♪ ♪ 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: 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.

♪ ♪ ANNOUNCER: To order this program on DVD, visit ShopPBS.

Or call 1-800-PLAY-PBS.

Episodes of "NOVA" are available with Passport.

"NOVA" is also available on Amazon Prime Video.

♪ ♪ ♪ ♪