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NOVA-S49E18-Zero to Infinity
NOVA-S49E18-Zero ao Infinito
Zero e infinito
Esses conceitos aparentemente opostos, óbvios e indispensáveis são invenções humanas relativamente recentes. Descubra a história surpreendente de como esses conceitos-chave que revolucionaram a matemática surgiram – não apenas uma vez, mas repetidamente, à medida que diferentes culturas os inventaram e reinventaram ao longo de milhares de anos.
NOVA PBS – 2022
NOVA-S49E18-Zero ao Infinito
Zero e infinito
Esses conceitos aparentemente opostos, óbvios e indispensáveis são invenções humanas relativamente recentes. Descubra a história surpreendente de como esses conceitos-chave que revolucionaram a matemática surgiram – não apenas uma vez, mas repetidamente, à medida que diferentes culturas os inventaram e reinventaram ao longo de milhares de anos.
NOVA PBS – 2022
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00:00We live our lives surrounded by numbers, tipping the scales at a whopping 14 pounds.
00:14The price? $400. But 145,000 new infections.
00:18But they didn't all arrive at once. Why did it take so long?
00:22Its utility in mathematics is undisputed.
00:25For one number in particular, it's more like a concept than a number.
00:29To gain full numberhood.
00:31I'd call it a very significant number.
00:33What's so scary?
00:35You divide a number by it, you blow up.
00:38About zero.
00:41In science and mathematics, the simplest ideas end up the most influential, the most profound.
00:46From zero, where do numbers lead?
00:50Can we follow them all the way to infinity?
00:54Infinity and zero are two sides to the same coin.
00:57Can one infinity be bigger than another?
01:00How much that is to understand that's where all the amazingness of infinity is.
01:07What happens when mathematicians mix the clout of zero with the power of infinity?
01:14It's all one big principle.
01:16Nothing less than our modern world.
01:19Come join me, Talithia Williams, as we dance with two of the strangest beasts in all of mathematics.
01:29It's nothing and everything.
01:32Zero to infinity.
01:35Right now, on NOVA.
01:38There.
01:39No.
01:40Yes.
01:41No.
01:42We'll be back.
01:44It's nothing.
01:45What do you mean?
01:46Maria Dena.
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07:46Nearby, and at about the same time, were the ancient Egyptians.
07:51They developed sophisticated mathematics, geometry, and astronomy.
07:57They also had their own hieroglyphic numeral system that evolved over time.
08:04And just like the Mesopotamians, the ancient Egyptians didn't use the number zero.
08:10Neither did the Greeks, neither did the Greeks, nor the Greeks, nor the Romans.
08:18Now remember, we're talking about zero as a number.
08:21For us, zero also acts as a placeholder from 404.
08:31Some ancient numeral system, but they weren't seen as a number.
08:38But they weren't seen as a number.
08:41They were just a way to keep things organized.
08:46In fact, as far as historians can tell, using zero as a number has only turned up twice.
08:53The Mayans had the idea.
08:55They represented the number zero with a shell.
08:58But the zero that we commonly use today came from another part of the world.
09:05The Indian subcontinent has been home to many societies, cultures, and traditions.
09:16Some dating back hundreds, if not thousands, of years.
09:20For example, the colorful Festival of Holi, which celebrates the divine love of Radha and Krishna.
09:29And it was here in India, about 1700 years ago, that one of the most powerful ideas in all of mathematics
09:40is thought by some to have taken hold.
09:43Zero.
09:44To learn more about India's critical role in zero's history, I've traveled to Princeton University
09:58to speak with one of the most highly-regarded mathematicians in the world, Manjul Bhargava,
10:07also an accomplished player of the primary percussion instrument in Indian classical music, the tabla.
10:14Anjul, we've had number systems for thousands of years, from the Egyptians to the Babylonians,
10:30but they didn't seem to have a need for zero.
10:32Why do you think it started in India at this time?
10:36Yeah.
10:37The concept of zero started off in philosophical works, the state of zero-ness, the state that
10:45we all try to achieve when we meditate.
10:48In the Hindu and Buddhist traditions, both with deep roots on the Indian subcontinent, the concept
10:57of emptiness plays a key role.
11:02Emptying the mind of all sensations, of all temptations, of ego, of thoughts, of emotions.
11:10And so that really put zero in the air as an important concept.
11:14But the first symbolic representation of a zero actually happened in the field of linguistics.
11:20In about the fifth century BCE, an Indian scholar, Panini, laid out the linguistic rules of what
11:27came to be called classical Sanskrit.
11:30Sometimes, when you're pronouncing things, you like to leave out a sound when you're pronouncing
11:37quickly.
11:38So, Panini is one of the great grammarians of India, had a special symbol when a sound gets
11:45deleted.
11:46That was called a lopa.
11:47And that's like a linguistic zero, very parallel to the modern apostrophe in the English language.
11:54Traditional Indian music, of the type Manjul plays, is greatly influenced by the poetic traditions
12:04of Sanskrit.
12:06It too will sometimes omit sounds.
12:10So, when the lopa came to music, that void is considered just as important as an actual
12:18sound, and can be just as powerful.
12:20So, occasionally, to emphasize the downbeat, you won't play it.
12:23So, it'll go and so that's how a musical zero came about, and the musical zero can be very
12:29powerful.
12:30Zero is like any other note.
12:32Din-din-da-da, din-din-da-da, din-din-da-da.
12:38And so that's how a musical zero came about.
12:41And the musical zero can be very powerful.
12:42Zero is like any other note.
12:44You can use it in very important moments
12:46and just put the void there.
12:50The centrality of emptiness
12:52in Indian philosophical traditions
12:55and the symbolic linguistic zero
12:58may have set the stage for the number zero.
13:01Many scholars date its development
13:04to sometime in the first half of the first millennium
13:08between the third and fifth centuries.
13:10But that opinion was originally based on indirect evidence
13:14because no hard physical proof had ever been found.
13:20Some believe that changed in 2017
13:24when Oxford University's Bodleian Libraries
13:27made a surprising announcement about one of their treasures.
13:32Now scientists from the University of Oxford
13:34have found a manuscript that originated in India
13:37and pushes back the discovery of the concept of zero
13:40by at least 500 years.
13:42The Bakshali manuscript, about 70 birch bark pages
13:47of mathematical writings in Sanskrit,
13:50had been dated to around 800 CE.
13:53But new carbon dating of one of its pages
13:57pushed that back about 500 years.
14:00The page shows a dot which has been interpreted to represent zero.
14:04There we see the zero used in the Indian number system
14:08just the way that we write them today.
14:11With one difference is that the zero is written as a dot.
14:14If the dating is correct,
14:17the manuscript is now the earliest evidence
14:20of zero's use as a number.
14:23Not all scholars agree, however.
14:27And the assertion that the writing is that old
14:30is hotly contested.
14:34However, there is little question
14:36that zero was in use in mathematics in India
14:39by the seventh century,
14:41in the time of the great astronomer
14:43and mathematician Brahmagupta.
14:46Ramagupta came around and he said,
14:49well, zero is a number just like any other.
14:52So he actually goes and writes down rules
14:54for multiplication and addition and subtraction of zero.
14:58So he's the first person to have thought of
15:00how we work with zero today.
15:02Right, right.
15:03Along with zero, Ramagupta also investigated negative numbers.
15:09Today when we place zero at the center of the number line
15:14between positive and negative numbers,
15:17that is a legacy of his work.
15:20So when we talk about the history of the zero
15:22from a mathematician's point of view,
15:24this was the grand moment where zero became
15:27a full-fledged number as part of our mathematics,
15:29and that really changed mathematics.
15:32Do you think it's the best idea ever in mathematics?
15:36In science and mathematics,
15:38it's often the simplest and the most basic ideas
15:41that end up becoming the most influential,
15:47the most profound, like the wheel.
15:50And it really did change mathematics and science.
16:00Before the Indian system became widely adopted,
16:03the main purpose of written numerals was for recording numbers,
16:07not calculating with them.
16:09Instead, calculations were done
16:11with a variety of techniques and devices,
16:14such as abacuses or counting boards that used pebbles.
16:20Numerals were only for storing the results,
16:24but the Indian system uses the same numerals
16:27for calculation and storage.
16:30Like the number zero, that's a fundamental breakthrough
16:34we all just take for granted.
16:37The innovative Indian system would eventually become
16:40the most popular in the world,
16:43but not immediately.
16:47A crucial step in that journey
16:49came out of the remarkable rise of the Islamic Empire,
16:53originating in the Arabian Peninsula in the 7th century.
16:57After only about a hundred years,
17:00it had reached India in the East and Spain in the West.
17:05To learn more about the key role of Islam
17:08in the spread of Indian numerals and zero,
17:11I'm visiting the Hispanic Society of America
17:14in New York City,
17:16which houses perhaps the most influential work in that journey.
17:20I'm joined by Waleed El Ansari.
17:26He's an expert in Islamic studies
17:29and the intersection of religion, science and economics.
17:33And like me, eager to see the rare manuscript.
17:37Its roots go back to what was then a recently constructed city
17:41and a new political and cultural center of Islam, Baghdad.
17:47So Baghdad was designed in a circular shape after Euclid's writings.
17:55And the circle is viewed as a perfect shape,
17:58and therefore it's a symbol and a sense of God.
18:03Strategically located at the crossroads of several trade routes,
18:07the city quickly grew.
18:10And it became the largest city in the world.
18:13It's really quite amazing.
18:15This center for trade on one hand,
18:18as well as intellectual trade,
18:21the transfer and transmission of ideas.
18:24Scholars translated texts that had been gathered
18:28from across the Islamic world and beyond,
18:32including those about Indian mathematics.
18:36They viewed all knowledge coming from these other civilizations
18:41that was consistent with the unity of God as being Islamic
18:45in the deepest sense of the word.
18:47And so it was very easy for the Muslims to integrate that into their worldview.
18:52Sounds like they were also the curators of this knowledge.
18:56And once they sort of brought it together, they then built on it as well.
19:00That's right. It wasn't just Aristotle and Arabic.
19:03That's right.
19:04It was more than that.
19:05In the early part of the ninth century,
19:15Muhammad ibn Musa al-Khwarizmi,
19:18a Persian scholar in a variety of subjects,
19:22wrote several hugely influential books.
19:25Two had a powerful impact on mathematics.
19:31In one, he laid out the foundations of algebra.
19:35In fact, part of the title of the book would give the subject its name.
19:41Another of his key works in mathematics,
19:44which only survives today in a 13th century Latin translation,
19:49is what's brought us to the Hispanic Society of America,
19:53home to one of the oldest and the most complete version.
19:58This is a gem.
20:00And so you can see here the Indian Arabic numeral system
20:05with zero, one, two, three, four, five, six, seven, eight, nine.
20:12And some of them are shaped very similar to what we have today.
20:18Some of them are not.
20:19Mathematics today, the foundation is right here in front of us.
20:24That's right.
20:25Which is unbelievable.
20:30The purpose of the book was to promote the Indian numeral system
20:35and explain its key innovations, zero,
20:39and the use of the numerals for arithmetic.
20:45The book also included procedures for computation that would come to be known as algorithms,
20:52a corruption of Al Khwarezmi's name.
20:56So it's a little manual to show people how to operate with these.
21:02And we learned this as kids, so in some ways we take it for granted.
21:05But you're right, someone had to say this is the process that we're going to use
21:10in order to build this mathematical knowledge, and here it is.
21:13That's right.
21:14Wow.
21:15That's right. So this is very foundational.
21:17Al Khwarezmi's work, along with that of other Islamic mathematicians,
21:23helped spread the Indian numeral system throughout the Islamic world and eventually beyond.
21:30The Islamic promotion of the Indian numeral system was so successful,
21:35the numbers would even come to be known as Arabic numerals,
21:39somewhat obscuring their Indian origins.
21:42So what we're looking at here is something that is now not only used in the Islamic world in the West,
21:50but really is the most important numeral system for the entire world.
21:54And so I can hardly overemphasize the significance of this text.
22:04In Europe, the Indian-Arabic numeral system, with its revolutionary zero,
22:11would eventually have a powerful role in the advancement of science.
22:17But the earliest users were Italian merchants who saw its immediate advantages
22:23for calculations and business records.
22:26In fact, in 1202, the son of a merchant, Leonardo of Pisa, better known today as Fibonacci,
22:36wrote Liber Abaci, an influential book about the new numerals advocating for their use.
22:44Ultimately, it would take hundreds of years for the new numerals to displace
22:50both the existing systems for recording numbers, such as Roman numerals,
22:57and the various devices and techniques used for calculating.
23:03But by the late 16th century, in part aided by the advent of the printing press and growing literacy,
23:10the new system had been widely adopted in Europe.
23:17Because of the European Renaissance, it started becoming impossible to really make those huge scientific leaps
23:23without switching over to zero in the Indian system of numeration,
23:28a system that allowed you to really do computations easily.
23:32And so it started becoming impossible not to use them.
23:35And so by the 17th century, they started becoming in regular use in Europe and then around the world.
23:41And the rest is history.
23:53Treating zero as a number transformed mathematics.
23:57But it did take some getting used to.
23:59Because in some ways, zero isn't like any other number.
24:04First of all, it has unique properties.
24:07Zero has some properties of number, but also some properties that make it more like a concept than a number.
24:13In addition, subtraction, and multiplication, zero behaves differently than every other number.
24:21But where zero really creates havoc is in division.
24:26You get to division and all of a sudden, it's the first time that you're sort of told, like, well, that's impossible.
24:32You can divide any number by every other number, except zero.
24:38When you divide a number by zero, for example, you blow up.
24:42Three, two, one, zero.
24:44I have no apples, and I share that among six students.
24:49When everybody gets zero apples, there are no apples to share.
24:54But if I have six apples and they are shared among zero students, the concept becomes messy now.
25:01How do we make sense of that?
25:03The problem is, you can't.
25:06Think of it this way.
25:08Dividing six by zero is the same thing as asking what number multiplied by zero will give you six.
25:18Since everything multiplied by zero always equals zero, there's no solution.
25:24So mathematicians officially consider the answer as undefined.
25:30Now, you might wonder, is that sort of hole in the bucket of division a problem?
25:36Does it get you into trouble?
25:39Turns out, it certainly does, under the right circumstances.
25:44In fact, a Greek philosopher who lived thousands of years ago, before zero even came to be, invented a paradox that captures the problem.
25:55His name was Zeno of Elea.
25:58And the paradox was about an arrow.
26:06To help me demonstrate Zeno's paradox, I've turned to Eric Bennett from Surprise, Arizona.
26:13VF is what we're looking for.
26:15He's a physics and engineering teacher at a local high school.
26:19And he's a Paralympian in archery, four times over.
26:25So Eric, what does it feel like to have participated in the Paralympics four times?
26:29It makes me feel old a little bit.
26:32But it's amazing.
26:34I've been competing at a really high level for 15 years.
26:37Wow.
26:38Wow.
26:39So how far away is the target here?
26:41The target is the standard Olympic competition distance of 70 meters, which is about three quarters of a football field.
26:48No way.
26:49Yes, actually.
26:50It's pretty far.
26:51Okay.
26:52All right.
26:53I want to see you shoot this.
26:54At 15 years old, Eric lost an arm in an automobile accident.
27:01So he draws the bowstring back with his teeth.
27:07The arrow finds its mark.
27:11Wow.
27:12That's awesome.
27:14All right.
27:15So you're going to show me how to use one of these?
27:16Absolutely.
27:17Yep.
27:18From 70 meters?
27:19No.
27:20I can try.
27:21You're trying to say I can't hit it?
27:23No.
27:24I just want to make sure that you're super successful on your first try.
27:26I appreciate that.
27:27Yeah.
27:28I appreciate that.
27:29Eric offers me a try with a beginner's bow and a target about 20 yards away.
27:37Let go, and it'll go right in the bullseye.
27:40So I channel my inner Katniss Everdeen from the Hunger Games.
27:55And as a statistician, may the odds be ever in my favor.
28:01Whoa!
28:02What?
28:03I don't know.
28:04Where'd it go?
28:05That is like a hundred yards down there.
28:06We'll go find it.
28:07You got a lot of work to do, Eric.
28:08Yeah.
28:09Well, I think it's going to be a while before I'm ready to compete.
28:10I had a lot of power, Eric.
28:11Yeah.
28:12You know?
28:13And so, um...
28:14But back to Zeno and that paradox.
28:30All of Zeno's original writings have been lost.
28:33According to a later Greek philosopher, Zeno suggested that we consider an arrow in flight at any instant in time.
28:44And at that instant, that now moment, the arrow is frozen in space, motionless.
28:54It's neither arriving nor leaving.
28:58And if you consider the entire flight, there's an infinity of those motionless, frozen moments in time and space.
29:09So Zeno asked, is the flight of the arrow and all motion really just an illusion?
29:21His radical conclusion is that motion is impossible.
29:24At a given instant, that arrow is some place.
29:29And then click time forward, it's at some other place, but at no moment was it moving.
29:37Okay.
29:38Let me get ready to let go.
29:42What?
29:43Did you hear that?
29:44Did you hear that?
29:45Well, the motion of an arrow looks real enough for me.
29:49That's right, Katniss.
29:51Got nothing on me.
29:53But you can see why Zeno's timeless, frozen moments are so problematic.
29:59Our whole notion of speed depends on time.
30:04Here's the formula.
30:06Distance traveled divided by length of time equals speed.
30:12But Zeno's frozen moment has a length of time of zero.
30:18That means trying to divide by zero, which is against the rules of division.
30:24But at the same time, we often want to know the speed of something in motion at a particular instant.
30:31One solution to the problem of instantaneous speed is a concept called a limit.
30:41Let's consider a stick figure who walks half the distance to a wall.
30:46And does that again.
30:49And again.
30:50And again.
30:51If the stick figure keeps going half the distance to the wall, they'll get closer and closer.
30:58But the steps will get smaller and smaller.
31:02And they'll never reach the wall.
31:04The wall is an example of a limit.
31:07As the number of steps heads to infinity, the distance to the wall decreases towards zero.
31:13But the figure will never reach the wall.
31:18You're getting infinitely close to a limit as far as you're going to get.
31:21But you never actually get there.
31:23Which, yeah, it's one of those concepts that bothers a lot of people.
31:26Even mathematicians, it bothers, I think.
31:28I can never start with a whole number and divide it by something to get zero.
31:32There's nothing, there's no way for me to ever get to zero.
31:36Even if you have an itty bitty bit and you divide it in half, you still don't have zero.
31:44Harnessing the power of infinity through limits gives mathematicians a workaround to the problem of dividing by zero.
31:53And in turn, opens the door to a world of solutions to some extremely difficult problems.
32:00It helped create a new field of mathematics, calculus.
32:05And that's really the big idea at the heart of calculus, as understood in modern terms, this idea of a limit.
32:11That you're supposed to think, how far did I go over a microsecond?
32:16That gives me an approximation to my instantaneous velocity.
32:20You know, the distance traveled divided by that duration.
32:23But that's not yet an instant.
32:24So rather than a microsecond, I think now a nanosecond, a thousand times shorter.
32:28How far did I travel then?
32:30That gives me a better approximation.
32:32And then this limit as the duration of time goes to zero, you often find you'll get a well-defined limiting answer for the speed.
32:41And that limit is what's called the instantaneous velocity.
32:44It sounds like a clever trick, but does it get the job done?
32:51To find out, I traveled to New York City to the National Museum of Mathematics, MoMAH.
32:59May I?
33:00Please, thank you.
33:01Take your pick.
33:02Here, Cornell University mathematician Steve Strogatz is enjoying a year as a distinguished visiting professor.
33:10Thirteen points, thank you very much.
33:12He shows me around.
33:14But I'm here for a specific reason.
33:18Steve is going to demonstrate the problem-solving power of limits and infinity.
33:25Though, as it turns out, we're missing the key component.
33:33If you want to understand what infinity can do, we're going to need pizza.
33:37Pizza?
33:39Yes!
33:40There's a science to making pizza.
33:43We don't typically associate pizza with infinity.
33:47Yeah, yeah, yeah, yeah.
33:48So how can New York City's most famous food help solve one of the most elusive mysteries of early mathematics?
33:57So, Steve, how is this pizza going to help us understand infinity?
34:09Huh.
34:10I would say it the other way.
34:11Infinity and the pizza are going to help us understand one of the oldest problems in math.
34:17Mm-hmm.
34:18What's the area of a circle?
34:19Which is not intuitive.
34:21No.
34:22You know, what's hard about it?
34:23You might think a circle is a beautiful, simple shape,
34:25but actually it's got this nasty property that it doesn't have any straight lines in it.
34:29Right.
34:30Ancient civilizations didn't know how to find the area of a shape like that.
34:37How to find the exact area of a circle isn't obvious.
34:42For a square or rectangle, you just multiply the sides.
34:47But what do you do with a circle?
34:50So what do they do?
34:51Well, they came up with an argument that you can convert a round shape into a rectangle if you use infinity.
34:57So we're basically going to kind of deconstruct this pizza and make it into a rectangle.
35:01Beautiful.
35:02And then we're going to know the area.
35:03That's it.
35:04So I'm going to start with four pieces.
35:05Okay.
35:06To do that, I'm going to go one point up and one point down.
35:10And then one point up and one point down.
35:14And yeah, like that.
35:16How'd you do in geometry?
35:19You don't think that looks like a rectangle?
35:21That is not close to a rectangle.
35:22No, it's not.
35:23It's not.
35:24But come on.
35:25I'm only using four pieces.
35:26If I use more, I can get closer.
35:28Okay.
35:29So we've got to cut these babies in half.
35:30Let's cut them.
35:31Let's rearrange them.
35:34Same trick.
35:35Alternating point up and point down.
35:38There's one up and one down.
35:41And one up and one down.
35:44There.
35:45We are ready.
35:46That is looking a lot better.
35:47Aww.
35:48What do you think?
35:49Is that a rectangle?
35:50It's not quite a rectangle, but it's getting closer.
35:53It is, right?
35:54Yeah.
35:55In both the four-piece and eight-piece versions,
35:59half the crust sits at the top and half at the bottom.
36:03But with eight pieces, the edge becomes less scalloped,
36:07closer to a straight line.
36:09So we need to go at least a step further.
36:11Let's go more.
36:12We've got to do 16.
36:16So we have to just change every other one.
36:18Am I going to mess this up?
36:20I mean, that's a parallelogram that's aspiring to be a rectangle.
36:24That's got aspirations.
36:25Yeah, it's got high hopes.
36:26It's got high hopes, I tell you.
36:28From four slices, to eight slices, to 16 slices, and even 32 slices,
36:42there's a clear progression towards a rectangle.
36:45With one piece out of 32 cut in half to create vertical sides,
36:50the rectangle is almost complete, except for the wavy top and bottom.
36:55But as the number of slices increases, the straighter and straighter those edges would become.
37:02And the argument here is that if we could keep doing this all the way out to infinity,
37:07so that this would be infinitely many slices, infinitesimally thin,
37:11this really would become a rectangle.
37:13Yeah.
37:14And we can read off the area.
37:15That's right.
37:16It's this radius, that's the distance from the center out to the crust, times half the circumference,
37:23which is half the crust, half the curvy stuff.
37:26And that's a famous formula.
37:27Half the crust times the radius.
37:29Yeah.
37:30One half CR.
37:31That's where the C is.
37:32Usually C for circumference, but you could see its crust.
37:35So at the limit, once we got all the way out there, it's going to look like a rectangle.
37:38It would be a rectangle, and that is actually the first calculus argument in history.
37:43Yeah.
37:44Like 250 BC to find the area of a circle.
37:47Who knew you could learn so much from pizza?
37:49Infinity is your friend in math, and that's the great insight of calculus,
37:54that you can rebuild the world out of much simpler objects,
37:58as long as you're willing to use infinitely many of them.
38:07By embracing infinity through calculus,
38:10mathematicians created one of their most powerful tools.
38:15For this professor of applied mathematics, it is part of how he sees the world.
38:27Do you remember that movie, The Sixth Sense, where the kid says,
38:30I want to tell you my secret now.
38:34I see dead people.
38:38Sort of what I feel like, except I see math.
38:45When I go out and see the New York skyline,
38:48I see all the rectangles and pyramids and the skyscrapers.
38:57I see the patterns of geometry.
38:59I see hidden algebraic relationships.
39:03There's traffic flow, and the cars look like corpuscles,
39:06which makes me think about blood flow in arteries.
39:09Laws of fluid dynamics and aerodynamics.
39:12Patterns of cylinders and the rings on the cylinders are spaced unevenly because of the way hydrostatic pressure works.
39:24There's so much math in the real world, and it's all one big principle.
39:30The whole world runs on calculus, and math is everywhere.
39:35Just can't help but notice it.
39:36I see math.
39:42Actually, I see dead people, too.
39:48Calculus is applied everywhere.
39:51And if you're looking for how infinity comes into play in the modern world, you need search no further.
39:58But even with the advent of calculus, infinity itself in mathematics remained poorly understood.
40:09It was only in the late 19th century that new mind-bending ideas helped tame that strange beast, infinity.
40:17When I asked my friend, author and mathematician Eugenia Chang, to discuss her thoughts on infinity,
40:28she 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.
40:45And this hotel is definitely an odd property.
40:54Well, the Hilbert Hotel is a pretty amazing hotel.
40:56It has an infinite number of rooms.
40:59Wouldn't that be great?
41:00You might think that you could always fit more people in.
41:03But what if an infinite number of people showed up?
41:06And then the hotel would be full?
41:08Oh, dear.
41:09Then if another person came along, what would you do?
41:12Well, if you weren't very astute, then you might just say, sorry, we're full.
41:17That's one solution.
41:19Or you might think, given there are an infinite number of rooms,
41:25you can just assign the late guest the room that comes after the one given to the last guest that checked in.
41:32You know, just farther down the hall.
41:34Just put this person at the end of the line. Why can't we do that?
41:37Where is the end of the line? Sounds like a philosophical question.
41:41But the thing is, you can't just tell them to go to the end.
41:43You have to give them a room number.
41:45And all the rooms are full.
41:47Hmm. Seems unsolvable.
41:51But 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.
42:02Okay, 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.
42:17Everybody has another room they can move into because everyone just adds one to their room number, and that will leave room one empty.
42:24So a new person comes. Welcome. You know what? We're just going to have everybody scoot over for you. Just scoot. Goes in room one.
42:31And then what if two people showed up? That's fine. Everyone moves up two rooms.
42:38What if five people show up? That's fine.
42:40But what if an infinite number showed up?
42:44Say, because of a fire, at a second nearby completely full Hilbert's Hotel?
42:55Is there room for a second infinity of guests?
43:01You've now got an infinite number of people.
43:03You can't just get everyone to move up an infinite number of rooms, because where would they go?
43:09There is a solution. The manager asks each person checked into a room to multiply their room number by two and move there.
43:19So one goes to two, two goes to four, three goes to six, and so on.
43:25Which means they will all move into an even-numbered room, and that will leave all the odd-numbered rooms.
43:30And that's an infinite number of rooms, and so all the new infinite number of people can move into the odd-numbered rooms.
43:39So then it feels like we've got twice the number of rooms, although we're still at infinity.
43:44Mm-hmm.
43:46In fact, the hotel can accommodate all the guests from an infinite number of infinite hotels.
43:54But you'll have to stop in to learn how.
43:56I guess here at Hilbert's Hotel, there's always room for one more.
44:05While 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.
44:22He built that understanding based on another area of mathematics he created, set theory.
44:30A 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.
44:41Cantor used sets as a way of comparing quantity.
44:48If 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.
44:58All of this may seem elementary, like counting with your fingers, but they are ideas that will carry you to some strange places.
45:07Counting in pure math is very profound, and it doesn't just mean list everything and label them one, two, three.
45:16It often means find some perfect correspondence 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.
45:28And so there are some really counterintuitive things we can do.
45:33Consider this. Which infinity is bigger?
45:37The set of counting numbers, one, two, three, four, etc.
45:42Or the set of just the even numbers, two, four, six, and so on?
45:47And intuitively we might go, well that's half of them.
45:51That's half, right. Yeah.
45:52But 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.
46:03And that will make a perfect correspondence.
46:07So the set of counting numbers and the set of even numbers are both infinite and both the same size.
46:15Cantor called these kinds of infinities, with a one-to-one correspondence to the counting numbers, countable.
46:25And 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.
46:38Cantor found the infinity of the prime numbers was also countable.
46:42And 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.
46:54Even that infinity was countable and the same size as the others.
46:58But now for the ultimate challenge.
47:14If you take all the rational numbers and add in the irrational numbers, like pi or the square root of two, numbers you can't represent as fractions using integers.
47:32You know, the ones that have decimals that go on forever without repeating.
47:39Then you have the real numbers.
47:42The complete number line.
47:45Every possible number in decimal notation.
47:49So is the infinity of the real numbers, just like the others, countable?
47:57Well, since the other sets of numbers are, this one has to be two, right?
48:03In 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.
48:14So to do that, you need to be able to list the infinity's members.
48:21Not literally. It's infinite and would take forever.
48:24But just the way the list of all the counting numbers marches off toward infinity, adding one with each step.
48:32Is there a way to list all the real numbers to prove that they're countable?
48:37Cantor demonstrated the answer is no, with an ingenious argument.
48:44Imagine you presented Cantor with what you think is a complete list of all the real numbers.
48:51To keep it simple, we will only do the ones between zero and one.
48:56And for consistency, a number that terminates exactly, like 0.5, will receive an endless series of zeros after the last digit.
49:05The list, of course, goes down the page infinitely and off the page to the right because the numbers are infinitely long.
49:15Cantor looks at your list and starts to construct a new number.
49:20He takes the first digit of the number in the first row and adds one to it.
49:25If it's a nine, it becomes a zero.
49:27Now he knows his new number won't match the one in the first row.
49:33Next, he takes the second digit of the second row's number and does the same.
49:39Now he knows his new number won't match the one in the second row.
49:43And he does the same thing with the third row's number.
49:48He 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.
50:03This famous diagonal proof shows that any attempt to list all the real numbers will always be incomplete.
50:13And if you can't create a complete list of the real numbers, they can't be counted.
50:21Cantor called the infinity of the real numbers uncountable, a bigger size infinity than all those countable infinities.
50:31Well, the idea of infinity had been around for a long time.
50:36But the idea that some infinities could be bigger than others, that's what Cantor's diagonalization argument demonstrated.
50:44And his argument is so simple.
50:46It's, again, one of those simple ideas that is just so profound.
50:50It's one of the most ingenious, innovative ideas ever inserted into the study of numbers.
50:56And our understanding of infinity is forever changed because of Cantor's incredible work.
51:01For humankind, the journey from zero to infinity has been extraordinary.
51:10Zero introduced thousands of years after the birth of mathematics, revolutionized it, enabling a new means of calculation that helped the advancement of science.
51:21Harnessing the power of zero and infinity together through calculus made many of the technological breakthroughs that we take for granted possible.
51:34And Cantor's work on infinity?
51:35He unveiled a new, strange vision of it for all to see.
51:41His ideas and methods laid a foundation for the development of mathematics in the 20th and the 21st centuries.
51:49But for me personally, I think his imagination helps us appreciate that we live in a universe of infinite possibilities.
51:59No doubt, new wonders still await us on the road from zero to infinity.
52:05To infinite.
52:062
52:08To infinite.
52:14To infinite.
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