Ideas - The most important numbers in the universe

Episode Date: May 29, 2026

Numbers get their due credit in this podcast. Even if we're not aware of them, numbers are essential to how we experience the world. IDEAS explores the most bizarre, surprising, mind-blowing and fundamental numbers in the universe. This panel discussion was recorded live at The Perimeter Institute for Theoretical Physics in Waterloo, Ontario.Guests in this episode:Asimina Arvanitaki is a particle physicist and the Aristarchus Chair in Theoretical Physics at the Perimeter Institute.Ben Webster is an associate professor in the Pure Mathematics Department at the University of Waterloo, and he’s an associate faculty member at the Perimeter Institute.Matthew Johnson is an assistant professor of physics and astronomy at York University.

At Desjardin Insurance, we know that when you're a building contractor, your company's foundation needs to be strong. That's why our agents go the extra mile to understand your business and provide tailored solutions for all its unique needs. You put your heart into your company, so we put our heart into making sure it's protected. Get insurance that's really big on care. Find an agent today at Dejardin.com slash business coverage. This is a CBC podcast. Welcome to Ideas. I'm Nala Ayyed. Our understanding of the universe keeps expanding, a little like the universe itself.

New discoveries and ever more powerful telescopes reveal a cosmos full of wonders that dazzle the eye and fuel the imagination. But when scientists look under the hood of the universe, they see equations, a number of numbers whirring away, underlying all of it. Some of those numbers are staggeringly huge or unimaginably tiny. Some, like Pi and the Golden Ratio, have been known for thousands of years. Others are still the subject of debate, and yet others still remain mysteries, or perhaps have just escaped our notice to this point. The numbers that shaped the universe was the theme of alive

event at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, where I moderated a panel of thinkers with deep knowledge and love of the numbers most fundamental to the cosmos. Asamina Arvanitaki is an internationally renowned particle physicist and the Aristarchist chair in theoretical physics at the Perimeter Institute. Ben Webster is an associate professor in the Pure Mathematics Department at the University of Waterloo, and he's an associate faculty member at the Perimeter Institute, as is Matthew Johnson, who's also an assistant professor of physics and astronomy at York University. I opened our discussion by asking Matt how he first came to appreciate how important math

and certain numbers are to the nature of the universe. I was a challenging youth. I would say I was not a stellar high school student. And then I got to university, I wanted to be a fiction writer. And so I took this course that integrated in a very creative way, fiction writing, ecology, biology, and calculus. Extraordinary. And I had never before in my life seen calculus. And for some reason, I don't know.

It was just so beautiful that I wanted to do more. And so I asked my professor, how do I do more calculus? And he said, oh, you might want to take a course in physics. And so then I became very interested in physics through that. But really it was through just the beauty of calculus, this mathematical structure, that you could predict observed physical phenomenon from this structure, this mathematical structure. It was fantastic.

And that's the difference between you and the rest of it. who when we saw calculus, we thought terrifying. Not beautiful. Thank you for that. Mina, hi. You have said in our conversations preceding this evening that math is the language of the universe. And I wonder when it was that you kind of realize that fact.

Maybe it's not mathematics per se. I can tell you an anecdote, okay? That I like to say when people ask me, what made you to physics, okay? So when I was in elementary school, in one of our books, there was a chapter that discussed the sun, the solar system, and it also had the speed of light.

So I calculated how long it took for light from the sun to reach us. Okay. And I calculated it took eight minutes. And two things made an impression on me from that number. One, eight minutes is a long time. And light travels faster than any car, than anything that we know of.

So for light to take eight minutes to come from the sun, the sun is far away. It's so far away that makes the solar system very large, much larger that we will ever realize. And as a result of the universe, so the Earth is actually tiny. So that's one. The other thing is that it made me realize

because the speed of light is, and we probably come back to me. We are going to talk about the speed of life. It's a constant. It means that we are bound to the past. The concept of simultaneously that we all always see things exactly when it

happens, it will never happen. So in some ways we're bound to the past of things just by the fact that light is the fastest thing that travel, the information that we get is reaching us at always at the finite time, not at instantaneously. What a beautiful and yet confounding kind of idea. Yeah.

Ben, you come out this kind of from a somewhat different angle than Matt and Mina. You're a mathematician by training and profession. How does that perspective or that perch affect the way you see the world and the universe or how you came to see it? I know. It's a big question. Yeah, well, and I mean, I fear. all through this. I'm going to be like, oh, I dispute the premise of your question.

And by the way, you're free to. I don't know that it affects, especially from the standpoint of whether you're a mathematician or a physicist, right? Because when I see the world, I'm still seeing the world through physics, right? Mathematics is indeed, it's sort of a language that you use to think about this. So, I mean, we, you know, we had had some discussions beforehand, you know, I got to reveal how the sausage gets made here. And, you know, one of the things you had mentioned was you sort of wanted to hear about, you know, personally, how did we get into this? And maybe my story, it's not quite the same as Matt's, because, you know, not to brag, but I was a stellar high school student. But the sort of point that came up when I was sort of thinking about that was also a class that I took in college on special relativity.

because, you know, when you're a stellar high school student, you know, you hear explanations of things like relativity on the level of what Mina just said, and that all sounds very cool. But then you're like, well, what does that actually mean? Like, we're bound to the past, but what do we mean by that? And mathematics is the language that lets you actually give that a precise meaning where you can really understand, okay, this is like what a light cone means. in. So I, you know, took a class on Spellas Show Relativity, I think maybe in my second year of university. And it was like, oh, all this stuff that was just words before, then words that sounded like kind of interesting concepts. No, it's formulas, right? Like, you know, either you're two points in space time, either you can get to them physically, possibly, by going less than the

speed of light, or you can't. And there's a formula that tells you whether you can or whether you don't. And there are lots of kind of interesting concepts. You know, maybe you've heard about things like the fact that people who've been on the International Space Station have aged more slowly than those of us who are on Earth, right? And there are formulas that tell you why that happens, right? So if you don't learn the formulas, that's just sort of a statement that you hear and you're like, oh, that's kind of cool. And if you want to understand why, you have to know the math. So keeping in mind that you might dispute the premise of this question, I'm curious. What is, if you had to think of a favorite formula or a number or a figure, what would that be, Ben?

Yeah, it's interesting because when I tried to do this, I kept coming up with, like, stories about why that's, like, not a good premise. So, you know, there was a very, very famous mathematician named Grotendik, who, you know, sort of revolutionized how mathematicians think about kind of, let's say, equations. You might not believe that that could be put in like a totally new perspective, but it can. And there was some very interesting work he was doing where you had to choose a prime number. And you were, you know, everything you were doing depended on this prime number. But it didn't really matter which one it was. You just had to choose one. And in a lecture, there was another famous mathematician from the older generation, Zariski,

who was getting very annoyed with him for being so abstract. And he was saying, oh, well, you keep writing P. What prime is P? And Grotendik was sort of like, well, you want me to name an actual prime number? And Zyrski was like, yes, what number is P? And he said, oh, fine, 57. Okay, I'm getting a little bit of, 57 is not a prime. It's divisible by three.

And if you look on Wikipedia, there's a Wikipedia entry for the Grotendik Prime, which is 57. So that's the number you're choosing? Sure, why not? Okay. Fair enough. Mina, do you have a favorite number or a favorite ratio or a favorite figure? So let me go back to numbers. I don't because for me, I'm a physicist.

So numbers, I have to say they are meaningless. What does it mean for something to be five? Five of what? Like $5, $5 million? That sounds better. So we always talk about ratios and or we always have to measure something for it to have a value or a number. So a great example, actually a cute example of this, and I think it's a bit timely because of something that will happen in August.

So the moon in size is roughly 400 times smaller than the size of the sun. At the same time, the distance of the Earth to the Moon is 400 times smaller than the distance of the Earth to the Sun. So far so good? Yeah. So what that means is that the angular size, the opening, the part of the sky that the moon covers is exactly the same as the size of the sun. So the fact that we have eclipses is based on this accident. There is no physical reason. There is no physics law that tells you this how to happen.

It's just that the size that the moon covers on the sky, relative to the size of the sun, they actually match to. to one part in a hundred, a percent. So that's why we get this beautiful eclipses, and there's gonna be one in August, right? So this tells me that the ratio of the distances, that there is this cancellation of the factors of 400, of how big the actual length size of the moon is relative to the distance.

So you're more interested in relationships between numbers rather than numbers in and of themselves. Exactly. That makes a lot of sense. And Matt, do you have a favorite number? or ratio or figure? Yeah, let's talk about the cosmological constant. Right into the deep end.

Let's go. So if we want to think about it in terms of ratios or comparisons to other scales, one number I could throw out is that the cosmological constant in terms of a density is about four hydrogen atoms per cubic meter. So it's a really tiny density. We could think about it in terms.

of a distance scale, and the distance scale associated with the cosmological constant is about 10 to the power 26 meters. I could think about it in terms of the size of galaxies, and it's like 10,000 galaxies in distance, so it's a really long distance scale. And I can think about it in terms of a fundamental unit that we think is associated with the quantum nature of the quantum nature of gravity. That's a pursuit of physicists for quite a long time. We haven't quite gotten there. And it is 122 orders of magnitude smaller than that scale. So that's a bunch of numbers. Now, what do they mean? And why is that interesting? So those numbers kind of connect to all of the important outstanding questions I would argue in fundamental physics today. Why is that number

so much smaller than this fundamental scale associated with quantum and gravity. In fact, you'd predict it to be about one on that unit scale. Why is it so much bigger than human scales? Why is the universe so big? And that number governs sort of the overall evolution of the universe. So the fact that that number is what it is tells us that we should just now in the history of the universe be able to start seeing it. So we're extraordinarily lucky in that sense to be alive right now. And it will tell us how the universe evolves in the future. So maybe take us back to the story of the cosmological constant. Yeah.

How did it, and I mean, I think it also is called the Einstein. Am I right about this? The Einstein cosmological constant? Is that true? Well, he called it his biggest blunder. Right. So Einstein came up with our current best theory of gravity, called the theory of general relativity.

And at the time, we had not seen out very far with our telescopes in the universe. And the concept was that the universe, kind of was always the same. It was static. That it didn't get bigger or smaller. It didn't get bigger or smaller. Right. And if I had a universe full of just stuff matter, matter attracts. And so you'd kind of expect things to be attracting. And there was a very natural solution to this problem, which was in the equation for general relativity to add a constant. Because that constant, what it does in the equations essentially is cause things to want to spread apart. And so you had a situation that Einstein presented, which was, okay, we have attraction, we have the cosmological concept, which wants to push

things apart, and they can perfectly balance and give you a static universe. And then, 1930 something, Edwin Hubble observed that, no, that is not quite right. In fact, things are expanding away from us. So the universe is expanding, and then he threw it out. He's like, Okay, that was a bad idea. Obviously, there's no static universe, and so let's give up on that. And then there was a long period of time where theoretical physicists argued about what it should be. The big development was quantum mechanics, because quantum mechanics actually told us that there is an energy associated with empty space, and that that energy is actually really significantly big.

And it looks just like this thing, the cosmological constant. And that's this 122 orders of magnitude difference in some fundamental scale we might predict it to be in what we observe it today, but the non-observation of it back then suggested it was really small. And so people spent a really long time arguing why it should be zero.

And there were lots of great theoretical ideas for why it is zero. And then in 1998, there was an observation of distant supernova, and we discovered that the universe was not just expanding, but it was expanding at an accelerated rate, and ah, that's exactly the kind of thing that a cosmological constant would do.

And so we were back to the cosmological constant, but now the puzzle was, okay, it's not as big as we think it should be. It's not zero, but it's this kind of random, really tiny number. So why on earth would it be this tiny number? And that kind of developed into what I would say, a very interesting crisis, and it had tons of implications. So when I came up as a PhD student, it was a little bit after the discovery that the universe was undergoing accelerated expansion. And theoretical physicists who had for a long time tried to argue it as zero or there's beautiful theoretical reasons why it should be just so, kind of lost their religion.

And we thought, man, well, what if it could just be anything? And this gave rise to something called the string theory landscape. And this was a period where we sort of as a community globally gave up a little bit on there being a fundamental explanatory reason behind this number. It just was anything goes. And it gave rise to this very uncomfortable situation where we had to explain the number we see because if we lived in a universe with a cosmological constant that was quite a bit bigger, the universe would have expanded so much that we never would get galaxies and planets and us. So there would be no life there to observe it. And this is something called the Anthropic Principle.

And oh, man, scientists were really uncomfortable with the Anthropic Principle. So this was another crisis. So, Meda, where did we land? Where do you continue that story? We haven't landed yet. We haven't landed. So what do you find interesting or puzzling about the cosmological constant? I think Matt nailed it right on the story.

ahead, in fact. See, our theory, there is a prediction. The moment we discover quantum mechanics and especially quantum field theory, which now is many people in this building study, and it's about, what, 80 years old now, almost. So the value of the

chronological constant is much larger than, that we calculate is much larger. In other words, if I were to put it in a size, the size of the universe would be that it predicted, at best, At best, it would be this big, like millimeter size.

At best, this is best case scenario, okay? So can you fit galaxies here? Yeah. The point is that our theory made a prediction. There was no reason why that prediction is wrong, and this is very important. We have no mathematics equations that can tell us why this number is so small, as Matt said. So this is a huge puzzle, continues to be a huge puzzle, and in fact, the multiverse idea, which is the multiverse,

and is the best one we have. So I like, in a way, the way Matt described how important this number is to the universe. I wonder just how what words you would use to describe just how fundamental and important the cosmological constant is to the universe as we know it now. So if we were to change the cosmological constant, even by a factor of, let's say, 10. which in the grand scheme of things, given that we're talking about 120 orders of magnitude,

a factor of 10 is small potatoes. So if we were to change it by, make it larger by that factor of 10, galaxies would never be able to form. They would have been ripped apart before they had any time to form. So we wouldn't have started. We wouldn't be around.

This is a force so powerful, it just rips things apart. It doesn't allow things to grow. So if the value of the cosmological constant was any bigger, we wouldn't be talking about it because we wouldn't even exist. So yes, that's a pretty important number. You're listening to a conversation about the numbers that shape the universe and ideas at the Perimeter Institute event with particle physicist Asamina Arvonitaki,

Matthew Johnson, who teaches physics and astronomy at York University, and University of Waterloo mathematician Ben Webster. This is Ideas. I'm Nala Ayyad. At Desjardin Insurance, we know that when you own a nail salon, everything needs to be perfect from tip to toe. That's why our agents go the extra mile to understand your business and provide tailored solutions for all its unique needs. You put your heart into your company,

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I know it. Beneva. Looks like people are starting to know Beneva pretty well. I knew that too. You're stronger with the right partner, Beneva. The cosmological constant is fundamental to our universe and our lives, even if most of us aren't aware of it. The next number we talked about in our panel is probably more familiar,

the speed of light, roughly 300,000 kilometers per second. I asked Ben to explain, why that number is so important in the life of the universe. Well, I mean, it sort of comes back to what I was saying about special relativity, right? Famously, it's the speed limit of the universe. If you try to go faster than it, you have to push harder and harder and harder to accelerate. You sort of try to break through it. And what happens is your mass keeps increasing.

You get heavier and heavier and heavier, which is why you have to push so hard to accelerate. And you just can't make it. But, I mean, it shows up in lots of other places, right? You know, people always call it the speed of light, but of course, that's also the speed that radio waves move. How was it theorized? Like, how do we even know that that is the speed limit of the universe? You know, very famously, Einstein had all of these kind of crazy thought experiments. So, you know, what had happened in the late 19th centuries, they started getting these strange observations of what was going on.

In the late 19th century, of course, the most excited. thing was electricity. We can use electricity to do things now. And Maxwell came up with these famous equations that describe how electricity and magnetism work. And if you sort of look at these equations, they don't work properly with the old version of physics that people thought was how the universe worked at the time. You know, this is sort of a common theme. You know, everybody in this building, And what we would love to do is kind of take gravity and quantum mechanics and make them work together. Well, before we knew about quantum mechanics and before we knew that gravity was as complicated as it is, people knew about the old Newtonian physics and they knew about electromagnetism.

And they needed to get them to work together. And Einstein was the one who sort of made that synthesis. I mean, the speed of light is fascinating because it imposes a fundamental definition of causality. There are things that can have cause and effect on each other, and there are things that cannot because they are simply too far away for light coming back to Amino's observation. As a child, that it takes eight minutes from the light from the sun to hit her eyes. The sun could have exploded. It took eight minutes for us to know.

So it imposes this fundamental definition of causality. It also illustrates that empty space has properties that we didn't think it had before. It has symmetries we didn't think it had. So if I have perfectly featureless empty space, you know, I could walk around. I could like go different places. It looks the same. And it forces us to think about space and time as this unified object. And that really laid the bedrock for trying to understand the force of gravity in a fundamentally geometric.

way in Einstein's theory of general relativity. But the other interesting thing is that the speed of light is not constant. If you've ever seen a rainbow, you know that that's true. Because the speed of light is the fundamental constant only in totally empty space. But when light goes through a raindrop, light of different colors travels at different speeds. And that's why it spreads out and we get to see a beautiful rainbow. Mina, you are nodding. Can you talk about the role of the speed limit in what we see on a nightly basis in the sky. Can I talk about the role? Can I talk about something else?

I did suggest before we started. That you don't need me tonight. Please. No, because there is something correlated with the importance of the speed of light and what Ben said before about how we got there. And it's not just Maxwell's equations. There was an actual experiment.

So Maxwell's equation told us that light behaves in many similar ways like waves on the surface of a lake. But at the same time, because people had experience with waves on the surface of a lake, they thought, okay, we need a lake. So they said we need something called the ether for this wave to propagating. So devised an experiment where they sent light in different directions that they tried to measure how the earth, should move through the ether. And they were expecting to see a signal that would tell them how fast is the earth moving with respect to the ether.

But that experiment found nothing. Meaning? Meaning that the only way to explain that experiment is with the Lorenz transformations that Einstein had found and they were coming from Maxwell's equations. So I like the speed of light and the properties of it

and special activity, because it also tells you something else, that in physics there are experiments that give you zero answers. Nothing interesting is happening, but it may have huge implications about our understanding of how the world works. Right. So, of course, my question with every one of these is, what would our world look like if the speed of light in a vacuum were different? If it was a different number, we wouldn't care. Yes, it wouldn't affect it.

It is the fact that there is such a thing. Exactly. That is the interesting part. Because it's always, it's a relative thing. So in my calculations, in fact, and I think in both of these guys' calculations, the set of speed of life to one. It's not three times 10 to the 8 meters per second. It's one.

So because we always compare it to the speed of a car. So the ratio of the two doesn't have any units. So going back to ratio. So numbers by themselves don't tell you much. It's what do you compare it to? That makes it interesting. Okay. So Mina, staying with you.

Speaking of comparisons, I'd like to talk about something that might be more familiar to our audience, which is the Higgs boson. Okay. Most people in this room probably recall the excitement when the existence of the Higgs boson was confirmed at CERN, which is the European Organization for Nuclear Research in 2012. But I'd like to venture to say that we may not remember why it was important that its existence was proven. Can you remind us why that discovery was so important? Oh, well, it's a bit of a long story. Because it completes everything.

So in some ways, the Higgs completes everything that we'll be doing in the 20th century. Quantum mechanics and our understanding of what the forces that tells us how matter behaves is, is what got completed with the discovery of the Higgs. So it gave us two numbers, but it was of a theory that had another 17 numbers in it. The theory of the standard model of particle. interactions. And I'm saying it's a journey because it started in the 19th century when J.J. Thompson discovered the electron. And then, rather than discover the proton. And then Fermi did radioactivity experiments and discover the weak force that is responsible for radioactivity and discover neutrinos.

And then as we were going, we discovered new particles. And then we discovered what is the structure of the strong force that keeps nuclei together. And we started piecing things together. And we ended up with, we call it simple because it has three forces. It has these particles called quarks that make up the proton and the neutron in the atom. It has these particles called leptons that are of three types. There is the electron and two brothers of it. And there is the Higgs. Okay.

And the way these guys interact with each other. So the Higgs was the last piece we needed. which was already needed from the structure of the theory, because the weak force to be unified with electromagnetism, we needed the higgs. So really it explains how matter forms. How matter interacts and behaves. Exactly.

We needed it to be there, and it was there. It's great. It explains why it was called God's particle. Yeah, I hate that word. Why do you hate it? Why do you hate God's particle? Why do we? I mean, I think it has to do with the fact that humans always look for meaning and we look for patterns and symmetries.

And I think sometimes it's just, it's mathematics. Mathematics, I don't think mathematics has a god. So what I understood was that the Higgs boson is roughly 130 times greater the mass of it than a proton. It's 170 something times. Why is that number significant? What does that mean? Oh, good. That's an excellent point. It has to do with a strong force. So we know of light, right? So light is an interaction. Light is the mediator of the electromagnetic force. Light interacts with charged particles like the electron. Okay. Now there is the strong force,

who's the analog for that force, the analog of the photon is the gluon. And the point is that because that force is so strong, it doesn't allow the particles that carry the charge of the strong force to roam around free. And the fact that they are glued together carries some energy. So that energy from the gluing together of the gluon and the quarks is the mass of the proton. And the proton actually contributes to the weight. So the way we interact with gravity is very little to do with electrons. It has mostly to do with protons and neutrons, a nuclei.

So that strong force tells us how strong our interaction it is with gravity of a human on the surface of the earth. Right. And yet, as certain as that sounds and as a big deal as it was back in 2012, there's still some uncertainty and controversy over this number as well, Matt. Is that true? So I think that's a case where actually the number is pretty interesting because the beautiful structure of the standard model of particle physics breaks. unless there is some new physics that comes in at a scale associated with that special mass of the Higgs boson, and trying to explain why that particular number connects with an idea that, you know, another beautiful idea that physicists had been working on for quite a long time, supersymmetry.

Supersymmetry is a theory that suggests that every kind of particle has a partner particle. It could potentially explain things like dark matter or the huge differences in the strengths of different forces, but it remains unproven. This is a way to take the standard model kind of to the next level and try and start to explain some of its properties at a more fundamental level. And so there were lots of predictions for what we might find at the large Hadron Collider that could be new particles associated with this thing called supersymmetry, and we have not found them yet. And every year we continue to not find them actually makes the number associated with the mass of the Higgs boson a little bit less explicable in terms of these beautiful ideas of theoretical physicists. It's a bit like the situation with the cosmological constant, but not as severe, that we're being driven to a place where maybe we need to think about the explanation as being this everything goes type explanation.

And it's a weird place to be in. Moving on to Plank's constant, Ben, there are a few numbers that are associated with German physicist Max Plan. He's actually one of the fathers of quantum theory and modern physics, and one of those numbers is Planck's constant. If any of you dabbled in physics, you would remember that. It gave me nightmares when I took physics. But what is it and why is it so important? All right. I fear I'm the wrong person.

I have to confess, I snuck in here. I'm not really a physicist. But, you know, quantum mechanics tells us that, for example, the energies that electrons can have inside atoms can't. can't be just sort of random, right? They're not in this anything-goes position where they could be anything. They have to hit certain levels. You can only have a certain number of atoms with some energy,

and once you have enough of those, you have to have one with a higher energy. Maybe you guys, when you took chemistry, heard about shells. That's what's really going on in shells. You can only have so many electrons with this energy. And Planx Constant describes amongst many other things, those jumps in energy that particles have to take. So it's sort of, it's describing some sort of resolution to the universe. You know, it's like like the pixels on your screen where you kind of can't go

below that. You can't, there's some distance that you can't really sensibly talk about distances shorter than that. So it's a decimal followed by 33 zeros. Do I have that right? If it were bigger, we would notice. So Plans Constant is important, as Ben said, because it has has to do with quantum mechanics and the wave nature of everything. So when you measure a wave, there is a wavelength, right? The distance between two maxima of the wave. And what plaques cost and tell you that for things like light, there is an energy associated with it.

So people can measure the wavelength of light, and they can measure the energy of light, comparing always to some units. again, I will say we set that H-bar to 1 when we do calculations because we always care about ratio. And C. Yeah, exactly. H-bar and C are 1. H-bar being the reduced Planck's constant.

That's Planck's constant divided by two times pi and C being the speed of light. But the point is that, yes, so you relate a length scale, a frequency, in other words, to an energy scale. okay so so this is how it was measured so it's the basics of of the fact that one of the important properties is the dynamics of waves that everything is a wave to some level i'm just curious if there are any numbers that you deal with that are even smaller than planks constant oh uh i mean in this the cosmological constant is the one we go back to everything goes back to that everything is that i mean people can imagine even with the multiverse actually i like them so the landscape, as people call it, when they found all these solutions where the value of the cosmological

constant can change by jumping from world to world. And world is a very generous statement, because it may be very empty. The amount of how many worlds does the mortivus have. So people came up with numbers like 10 to the 500. I don't even know what that means. What does that mean? I cannot understand it. Maybe Ben can. 10 to the 500. We have... I have trouble even fathoming what that number is. I want to cover one more constant before we move on to other things. And Mina, staying with you, there's the fine structure constant. I have to admit, I have never heard of this.

The fine structure constant. Yeah. It has to do with the charge of the electron. Okay. It's one over 137? Yes, exactly. And it tells you where strong electromagnetism is. and it's an important number because in the atom, for example,

the electron and the proton are bounded by electromagneticism. And it tells you how big the atom is, or how much energy it takes to excite atoms. So it determines there are properties. Like many, I mean, because we are held together by electromagnetic forces, mainly the way our bodies work are determined by the values of the fine structure constant, the electron mass and the proton mass. And just to give you an idea, let's compare it to gravity.

That's a very interesting comparison. So if you were to calculate the force of gravity in the atom, you would find that it is 40 orders of magnitude smaller than electromagnetism. I don't even know how to describe to you what 40 orders of magnitude is. What I can show you is the following. So every time you pick up something, what happens? So you're using the electromagnetic forces in your body to overcome the pull of the entire planet.

Think about this. You cannot eject yourself in space. We're not supermen or superwomen, okay, or superpersons. But we can beat the force of the entire Earth, but just lifting our hands. And this is related to the Higgs mass and the hierarchy problem. This is another way to say, if the natural value for all mass scales is, in fact, is the plank mass. So the fact that the Higgs mass is so much smaller than the Planck scale tells you that if it were at the plank mass,

the force of gravity would be at the same strength as everything else. So this is another way to say the puzzle of this hierarchy problem and why it is a puzzle and why people try to find theories like supersymmetry to explain it. But it's definitely true that the big ratios, that's what makes the universe interesting. Exactly. It would be really boring if everything was almost the same. I know. This is the thing.

So it's like it's these imperfections that make our universe interesting. It's like when you have, you know, that people buy diamonds grown in a lab, but they are worthless, right? The valuable diamonds are the one from nature because they have the imperfections and they have color in them. All the impurities that they have is what makes them beautiful. And in fact, it's all these ratios of numbers that make the universe so unique. I mean, three of the numbers that we've talked about are constants. And I'm wondering whether that suggests some kind of coherence or a governing principle. Just it would seem reassuring that there are specific numbers underlying the nature of the universe.

Does it make it seem more orderly or rational? I'll just punt it back to the relatives. I mean, there are physical phenomenon. and it's a wonderful thing that we can associate scales with different physical phenomenon, because that is really something that underlies the explanatory power of physics. If there was no such set of rulers, if there was no such large ratios or differences between scales, then this thing we call physics would have far less explanatory power. And yeah, the universe would be less orderly in that sense

because a lot of the concepts and ideas that underlie why applying mathematics to describing physical phenomenon is so unreasonably effective does come down to that. Do numbers make the universe more understandable, more coherent and profound? It does, and it's both about the beauty of mathematics. As humans, as I said, we always try to find patterns and purpose and symmetries, but many of the interesting fact comes out of this funny hierarchies, right, of these funny ratios of small numbers that appear.

For example, the Higgs particle gives mass to the quarks that made up the proton, but it's a tiny contribution, but at the same time gives the mass to the electron. So if I start changing, the Higgs mass by a little bit, then if you calculate what properties of nuclear properties you would get, you would get a universe that's very boring. It will either be filled with hydrogen, basically a proton and an electron going around it, or some weird doubly charged partner of the proton. No carbon, no beautiful diamonds, no wood, no humans.

So it is those hierarchies, those things that deviate from the natural pattern of what you expect, that make the universe so rich. Perhaps on the less reassuring side, and I guess, again, this is from the layperson's point of view, there are some numbers that just don't seem to make a lot of sense on the face of it, and like something called imaginary numbers. Make it even more complicated, but I'll start with you, Matt. What is an imaginary number?

It is a solution to equations that, do not have a solution if I restrict myself to normal numbers like one, two, three, four, five. Right. Math quiz. X squared plus one equals zero. What is X? X squared plus one. Yeah.

So I need X squared to be a negative number. Any guesses? That to square. Yeah. I. Excellent answer. You get an A plus. Yeah.

So that is I. That is an imaginary number. I is for imaginary and I squared is equal to minus one, which means that I is equal to the square root of minus one. Even though if you try to find out the square root of minus one on a calculator, you'll get an error message, or it'll say undefined. But here we are. I is for imaginary. And if I have a mixture of real numbers, the ones we're used to, and imaginary numbers are call that a complex number. And these are just wonderful tools that underlie the mathematics

allowing us to describe waves, for example. A lot of the mathematics underlying physics, in particular quantum mechanics, relies very heavily on this concept of imaginary numbers. It's a really wonderful mathematical tool that allow us to describe many phenomena in nature. Ben, as a mathematician here, you make of imaginary numbers? Well, I mean, you know, so somewhere around, I don't know, a third or fourth year of university, you start assuming that students are maybe more comfortable with complex numbers than real numbers. That's maybe a little early, but, you know, one of the remarkable things is a lot of mathematical things actually become much easier if you think about complex numbers rather than real numbers. So, for example, right, Matt was sort of saying, oh, well,

how do we get complex numbers? Well, we look at x squared plus. 1. Well, x squared minus 1, there are two solutions to that 1 and minus 1. X squared plus 1, there's no solutions. Well, that's really upsetting. But if we go to the complex numbers, there are two solutions, because our lovely audience supplied one solution to x squared plus one, but there's another one, which is minus i. So once you incorporate complex numbers, any polynomial equation like that where you have x, right? What if you had x cubed plus 1? Well, there are three of them, and they're perfectly good complex numbers. And there are lots of other sort of more complicated concepts that follow on from that, where, you know, if you try to describe what

happens, if you use rational numbers, you, you, something very complicated happens, right? It's a very complicated question when an equation like that has a rational solution. Rational means a fraction of two whole numbers. Or, you know, real numbers, the sort of usual numbers we use that are decimals, whereas with complex numbers, the answer is much simpler. So for I think almost all mathematicians, complex numbers are sort of the kind of default kind of numbers to use. So we have imaginary numbers that are essential to quantum physics, as you said. There are numbers that are foundational to the world like pie and the golden ratio that are irrational numbers, numbers that have decimals that go on forever.

What does it say about the nature of our universe, of research, of real? reality, that these numbers are so foundational. These are rational. I mean, it just says that geometry is a little more complicated than that, right? So, you know, the Greeks famously wanted everything to be a rational number. And there's even a story about a Greek, you know, jumping off a ship because he was so upset that the square root of two was an irrational number. But, yeah, geometry is more complicated. One of these things that's sort of hard to think about now in the modern world, or at least for those of us who, you know, took university classes on this stuff, for a long time, people thought that, like, numbers and geometry were separate things. Like the Greeks,

there's sort of Euclid who, you know, did many, many books on geometry, and all his geometry was purely in terms of, oh, when these two lines intersect and yada, yada, and then there there were other people doing, I think maybe even the same people, but they were doing it separately, doing, you know, solutions to algebraic equations. And it was like a huge brain to be like, oh, we can describe the points in a plane by pairs of numbers and we can understand geometric shapes by using equations. That was a huge advance in the 1600s. But once you start doing that, you immediately are like, oh yeah, irrational numbers. They're here to stay. We're not going to get rid of them. I'm just in that story, the person that you said jumped into the water because of not

getting their head around an irrational number. How have mathematicians dealt with, and scientists with these irrational numbers and imagining numbers through the ages? I think there's a saying, you know, you don't really come to understand things. You just get used to them. You know, you have the tools to work with these things. You know, we get started, people started early now, right? And so it doesn't seem so upsetting that you take a square root and it's an irrational number. So let me ask this. We've been talking this evening about why these numbers are so important. But many of us are really kind of blissfully unaware of much of what we've talked about tonight. And I wanted to ask each of you just what are we missing out on by not

knowing about these numbers that basically rule our universe? Matt. Yeah, I mean, it is a beautiful thing that we can use mathematics to understand and predict the world around us. The fact that it is so unreasonably effective, it just gives you a new perspective on essentially everything, and it gives you a framework within which you can try and understand the world around you. And a lot of those concepts, I don't know, I mean, you might apply them to yourself, your view of the universe, life, everything, it's really a perspective, right? So you're missing out on that. Yeah, Mina.

Yeah, that was very nice, Matt, actually. Yeah, to that I would like to add, again, going back to symmetries and the fact that a lot of the equations, in principle, you can write equations and never talk about numbers. You can say my number that I'm missing is X, okay, and keep it to that. A lot of these funny numbers that we talked about, the hierarchies, why is the Higgs must the way it is, why is the universe so large, it didn't come from the structure of mathematics by itself,

if it came out from observation and observing the rich structure that we see around us, doing experiment. So the language of nature is mathematics, but the way we ask questions to nature is experiment. And that's something to remember. For us, for physics, those two come together to describe the world around us. And the other thing is, yes, you have these weird numbers 19 in the standard model,

you have the cosmological constant, how much dark matter there is, which we didn't talk about. All the rich structure that you see comes out of these things. But it's a small set of numbers, which I find remarkable that as a human race sitting in a little rock, going in a vast, empty space, we did pretty well. Yes. Ben. I mean, I guess the sort of thing that came to mind for me is, I think there's, There's a lot to be gained by just not being scared of numbers.

Numbers will not hurt you. And they follow very simple rules that, trust me, you can understand. I think, you know, there is some cultural block. Believe me, having, you know, gone around for 30 years of my life telling people, oh, I'm a mathematician. And you're quite often here like, oh, I hated math in high school. And, you know, I'm not saying you have to love math. But I think you should do your best to be comfortable with it and sort of not be scared by it.

So we've kind of done a very brief survey of those numbers that underpin or shape our universe. And I wonder just kind of as a final round to each of you, if you wanted to give the average layperson an entry point of the numbers that we discussed, what you think is the most important or fundamental or most interesting number that you would ask us to explore further. Of all the numbers we've talked about, what would that be, Ben? I mean, I'm still going to stand up for the speed of light. You know, I mean, Planck's constant is great, of course, but the speed of light, it's sort of, I mean, it's so important to what's going on around us, right? Like all of this technology, you know, like the cell phone in your pocket, it wouldn't work. If the speed of light was not a constant and it was changing.

And I think, you know, the special relativity, it's, again, it can sound a little scary, but some really amazing stuff happens. You know, this discussion about causality, this sort of. idea that there are sort of events because of where they are in space and time. They couldn't affect each other. I think that's all stuff that people can understand and it will really expand your mind. For me, because I know what Matt is going to say? No, I'm not going to say. On purpose, I'm not going to say it. That's boring now. I'm going to say that people should remember that gravity is a very weak force. It doesn't feel like that when you get off the couch some days or you go up.

upstairs. Every day. But it is a very weak force relative to everything else. And it's a huge puzzle. It shapes us. And it's something that's worth understanding. Thank you. Matt, should we even ask?

No, yeah. I'm going to go, I'm going to go small. We've gone big. I'm going to go small. I'm going to say you should understand H-bar. Because quantum, just like special relativity, underlies pretty much everything we know about the modern world. And so you don't have to go as fancy as quantum computers. You could go to the color of a fluorescent light. It's something that you should

know about. And it's very interesting. It's all really just wondrous. And I'm just so privileged to have had the chance to speak with you about it. Thank you so much. Matt, Mina, and Ben. Thank you. Thanks. Our panelists for ideas at the Perimeter Institute were Matthew Johnson, Assamina Arvanitaki, and Ben Webster. And if all this whets your appetite for more about numbers, ideas will soon bring you several episodes devoted to numbers, including 13, 12, 60, 27, and 4. Special thanks to Hillary Potts, Mark Healy, and everyone at the Perimeter Institute for Theoretical Physics, who helped with the event. This episode was produced by Chris Wadskow.

Our website is cbc.cai.cai. and you can find us on the CBC News app or wherever you get your podcasts. Technical production, Danielle Duval. Our web producer is Lisa Ayuso, senior producer Nicola Luxchich. Greg Kelly is the executive producer of ideas, and I'm Nala Ayyad.

For more CBC podcasts, go to cBC.ca slash podcasts.