Ideas - Echoes of an Empty Sound: The Story of Zero

Episode Date: December 30, 2024

It's nothing — and it's everywhere. Zero has confounded humanity for thousands of years. On IDEAS, we explore the infinite danger and promise of the void in a series called The Greatest Numbers of All Time. *This episode originally aired on Sept. 26, 2023.

Hey there, I'm David Common. If you're like me, there are things you love about living in the GTA and things that drive you absolutely crazy. Every day on This Is Toronto, we connect you to what matters most about life in the GTA, the news you gotta know, and the conversations your friends will be talking about. Whether you listen on a run through your neighbourhood, or while sitting in the parking lot that is the 401, check out This Is Toronto, wherever you get your podcasts. This is a CBC Podcast. In everyday life, where do I see zero?

Okay, um... Welcome to Ideas, I'm Nala Ayyad. Well, I get up, I go and have a shave. So, you know, my alarm clock goes off at exactly zero minutes past seven o'clock. I go downstairs to breakfast and at that moment, the amount of breakfast on the table is definitely zero. Today we begin a special series, the greatest numbers of all time. As a mathematician, I am absolutely attached to zero because I can't do anything really without it. It's nothing at all, but when you add it onto a number, it makes it bigger.

Zero, once it made its way into mathematics, it really changed the world. Zero. It's nothing. And yet, everything. We now believe that the universe started with a big bang. Then there was nothing. And then suddenly there was something. And there was this universe that suddenly exploded out of nothing. We often take it for granted, but it's one of the greatest inventions of all time. This is the story of Zero.

Are you ready, Charles? I am indeed. Ideas producer Annie Bender will take it from here. Where do you feel Zero in your everyday life? I think for me, Zero, I feel the most is kind of lying on my back in a very dark night and looking up at the Milky Way and the stars and seeing the vastness of the cosmos and realizing, despite all this beauty,

these lights that are twinkling, that what we're seeing is the vast minority of what is out there. And the flip side is true nothingness. First thing I'm going to do is get you each to say your name and how old you are, okay? Hi, I'm Silas. I'm 11. Hi, I'm Elijah, and I'm 10. Hi, I'm Mika. I'm 11. Hi, I'm Elijah and I'm 10. Hi, I'm Mika and I'm 7. What do you think about when you think about the number zero?

What picture comes to mind? I think of it sort of as an age. Like, let's say you're zero years old. That's the beginning of your life. Zero years old, that's the beginning of your life. And I think without the number zero, it would just be like you're one year old. Like when you're born, you've already had one year of your life. So how old would you be then without zero?

Um, I think I would be six. Would you rather be six? I don't think so. We owe a lot to zero. It's the origin of every origin story, the place where each of us gets our start. But it's also easily dismissed. It is not a number that most people would say is their favorite. Like, it's not a prime number.

You can't even multiply anything to get to zero. Like, schools try to teach you that, like, zero is important on the end of a number to multiply it by ten. But, like, on its own, it's just pointless. Zero doesn't amount to much. That's just a plain mathematical fact. And yet, zero manages to make quite a lot out of nothing. It's there when you use the vacuum, when you meditate, or go into surgery.

You wake up in another room with a clock on the wall telling you that 40 minutes of your life has just passed and you remember nothing. In fact, you can conjure up zeros just about anywhere. There's not a zoo in this backyard. There's not a car in this backyard. There's a lot of things there aren't. Like all the countries aren't here. There's not, like,

most of the stuff in the world is not in this backyard. Do you think Zero is the life of the party? Honestly? Yeah. But Zero wasn't always welcome at the party. For a long time, we didn't even know it existed. So which society was the first to start playing around with zero?

So it was the Sumerians and their descendants in the Fertile Crescent. The Sumerian notation used the wedge of a reed pressed in clay to denote various characters. Charles Seif is a mathematician turned journalist based in New York City. He wrote the book on nothing, literally. Can you say the title of your book? It's Zero, the Biography of a Dangerous Idea. And of course, no biography is complete without an origin story. The first zero appeared around 300 BC, we know, probably significantly earlier. The double wedge, there was kind of a slanted double wedge. That slanted double wedge wasn't quite zero as we know it today, but it was an important first step.

Before it was born as a number, it wound up as a placeholder. When you look at the number 100, it is a 1 followed by two 0s. Without 0, we wouldn't be able to distinguish between 1 and 10 and 100. You'd need that little place. Whether it's a, it doesn't have to be a zero, you can just have a little dot, or you can have an X, or something like that, just to denote that the 1 is in the first place or the second place or the third place. So it sounds like if there was a problem that zero was there to solve,

it was basically a problem of notation. That's correct. It was to give a flexible way of writing numbers. What do you think the world was like before zero? The world would be like very limited. If we had no zero, then we probably couldn't have had decimals. If we didn't have zero, we couldn't have negative numbers. If we didn't have zero, we'd have to write 10 as 11. The Sumerians' placeholder zero ultimately made it possible for us to write any number we want using just nine symbols. You can keep adding zeros.

You can keep having places out there and growing your number system ad infinitum. You don't run into an end because you're never having to invent new symbols. You just go out more places. You might run out of paper, but the concept is still there. In fact, with scientific notation, you can just say, go out 10 to the 79, go out 79 places and put a one there. And you can do that very, very easily. You can go up to any number unfathomably large, and you can get there with place value notation. It's fascinating. So part of the power of zero is that it allows us in some way to touch infinity.

Yes. Without the infinite power of that placeholder zero, early mathematicians could only get so far. The Greeks and the Romans, they didn't care as much about place. So what you would do is you would string symbols together, symbols that meant two, a symbol that made 300, and a symbol that meant 2,000. And those three symbols together would mean 2,302. But, I mean, actually, when you run out of symbols, you run out of numbers. And Archimedes actually had the problem.

He was actually in the island of Syracuse in the 3rd century AD. And he was the pinnacle of Greek mathematics. He's famous for figuring out some of the physics of buoyancy. You probably know the story. He sank into the tub and seeing the level rise when he sank in, he shouted Eureka

and ran through the streets of Syracuse naked. Right, of course. And he was calculating with things that were so large that he ran out of symbols that represented numbers in the Greek system. So he had to invent his own kind of expansion to the number system, which went very large, but it still was finite.

Zero emerged in the Fertile Crescent well within Archimedes' lifetime. But long after the Sumerians blasted off toward infinity, Archimedes was still stuck tinkering in the dirt with his limited set of numbers. When did mathematicians in what's now known as the West first become aware of what the Sumerians were doing with zero? Well, we do know certainly by the late classical period, so we're talking 200 AD, that Greek mathematicians were using Babylonian notation, Sumerian notation, in astronomical calculations. So there was definitely awareness of the notation, but it never caught on in any of the Greek-speaking or primarily Greek-speaking or Latin-speaking areas, as far as I know. And why not? Because there was a well-established way of doing things already.

Because there was a well-established way of doing things already. It's like, why are you still using Microsoft Word when there is a better word processor out there? Well, it's because we've done it this way for so many years. Old habits die hard. But the Greeks' resistance to zero went much deeper. Greek philosophy, and by extension, early Christian philosophy, more or less rejected the idea of nothing, which is kind of interesting because if you think about the creation story that is accepted by the Bible, is that the world, the universe, was created from nothing. The earth was formless and empty. Darkness was over the surface of the deep.

And God said, let there be light. But the Greek superimposition of the philosophy upon Christianity kind of shoved that concept under the rug. Among those fundamentally opposed to zero, Aristotle. For him, any mention of the void was itself null and void. The idea of nature abhors a vacuum comes from Aristotle's physics. Because it broke the whole system of physics that Aristotle had. And not just Aristotle's physics.

It broke metaphysics. It broke some of the religious beliefs that were associated with the way the world works. The Aristotelian and a later Ptolemaic picture of the universe required that the planets and the sun be moving in these spheres circling the earth. But motion cannot come from nothing. There is no nothing. So there has to be some motive force that is moving the outermost spheres. And this was referred to as the prime mover. And naturally, when Christianity came through Rome and the West, that prime mover was associated with the Christian God.

So in some ways, the existence of God was a refutation of the void. And so it sounds like the Greeks, basically their response to this idea of zero was in some ways to almost plug their ears and start humming. Yeah, it was dismissed. I mean, the idea was played with, but was dismissed as unphysical. That it just was something that we humans could think of like a talking horse, which doesn't exist in the real world. like a talking horse, which doesn't exist in the real world. Even Archimedes, one of the greatest mathematicians of all time, just couldn't see the talking horse.

No, no, he did not quite make it there. He never really dealt with the zeros. And neither did kind of other Greek philosophers. Should I do it now? But not everyone in the ancient world shared the Greeks' stubborn fear of zero. Can you help us to hear how the concept of zero was used in ancient India through tabla? The downbeat is the most important beat in a rhythmic cycle in Indian music. And sometimes on the downbeat, you'll play a very loud sound,

but sometimes you'll play a soft sound, and sometimes you'll play the empty sound. And that's recognized as a very powerful moment sometimes when you're anticipating this downbeat. Everything is building up to that. And then what do you play on that downbeat? You play the empty sound. It feels very powerful. I'm Manjil Bhargava. I'm a professor of mathematics at Princeton University.

It's a sound in itself, the empty sound, the zero sound, is a sound. In Western music it's often denoted as a rest, where it's not a sound, you're just like, you're just resting. But in Indian music, you recognize the empty sound as you're actually doing something, you're actually playing a note there, but it's the empty note. What does that tell us about the way that people in ancient India were thinking about emptiness or in effect, by extension, thinking about zero? Right. So, yeah, the philosophical view that emptiness is something, that nothingness is something to be discussed, to be studied, was there in the air philosophically, and then it made its way into linguistics, and then it made its way into music. And so this was a natural progression.

You first had zero there philosophically, then zero there as a symbol and something you can hear, something you can feel. While the ancient Greeks plugged their ears to zero, ancient Indians were chasing the empty sound. So there are sort of three landmarks in Indian history that led to the way we think about zero today. And the first one is zero as a concept, which then eventually led to zero as a symbol. And then it finally led to zero as a number. And then it finally led to zero as a number. But zero as a concept was something that was in the air in India even thousands of years ago. There was the concept of shunyata. Shunya means zero. So shunyata means zero-ness.

The concept of emptiness, the concept of the void, nothingness as being something. And it was very important in Indian philosophy because, for example, in meditation, the goal of meditation was stated as being the desire to achieve nothingness, chunyata, emptying one's mind of sensations and emotions and ego and thoughts. That was the goal of meditation. And so the concept of zero was in the air philosophically. How else was this idea of emptiness present in the culture of the time? Linguistics is sort of the next stop where zero made its appearance. And in linguistics, oftentimes when you're speaking,

you make pauses, you stop for a little bit. And in Sanskrit, actually, there are words. When words combine, you actually introduce this empty symbol and this empty sound. The notation was called an avagraha, and the sound was called a lopa. The avagraha is written as this S-shaped symbol that's just as big as any other letter. It's just treated like any other sound. It was a linguistic zero.

So in a sense, the first written zeros in India weren't numbers at all. They were more like letters. That's right. Yeah, they were letters representing the empty sound. And so that's kind of the first movement from zero as a concept to zero as something that's concrete that you can write down. Skip ahead a few centuries, and you start to get glimpses of zero as we know it today. The first place that we see the numbers that we use today is in the Bakshali manuscript, which is dated to about the year 300. It's kind of a merchant's manuscript.

They're doing calculations about transactions, and the zero is used in the way we use it today, nine symbols to represent the numbers one through nine, and a zero, which is represented as a dot. So it's not exactly the way we use it today, in the sense that the zero is represented as a dot. So it's not exactly the way we use it today in the sense that the zero is represented as a dot. It's kind of a placeholder rather than a full-fledged number by itself. But you can see how it's starting to emerge as a daily tool, the zero, in the year 300. So how does zero emerge on the scene then as a full-fledged number?

Zero first became a full-fledged number just like any other number. It was treated that way first by the great and legendary mathematician Brahmagupta in the year 628. Before, when people thought of numbers, numbers meant 1, 2, 3, 4, and so on. But if you take, for example, 3 minus two, you get one. Fine. But if you take two minus three, you get something that's not a number anymore. And that was problematic to Brahmagupta. What if you take two minus two? What do you get? You have to give that a name. And he said, that is the concept of zero, and we have to treat that as a regular number. And that's really groundbreaking as a concept. But what's more, he didn't stop there.

And that's really groundbreaking as a concept. But what's more, he didn't stop there. He introduced the zero as a number and gave all the rules for arithmetic of zero. But then he said, well, what about if I take zero minus one? What happens there? And so his book is actually the first place where negative numbers were introduced. So that's the sense in which zero was kind of opened a floodgate of lots of other numbers. So in the sense what Brahmagupta did here was that he made it possible for mathematicians to engage with numbers that were abstract, like numbers that didn't necessarily need to have an analog in the real world. That's right,

that's exactly right. And that can really revolutionize mathematics. And it did. As zero started to take up more space in mathematics, it also started to take up more space on the page. The zero dot started increasing in size after Brahmagupta's work, from being a dot to a little circle, and then a little bigger. And by the 800s, just 200 years after Brahmagupta wrote his great work, the zero was starting to being written as the same size as all the other digits. Zeros popped up in other parts of the world too.

Yeah, the Mayan civilization in Central America, sort of Mexico region, probably basically in the year zero, roundabout then, they already had a symbol for zero in their numbers. I'm Ian Stewart. I'm a retired mathematics professor at the University of Warwick. I'm Ian Stewart. I'm a retired mathematics professor at the University of Warwick. The Khmer civilization in ancient Cambodia probably got its system for numbers from India, but developed its own version, and they had a system for zero. By about 600 to 800 AD, everywhere in that region that had numerical systems had tended to have some

variant on the Indian version, which led in fullness of time to what we use now. But like the stubborn Greeks before them, the Christians of medieval Europe still wanted nothing to do with zero. Case in point, Arabic numerals, which by the year 1200 were taking off among European bankers. When it made its way to the Arab world, that's when it really started to spread. But then in 1299 in Florence, the church banned the use of these numerals, saying that they represent Satan. And so Roman numerals continued to be used, despite the mathematical advantages of using this system. And it wasn't really until the Renaissance that scientists and merchants really realized

we're not going to be able to make progress without switching to this system of enumeration. And of course, in the Renaissance, the great scientific discoveries started happening, and the switch to these numerals was made at that time. And the rest is history. started happening and the switch to these numerals was made at that time. And the rest is history. But it took hundreds of years to be accepted. And if those numbers weren't adopted at that time,

maybe the Renaissance would have happened hundreds of years even later than that. So by the 14th century, zero as we know it today had finally arrived. And once zero took the stage, nothing would ever be the same. There are certain avenues that are very difficult to wind your mind down without that zero. Things like calculus, things like the vacuum in physics. What it set the scene for was abstraction, was taking mathematics to a higher level, because once you've kind of got out of your brain or off the counting table, that was an essential precursor, really, to kind of the next stage, which is writing down things like algebra.

My name is Sarah Hart, and I'm a professor of mathematics. Solving more complicated situations and maybe thinking about higher dimensions and all of these things that have led to more modern mathematics. Everything that's going on in your phone, on your computer, is based on the arithmetic of computers and what the computers use. They use binary numbers, and every single binary number is written with zeros and ones. Those are the two absolute foundational numbers with which you would not have computers or any of the technology that we have really today.

Numbers with which you would not have computers or any of the technology that we have really today. It is transformative. And I think the way that we look at a world with zero and with a void is very different from one where we're afraid to incorporate those. But let's not forget the title of Charles Seif's book. It's Zero, the Biography of a Dangerous Idea. It's got these strange properties. When you try to do these very simple arithmetical operations with zero, it sometimes goes badly wrong. Humanity may have made its peace with the void, but zero still has plenty of tricks up its sleeve.

You know, our assumptions that we try and make just can get turned on their heads. And that's why calculators will, like they've been ordered, stay away. Don't get involved with one divided by zero or anything divided by zero, because then someone will say, what zero divided by zero? And then we really are in trouble. You're listening to Ideas and to a documentary by producer Annie Bender about nothing.

Ideas, of course, is very much something. We're a podcast and a broadcast heard on CBC Radio 1 in Canada, on US Public Radio, across North America on Sirius XM, in Australia on ABC Radio National, and around the world at cbc.ca slash ideas. Find us on the CBC Listen app and wherever you get your podcasts. I'm Nala Ayyad. My name is Graham Isidore. I have a progressive eye disease called keratoconus.

Unmaying I'm losing my vision has been hard, but explaining it to other people has been harder. Lately, I've been trying to talk about it. Short-sighted is an attempt to explain what vision loss feels like by exploring how it sounds. By sharing my story, we get into all the things you don't see about hidden disabilities. Short-sighted, from CBC's Personally, available now.

Hidden Disabilities. Short-sighted from CBC's Personally. Available now. If zero could make a sound, what would it be? Oh. Yeah, that's a very good question. White noise. I think kind of like a buzzing noise.

I think maybe it would be a very, very low hum kind of sound, like you get in mournful music. A plaintive, low sound like that. Tailing off as well into nothing. Yeah, I would think that the sound of nothing would have to be no sound at all, which is something that none of us really truly encounter. In ancient India, one of the earliest representations of zero or nothingness was a sound, the empty sound. It's a sound in itself, the empty sound, the zero sound is a sound.

A piece of music, you might think that the zero is actually represented by the absence of sound, you know, a rest, a pause. That's zero sound happening. But if we were to try and actually make zero sound, it's impossible. This ancient number, zero, has proven itself to be infinitely confounding. You can't really think about nothing because if you're thinking about nothing, you're thinking about the word nothing, maybe. There's a lot of very confusing things, like sort of like a riddle, you know, like things won't make sense because of like zero and like negative numbers and all that stuff.

It's elusive. It's hard to track down. It's hard to pin down. And it is a bit of a trickster. I mean, maybe it's the Loki of the Marvel Universe of Numbers because you just can't quite be sure it's not going to mess things up for you in many ways. But that hasn't stopped us from trying to pin it down. The quest for zero, even if you can't get there, still turns up lots of interesting things. As part of our series, The Greatest Numbers of All Time, our quest for zero continues. With producer Annie Bender.

We've come a long way from thinking zero is ungodly. But as we've embraced it, other dangers have emerged. As we've embraced it, other dangers have emerged. As a mathematician, you get to find that there are pretty weird things happening around zero. Sarah Hart is a professor of geometry at Gresham College in London. She also teaches mathematics at the University of London. You can multiply by zero all day long, and you still end up with zero. Whereas if you add zero to something, then you leave that thing completely unchanged. So that's the sort of two universal qualities of the number zero,

which other numbers don't have. Let's talk about what happens when you input one divided by zero into a calculator. In fact, I'm going to do this right now. I'm opening up my calculator on my phone. I'm putting it in one divided by zero. It will say something like error or something like this. It won't do it for you. Yes, a big, bold error just popped up on my calculator here. What's going on here, Sarah? Have we broken math? Well, we might have slightly dented it. So the issue is with really what we mean by dividing. So when we divide 1 divided by, let's say, 0.1, right, that's a tenth. What we find is that we get 10. It's a bigger number.

If we divide 1 by a small, tiny fraction, the outcome will be bigger than 1. And you can keep doing this and dividing by smaller and smaller and smaller things. and if you try this on a calculator dividing by tiny tiny tiny numbers the outcome gets bigger and bigger and bigger but if you extend that and sort of imagine in your head what will happen when i extend that and make that number so small that eventually it becomes zero then so you could say well maybe one divided by zero is infinity. Maybe that's the answer. But then you have to get into, well, what's infinity then? Which is a whole other can of worms. Even Brahmagupta, ancient India's biggest ambassador for zero

all the way back in the 7th century, could sense the looming threat. all the way back in the seventh century, could sense the looming threat. He started to realize that when you multiply by zero, when you multiply any number by zero, it becomes zero. So zero kind of consumes everything. When you divide by zero, though, it doesn't have an obvious meaning. The framework of the mathematics begins to shudder. And if you're not careful, it can actually destroy the logical foundations of mathematics. Yikes. You have this line that I like where you write, basically, when Brahmagupta was faced with this problem of dividing by zero, that he basically just, quote, waved his hands and hoped the problem would go away. Yeah, yeah, this is what mathematicians do. They say, this is an odd case, we'll turn to it later, and they forget about it, or hope it doesn't come to bite them in the bottom.

The further you go down this rabbit hole, the weirder things become. further you go down this rabbit hole, the weirder things become. There are other issues around zero, like anything divided by zero, perhaps we think it's infinity. But then what would zero divided by zero be? Because we know that anything divided by itself is one. So now have we proved that one equals zero? We can't have done. And that's when you start getting a bit, oh no, what's zero done to us now? It's hard to do division on the radio. So I thought maybe we could try it somewhere else. Can you do this in cows instead of numbers? Well, if you divide the cow by zero, you might get too much meat all over the universe.

Oh, God. Certainly, you would invert the cow. That could be messy. I mean, if you accept division by zero, you could say two cows equals one cow, because two cows times zero equals one cow, because two cows times zero equals one cow times zero. You divide by zero. That means two cows equals one cow. Right.

So automatically, any cow you have is two cows. They clearly divide like amoebas. They just split like that. And then we have four cows. Hi there. Are you still with us? Is your head still on?

Assuming it is, there is a good chance it's spinning. I've learned to expect that from Zero. But don't let your guard down yet. We're just getting started. And then you can prove that, say, if a cow times zero equals a chicken times zero, you can see that a cow is a chicken. So you've got a cow, because it's two cows, is equal to a cow and a chicken, which is equal to two cows and a chicken, which is equal to a cow plus two chickens. And so you can cure world hunger by dividing by zero. Well, I was just going to say, I mean, you can understand why the Greeks were so

threatened by a number that can turn a cow into a chicken. But you're right. There could be some upside to that as well. That's right. Over the span of a few thousand years, zero took us from cows to calculus. The art of properly dividing by zero is really calculus. That is what calculus is based upon. The whole of calculus rests on numbers that get smaller and smaller and smaller and

approach zero as close as you like, but don't actually get there. Because if they got there, the calculation you're trying to do wouldn't make sense. But if you assume they're very, very, very small, you can do the calculation and then you can say, well, suppose we kept making these things smaller and smaller and smaller, what would happen? The speed of a car, average speed. If I drive 100 miles in one hour, then the average speed would be 100 miles an hour. But the car would not actually be doing 100 miles an hour all the time. For a start, it would be zero speed when it started off so if you wanted to know how fast is the car going now what you'd like to do is say take a very small time and see how far the car travels in that small time and then convert it

into miles per hour and you might get 60 miles an hour approximately and they say well yeah but that's just the average over one second. What about half a second, tenth of a second, millionth of a second? And in calculus, you take that point of view and follow it to its ultimate conclusion, which is let the interval of time tend to zero, move towards zero, get as close as you like to zero. And this is why calculus is quite hard.

But it is immensely important in engineering and everything. But that's fascinating. In a sense, all of calculus is built on this kind of principle of almost but not quite looking at zero. That's right. It really does sound to me like, in some way, the solution to zero's odd behavior is to almost kind of treat it like the sun, to say, you know, as long as we don't try to look at it directly, we'll be fine. Do you think that's right?

I think that's right. All the tools that we have developed are almost like protections, putting ourselves out of harm's way in some sense. We don't touch it with our bare hands, you know. We've got all these tools and equipment in place to allow us to very carefully manipulate things that are close to zero and that might suddenly shoot off to infinity if we try and divide by things. So we've developed a lot of machinery to help us kind of deal with it safely, really. Yeah, you're right. Zero is like the hot potato of mathematics. Absolutely.

of mathematics. Yes, absolutely. So if zero is like the sun, then calculus is a sort of beach umbrella with solar panels, an umbrella that's allowed us to harness zero's power in all kinds of remarkable ways. It's everywhere.

All of our technology, physics just couldn't be what it is today without calculus because physics when you're looking at you know how the planets are moving around the sun understanding where they're going to be you do that with calculus understanding the properties of the motion of planets and of and of bodies in space more generally, to see why those rules hold, you have to use calculus to do it. We couldn't, you know, put a satellite into space without using calculus to understand where will it be, you know, if I give it this amount of thrust, if I put this much fuel in my rocket, where will that get me to? Those calculations are all done using calculus and we couldn't

do it without them right i mean i've got my own list here some of the technologies that i've been told wouldn't exist without calculus gps yeah ultrasound technology all of space travel as you just said and then of course there's cell phones computers even dare i say it radio does that mean that i owe my career to zero yes, absolutely you do. As a concept, Xero is everywhere. Without empty spaces to fill in the gaps, life wouldn't make any sense. There are a lot of things you wouldn't be able to say.

There are a lot of things you wouldn't be able to say. Like we were discussing the things in our backyard that aren't here. You wouldn't really be able to say that. So your backyard would be like everything in the world. And if someone else's backyard had something that your backyard would be like everything in the world. And if someone else's backyard had like something that your backyard didn't, then they just have like extra of everything. It gets confusing without zero, doesn't it? Yes.

Definitely. I think it can be very confusing sometimes. Yeah. Despite its essential service and its ubiquity, the Loki of numbers isn't so easy to track down in the lab. If we look at it in a purely material sense, how attainable is total nothingness? It's essentially, in most cases, it's completely unattainable. Oh, come on. Let's just give it a try.

Okay. If we go into space. Right. The air disappears and it's a vacuum right now a vacuum a perfect vacuum would be a region of space with absolutely nothing in it not a grain of sand not a single molecule not a single atom not a single electron basically that doesn't exist. Even in the vacuum of space? Even in the vacuum of space, in the dark between the stars, way out, there is probably one atom of hydrogen in every cubic meter of space. But in between those particles of hydrogen? In between, of course, first sight, that's empty. But no, there's electromagnetic radiation, there's light, there's x-rays, there's stuff passing through. It may not be matter as such, but it's still not the complete absence of anything. But there could be at least one exception. So if you could go inside a black hole, which I don't recommend, don't do this at home,

and if the physicists are absolutely right, you would find what they call singularity, which is an infinite amount of mass concentrated at a zero-sized point. Well, that's pretty hard to wrap your head around. But even in the mind-melding realm of zero-sized points, there are no guarantees of true, absolute nothingness. That's hard, yeah. I think what everyone in physics thinks, the theory probably breaks down. The real world knows something that the theory doesn't.

down. The real world knows something that the theory doesn't. And that's exactly why the ancient Greeks rejected zero. They argued that if a number weren't present in the natural world, it couldn't possibly exist. So I don't blame them for wanting to steer clear of zero. If they had, they would have had to grapple with negative numbers and just try to imagine a negative number in nature. It gets weird fast. There's not a zoo in this backyard. What if you had, instead of zero zoos in this backyard, one zoos like i'm almost picturing it would be the exact opposite of a zoo it would be a alternate plane of existence made of antimatter that only had humans that were not in cages and there it it maybe it costs the exact negative of the price it would normally cost to go to a zoo.

And who knows, maybe it would be called an ooze. This is the kind of stuff I'm talking about, like how things can get confusing. As Aristotle so famously put it, nature abhors a vacuum. It turns out there is a lowest possible temperature that you can get to, or that could in principle exist. The temperature of an object is to do with how energetically its atoms and molecules are vibrating. And if you slow the vibrations down, it gets colder and colder and colder. And the lowest you can go is when it's not vibrating at all. You can't have a negative amount of vibration, a negative energy.

So this is called absolute zero. And it's about minus 273 Celsius. So it's very, very cold. And physicists over the centuries have tried to get closer and closer to it, partly because it's interesting, partly because they just want to see what happens when you get really close to absolute zero. Why not? Yeah. Yeah. And one of the laws of thermodynamics, which is very basic in physics, says you can get, in principle, as close as you like to absolute zero, but you can never actually get there. Oh, why?

Because if you have a system which is in its zero energy state, it has to be totally out of contact with the rest of the universe. Otherwise, a little bit of vibrations, a little bit of energy will leak into it. So in modern times, physicists have got down to very, very tiny fractions of a degree away from absolute zero, but never actually there. It would be quite dangerous to get there. Yeah. Call it the golden rule of zero. Get too close and one way or another, it's going to cause trouble. When scientists get as close as they possibly can to this unattainable zero in temperature, absolute zero, what happens?

Very strange things happen. The classic example is if you liquefy helium, the helium, the gas, and take it right down to very, very low temperatures, it starts to, if you did this in a glass beaker, it starts to climb out of the beaker of its own accord. Wow. And materials at very, very low temperatures become superconducting. They can conduct electricity with no resistance. So, you know, if you have a wire and you pass electricity along it, it's a bit like trying to push something through a tube. There's a resistance to the flow and it heats the wire up a bit. So the resistance in the wire creates heat and shows up

and some of the electricity gets lost because it's converted to heat. Now, in things like computers or power line transmissions, you'd like not to have that happen. You don't want to lose your electricity. You want to hang on to it until it gets to where it's supposed to go. So superconductors would allow you to do that, which is great, but it means you have to keep your computer at very, very close to absolute zero, which is actually feasible, but you wouldn't have one on your desk, you cannot keep your electric transmission power lines close to absolute zero.

So there have been a lot of efforts to find materials that will superconduct at higher temperatures, and they do exist, but not really practical for those purposes yet. It does seem like there could be some very powerful applications for this if we could find a way to make it practically possible. This is right and I think what we're seeing here is that the quest for zero even if you can't get there still turns up lots of interesting things. I find it pretty incredible that simply by allowing for the possibility of zero to exist and edging closer to it, zero has allowed humanity to advance in some very real, very profound ways. It is remarkable how a number that literally represents nothing can be responsible for the creation of so many different

tangible things, technologies, ideas. Yeah, it's the paradox of zero, but it's the start of everything, right? So, you know, you can have, if we don't have zero, we have nothing to build on. This is something that in mathematics, we see all the time that you've got to have a starting point to step off from. You have to have a firm foundation. Otherwise, you can't build anything. You've built your house on sand if you don't have that beginning point. So if we think of it as the beginning of everything, it's less of a surprise that it's so vital. The beginning of everything.

The beginning of everything. For all the complexities of nothingness, you don't have to be a mathematician to recognize the power of zero. Let's say you're zero years old. That's the beginning of your life. And it's not just a starting point. It's also the middle. It is both a negative and a positive at the same time.

And the end. What if, when you die, you become zero? It's the first and the final chapter of the universe itself. On a physics level, it's really very literally true that we began from a singularity, from a Big Bang, where the entire universe, it's all its matter and energy

and all its potential is in a zero point. It's nothing. There's nothing there. It explodes and we come to be. is in a zero point. It's nothing. There's nothing there. It explodes and we come to B. And if you trace the universe forward, even before the cosmological revolution of the past couple of years, we knew that there were only a few ways that the universe was logically going to end

and all of them involved zero in some way. Whether we were going to compress back into a point or expand forever and everything would slowly dissolve into nothing, nothing was the end of the universe. In many ways, nothing is the foundation that built the modern world. This invention of the zero and the way we write our numerals today is what is now the basis of all modern technology and modern innovations. They just wouldn't be possible without it. And we often take it for granted, but it's one of the greatest inventions of all time. Modern mathematics could not operate without zero, without understanding what zero is,

without using that symbol just everywhere, you know, in every computer. I can't imagine a world in which we didn't have zero. Zero, once it made its way into mathematics, it really changed the world. Curing diseases, antibiotics, all of this sort of thing, things that we kind of take for granted but which actually make our lives in many ways much more comfortable than they used to be. Without zero and the mathematics and physics and everything else that goes with it,

we would still be stuck back in the Middle Ages. It's a mind-bending concept. It's awe-inspiring. I mean, it's not a coincidence that the science sections of the papers, whenever there's a story about a black hole, it gets a lot of headlines because we are fascinated by the nothingness. It's the ultimate destructive power in some ways, yet it is still nothing.

And so getting our heads around that duality is very difficult. And so I think we kind of like tantalizing ourselves, kind of walking along that edge of the cliffs, looking down into the maw of zero without ever truly falling in. If history is our guide, we'll be walking along the cliff edge of zero for the rest of time. Try not to fall in. You've been listening to Ideas and to a documentary called Nothing, Echoes of an Empty Sound. This episode of Ideas was produced by Annie Bender.

Next up in our special series on numbers is the number three. Our web producer is Lisa Ayuso. Technical production, Danielle Duval, Emily Cervasio, and Orande Williams. Nikola Lukšić is the senior producer. The executive producer of Ideas is Greg Kelly, and I'm Nala Ayed. Let's switch things up. I'm going to ask you a question. Okay. What's half an infinity? Okay. Now, half infinity is actually infinity. And the reason why half infinity is infinity is that two times infinity is the same as infinity.

So I can split the infinity of numbers into two infinite pieces, each the same size as the original. And that's where you start to say, hmm, infinity is weird, isn't it? And the answer is, yes, it is. Something that I've always wondered is, what would infinity divided by zero be? Oh, wow. Okay.

Wow. I'm a bit scared, you know, in case my membership card for mathematics gets revoked. But if I had to, you know, at some point say what infinity divided by zero was, I would say it's infinity if I had to, but I'd be quite nervous about it. This is going on the radio, right? What channel? CBC Radio 1. 1? Not zero.

Oh, okay. Okay.