And this hotel is definitely
an odd property.

♪ ♪

Well, the Hilbert Hotel
is a pretty amazing hotel.

CHENG:
It has an infinite number of
rooms.

Wouldn't that be great?

You might think that you could
always fit more people in.

But what if an infinite number
of people showed up?

WILLIAMS: Mm.
CHENG: And then the hotel would
be full.

Oh, dear!

Then, if another person came
along,

what would you do?

Well, if you weren't very
astute,

then you might just
say, "Sorry, we're full."

WILLIAMS:
That's one solution.

Or you might think,

given there are an infinite
number of rooms,

you can just assign the late
guest

the room that comes after
the one given

to the last guest that checked
in,

you know, just farther down
the hall.

Just put this person
at the end of the line.

Why can't we do that?

Where is the end of the line?

Sounds like a philosophical
question, but the thing is,

you can't just tell them to go
to the end.

You have to give them a room
number.

And all the rooms are full.

WILLIAMS:
Hm, seems unsolvable.

But luckily, any manager of a
hotel

with an infinite number of
rooms,

and an infinite number of
guests,

has to have an infinite number
of tricks up their sleeve.

CHENG:
Okay, how about the person in
room one moves into room two,

and the person in room two moves
into room three,

and the person in room three
moves into room four, and so on?

Everybody has another room they
can move into,

because everyone just
adds one to their room number.

And that will leave
room one empty.

WILLIAMS:
So, a new person comes.
CHENG: Mm-hmm.

Welcome-- you know what?

We're just going to have
everybody scoot over for you.

Just scoot, goes in room one.
Mm-hmm.

And then what
if two people showed up?

Mm.
That's fine.

Everyone moves up two rooms.

What if five people show up?
That's fine.

WILLIAMS:
But what if an infinite
number showed up?

(bell ringing)

Say, because of a fire

at a second, nearby, completely
full Hilbert's Hotel?

Is there room for a second
infinity of guests?

You've now got an infinite
number of people.

You can't just get everyone to
move up

an infinite number of rooms,
because where would they go?

WILLIAMS:
There is a solution:

the manager asks each person
checked into a room

to multiply their room number by
two, and move there.

So one goes to two, two goes to
four,

three goes to six, and so on.

Which means they will all move
into an even-numbered room,

and that will leave all the
odd-numbered rooms,

and that's an infinite number of
rooms.

And so all the new infinite

number of people can move into
the odd-numbered rooms.

WILLIAMS:
So then it feels like we've got

twice the number of rooms,

although we're still at
infinity.

Mm-hmm!

WILLIAMS:
In fact, the hotel can
accommodate all the guests

from an infinite number of
infinite hotels.

But you'll have to stop
in to learn how.

I guess here at Hilbert's Hotel,
there's always room

for one more!

While Hilbert's Hotel is named

for the person who conceived
of it,

the ideas it plays with came
from Georg Cantor,

a German mathematician who,
in the late 19th century,

introduced a radically new
understanding of infinity.

He built that understanding

based on another area of
mathematics he created:

set theory.

A set is a well-defined
collection of things,

like all the bright red shoes
you own,

or all the possible outcomes

from rolling a typical six-sided
die.

Cantor used sets as a way
of comparing quantity.

If you can match up the die
roll possibilities

in a one-to-one correspondence
with your shoes,

with none left over in either
set,

then you know they have the
same quantity.

All of this may seem elementary,

like counting with your
fingers,

but they are ideas

that will carry you to some
strange places.

Counting in pure math is
very profound,

and it doesn't just mean

that, list everything and label
them one, two, three.

It often means, find some

perfect correspondence...
Mm-hmm.

...in the ideas

so that you don't have to list
them all,

but you can know that they match
up perfectly

without listing them all,
and so, there are some

really counterintuitive things
we can do.

WILLIAMS:
Consider this: which
infinity is bigger,

the set of counting numbers--

one, two, three, four,
et cetera--

or the set of just the even
numbers--

two, four, six, and so on?

And intuitively we might go,
"Well, that's half of them."

That's half, right, yeah.
Right?

But we could still perfectly
match them up

with all the numbers, because
all we have to do is

multiply each of the ordinary
numbers by two.

And that will make a perfect
correspondence.

WILLIAMS:
So, the set of counting
numbers

and the set of even numbers are
both infinite

and both the same size.

Cantor called these kinds of
infinities,

with a one-to-one correspondence
to the counting numbers,

countable.

And he investigated other kinds
of infinities,

like that of the
prime numbers,

whole numbers greater than one

that can only be evenly divided
by themselves or one.

Cantor found the infinity of
the prime numbers

was also countable.

And even the infinity of the
rational numbers--

all the negative and all
the positive integers,

plus all the fractions that can
be made up from them--

even that infinity was countable

and the same size as the
others.

♪ ♪

But now for the ultimate
challenge.

If you take all the rational
numbers

and add in the irrational
numbers,

like pi or the square
root of 2--

numbers you can't represent as
fractions using integers.

You know, the ones that have
decimals

that go on forever without
repeating.

Then you have the real numbers,
the complete number line.

Every possible number in decimal
notation.

So is the infinity of the real
numbers,

just like the others,
countable?

Well, since the other sets of
numbers are,

this one has to be, too, right?

In Cantor's work, for an
infinity to be countable,

it has to have a one-to-one
correspondence

with the counting numbers,

like we saw with the infinity
of the even numbers.

So to do that, you need to
be able

to list the infinity's members--
not literally.

It's infinite and would take
forever.

But just the way the list of all
the counting numbers

marches off toward infinity,
adding one with each step,

is there a way to list all the
real numbers

to prove that they're countable?

Cantor demonstrated the answer
is no

with an ingenious argument.

Imagine you presented Cantor
with what you think

is complete list of all the
real numbers.

To keep it simple, we will only
do the ones

between zero and one.

And for consistency, a number
that terminates exactly,

like .5, will receive an endless
series of zeroes

after the last digit.

The list, of course, goes down
the page infinitely,

and off the page to the right,

because the numbers are
infinitely long.

Cantor looks at your list,

and starts to construct a new
number.

He takes the first digit of the
number in the first row,

and adds one to it.

If it's a nine, it becomes a
zero.

Now he knows his new number
won't match

the one in the first row.

Next, he takes the second digit
of the second row's number,

and does the same.

Now he knows his new number

won't match the one in the
second row.

And he does the same thing with
the third row's number.

He continues down the list,
moving diagonally,

building the new number,

making sure that in at least
one position,

a digit will be different when
compared

to any other number on the list.

This famous diagonal proof
shows that any attempt

to list all the real numbers
will always be incomplete.

And if you can't create a
complete list

of the real numbers,
they can't be counted.

Cantor called the infinity of
the real numbers uncountable,

a bigger-size infinity than all
those countable infinities.

Well, the idea of infinity had
been around for a long time,

but the idea

that some infinities could
bigger than others,

that's what Cantor's
diagonalization argument

demonstrated, and his argument
is so simple.

It's one, again, one of those
simple ideas

that is just so profound.

It's one of the most
ingenious,

innovative ideas ever inserted
into the study of numbers.

And our understanding of
infinity is forever changed

because of Cantor's incredible
work.

WILLIAMS:
For humankind, the
journey from zero to infinity

has been extraordinary.

Zero, introduced thousands of
years

after the birth of mathematics,

revolutionized it,

enabling a new means of
calculation

that helped the advancement of
science.

Harnessing the power of zero
and infinity

together through calculus

made many of the technological
breakthroughs

that we take for
granted possible.

And Cantor's work on infinity?

He unveiled a new strange
vision of it for all to see.

His ideas and methods laid a
foundation

for the development
of mathematics

in the 20th and the 21st
centuries.

But for me personally,

I think his imagination helps us
appreciate

that we live in a universe of
infinite possibilities.

No doubt new wonders still
await us

on the road from zero to
infinity.

♪ ♪

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♪ ♪

♪ ♪