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Wait, you're listening? Okay. Alright. Okay. Alright. You're listening to Radiolab. Radiolab. From WNYC. WNYC.

Hello. I have to find your window. Hi. Are you tired? No. No, I'm all right. How are you? Oh, good. I'm good. I'm excited for this random little thing. Hey, I'm Latif Nasser. I'm Lulu Miller. This is Radiolab. Okay, well, a mystery guest is going to appear momentarily. Mm-hmm.

Oh, we see you. Hey. I can hear you both. Perfect. It worked. Hi. Well, okay, so Kareem Latif. Hi. It's very nice to meet you. My pleasure. Where are you? I'm in Alexandria, outside of D.C. Okay, so I guess the best way to set you up is that Kareem is here because he has broken one of the most forbidden rules ever.

of the universe are you a cannibal is that what i'm about to learn i haven't i haven't broken anything it's the question of you haven't yet no no it's well that's the question it's the question of whether to break it and how to break it what are the consequences of breaking it could you break it should i should i try to should whoa you seem like you're on a precipice brother okay so what is this rule that the rule yeah what's the rule in mathematics

You're allowed to do everything for the most part. You can multiply, you can divide. But as you may recall from school, there's one thing in mathematics you're not allowed to do. Do you remember? Just dividing by zero? Dividing by zero. We have this entire structure of mathematics that is incredibly useful. It's incredibly powerful. But it all kind of hinges upon our agreeing to not go through this one door that has on it

There is a sign on this door that says, division by zero, don't open this door, because what's on the other side of this door is all sorts of craziness. An infinite loop. To see a world in a grain of sand. Where everything is the same. Don't divide by zero. When you make the two into one, and when you make the inner like the outer, then you will enter the kingdom.

It's like this sign you hang on the elevator that's not working, you know? Like, it's like, out of order. Like, please, do not go through here. Well, here's the thing, though. It isn't that the elevator is out of order. It's that the elevator goes to a dimension that is so problematic to our way of thinking in this dimension that as long as you agree to not go into that kind of elevator shaft wormhole, we're good. You can have your airplanes. You can have your computers.

So today we've got a story about a paper that Kareem Ani wrote almost 20 years ago about dividing by zero. I happened across it about 10 years ago, and it really tickled something in me. Over the years, I would think about it. I'd wonder whatever happened to this guy who wanted to divide by zero. I'd wonder if there were consequences for math. I'd wonder if there were real consequences for reality, for my reality, for his reality. I didn't know.

But I thought that as we ourselves are rounding the clock of the calendar year, passing through zero to start anew, I thought now might be a nice time to call him up and try to understand. So my friends, leave your calculators at the door because we are going to try to enter a new kind of math.

Here we go. Well, I think what a mathematician would say is... Are you a mathematician? Yeah, sorry. We should say who you are, by the way. Who are you? What do you do? Like, what do you do? So...

I am the founder of Citizen Math. And what we do is we write lessons for middle and high school classrooms around real issues. So students are using mathematics to discuss, should the federal government increase the minimum wage? Or why do airlines oversell their flights? Using mathematics as a tool for discussing and analyzing the world around us. Love it. But coming back to this idea of division by zero. Yeah. Okay. So...

Maybe we just start with why can't you divide by zero? Like, why is that such a hard and fast rule? Okay, well, so one reason is because it violates a mathematical principle that every operation needs to be undoable. Anything you do, you need to be able to undo. Okay. Let's say you start with 10.

And you divide by five. And so now you're at two. Now you need to be able to get back to 10, though. And so you can go from two and multiply it by five to get back to 10. 10 divided by five gets you to two. Two times five gets you back to 10. But if you now try that with zero. OK, 10 divided by zero is some number.

Well, to go backwards, now that some number times zero, how can that get you back to 10? So it violates... Because zero times anything is zero. It kind of sucked back into the black hole of zero-ness. Right. So that's the, it violates this custom, let's say, or law. Because there'd be no thing you can multiply by zero to get to the number 10. Exactly. Once you do... Right. Right.

So mathematicians created this rule, this kind of barricaded door that basically says do not try to divide by zero because the answer is undefined. There is no answer. You can't do it. However, there have been people who have...

gone through that door. Hi there. Hi, Steve. Lulu, I don't think we've ever done this together. I know, isn't that wild? I have never actually gotten to meet you. This is so nice. Oh, it's nice. Yeah, hi. Okay, this is Steve. Steve Strogatz, and I'm a mathematician and a math professor at Cornell University. And Steve,

And Steve has not walked through the door of dividing by zero, but he says that these sorts of rules, these sorts of barricades, in math, it's always been important to break them. Exactly. That's actually some of the most fruitful parts of math.

that when you try to do something that seems impossible, it often leads to the creation of whole new universes. So, for example, Steve was like, okay, let's think about square roots. So if you take a number, like the number three. Okay, so three times three, that'd be, in the jargon, three squared. Three times three, three squared is nine. So the undoing of that is that the square root of nine is three. But now, let's say you wanted to take the square root of...

of negative 9. You know, your first thought would be negative 3 maybe is the square root of negative 9, but it doesn't work. If you do negative 3 times negative 3...

You get positive 9, not negative 9. Because in math, and we're not going to go into why, if you multiply two negative numbers, you get a, boom, positive number. So you can't do it. You can't take the square root of negative 9. There is no number that will work. So a long, long time ago, mathematicians were like, OK, there is a rule, no square roots of negative numbers. But then, in like the late 1500s, a bunch of new rambunctious upkeepers

coming disobedient mathematician said, well, what if we just broke that rule? And

To make the math work, we just invented a whole universe of new numbers. That is so bizarre that mathematicians called these imaginary numbers. Numbers that are not technically negative and they're not technically positive. They were sometimes called fictitious numbers. But they allow the math to work in such a way that you can start doing square roots of negative numbers. Because you just wish it to be so. Just invent some new numbers? Yeah.

It's invention. Exactly. It's invention in the artistic sense. You can invent something.

that didn't previously exist. But was anyone like, no, we have a rule. You can't take a square root of a negative number. Yes, absolutely. It's like anything else that human beings do. They're always reactionaries. There are always people who say, you're muddying the waters. You're messing up the pristine and beautiful world of math with your ugly ideas because these ideas have a lot at stake intellectually and there's always resistance.

But that's where the breakthroughs happen. You take something that earlier generations say was impossible and you say, what if? And then you try it and you figure out a way to do it. And that's where the progress happens. But like what is like an imaginary number give us? That gives us the modern world. Like concrete stuff. I'm going to tell you. OK. I mean, imaginary numbers.

Okay, so if we fast forward to the 20th century, this is not why imaginary numbers are invented. They're invented much earlier than that. But in the 20th century, when the theory of the atom starts to be worked out, we learn how to describe what's going on with hydrogen atoms and helium and how light works. In other words, we invent – we, the collective of scientists in the 1920s invent quantum mechanics.

So, it's our most accurate physical theory there is. It gives us today everything. It gives us what we're doing right now, talking over the internet. It gives us lasers. It gives us transistors, chips, everything in the modern world.

has an underpinning in quantum theory and the electronic revolution that it made possible. The math of quantum theory is built on imaginary numbers. You can't do quantum mechanics without comfort with imaginary numbers. And it's crazy in that what was thought to be imaginary a few decades or really more like a few centuries later turns out to be the mathematics of reality.

And to Steve, this is sort of the beauty and the artistry of math. I mean that in math, we have creative freedom. We can do anything we want as long as it's logical. Mathematics in many ways is a chronicle of humans' understanding of reality and logic. Kind of a chronicle of

Like, it began with early humans coming up with the idea of what we call natural numbers, one, two, three, and so on. Then the Sumerians in Mesopotamia and the Mayans each independently came up with the idea of zero, which blows its way around the globe. And then a few thousand years later, the third century in China, negative numbers show up, and they too spread across the world, and math gets more and more complicated, and we

So we start to come up with rules and then we try to break those rules. And in the wake of that breakage, we often invent new numbers like imaginary numbers or rational numbers or real numbers or complex numbers. We come up with all these different tools that we've invented by pushing at the rules, pushing at the boundaries of math that then help us to better understand the world around us. But this is where division by zero is different.

categorically different because it's so beyond the like it it leads to these results that would undermine all of mathematics it would break math as we know it and this is where for me this becomes actually quite um existential when we come back we are stepping through the door

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This week on the New Yorker Radio Hour, Israel has occupied the West Bank since 1967. Will that occupation ever end? The longevity of this system depends on huge numbers of Israelis who really do not feel in their day-to-day lives the occupation that's just a dozen kilometers away. That's our subject on the New Yorker Radio Hour from WNYC Studios. Listen wherever you get your podcasts.

Lulu. Latif. Radiolab. We are back with Kareem and... Dividing by zero. All right, friends. It is time now to break the rule. We are going to divide by zero. We are going to grab our calculators and watch what happens next.

When we do. If divide by zero, does it catch fire? Because there are actually all these videos on YouTube. Try to divide by zero is awesome and dangerous. We're sweet. Where we show the machine dividing by zero. Nerdy men. We're going to watch as this calculator tries to divide by zero. We'll take these old mechanical calculators. I will just input a... Punch in some number. Dividend of one, two, three. Divided by zero, we hit equals. So here we go.

And what happens is the numbers on these calculators just keep rolling over and over and over. And what happens is that it gets into an infinite loop. And over. And it will never stop. And I guess it heats up, so eventually it would catch fire. Like the mechanisms driving that calculator just get stuck. In an infinite loop, loop, loop, loop.

And it is right here for Kareem. Where this becomes actually quite existential.

Because, he explains, to understand what's driving that looping, you have to think about the math going on. He said, you know, take, for example, the number 10. If you take 10 and divide it by 10, you get 1. 10 divided by 5 is 2. 10 divided by half is 20. The smaller the number in the bottom, the number that you're dividing by, the larger the result. And so by that reasoning... If you divide by zero, the smallest nothingness number we can conceive of...

Then your answer would be infinity. Why isn't it infinity? Infinity feels like a great answer. Because infinity in mathematics isn't actually a number. It's a direction. It's a direction that we can move towards, but it isn't a destination that we can get to. And the reason is because

If you allow for infinity, then you get really weird results. For instance, infinity plus zero is... Infinity. Infinity plus one is... Infinity. Infinity plus two is infinity. Infinity plus three is infinity. And what that would suggest is zero is equal to one, is equal to two, is equal to three, is equal to four. Ah!

And that would break math as we know it. Again, Steve Strogatz. Because then, as your friend says, all numbers would become the same number. Which, you know, for math... The whole vast interconnected web of it... Would be a problem. The world of fluid dynamics, calculus... Geometry, physics, all this stuff depends on numbers being individual, discrete things. But...

If you allow for division by zero, that all goes away. And you get into all of these strange consequences like one equaling zero, equaling two, equaling infinity, equaling four. And so in order to protect math and all the things we use it for, like making computers and planes and... And all modern technology. Mathematicians said that when you try to divide by zero, the answer... Is undefined. Undefined.

It's undefined. There's no sensible definition. And that's why they put up that barricaded door. Because what's beyond the door is, it just seems impossible. It seems very difficult to get our heads around. Because effectively what we're saying is, everything is one thing.

Now, Karim says... When I first started thinking about this 10 years ago or however long that was, it was something fun to think about. It was something fun to write a grad school paper about. But he says more recently, he's had this feeling that's grown and grown... Of this isn't complete. There's something else here. Now, maybe...

This is something you have felt at some point in your life. Maybe you're even feeling it right now that the daily stuff of it isn't all there is, that there's something else out there.

And for Karim, he's like, look. I'm not religious. He's devoted basically his whole life to math. And mathematics is kind of a representative of one way of thinking about not just the world, but one way of thinking about reality. And so to Karim, it perplexes him. It sort of tugs at him to see math itself saying when you actually follow out the operation of dividing by zero. You end up in a completely different world.

Where one equals two equals three equals infinity. That all of these numbers are one and the same, that everything is effectively one thing. Everything is equal to everything else. And this world of division, I don't mean political division, but that too, this world of duality, of differences, of things being discrete from one another, that all goes away.

And Karim can't help but to notice that's the sort of stuff you hear from Jesus said to them. People like Jesus. When you make the two into one. And Buddha. Or people who follow Taoism. Or people who have done intense meditation or intense hallucinogenics. Oftentimes those people come back and the thing that they say is, I felt like I was one with everything.

So you see in these like religious texts, you see literally like the collapse of the integer system. I'm seeing math being a way of thinking about reality and thinking about the nature of nature. And to Karim, because the math itself leads to this undefined place where numbers work really differently. Where all of these numbers are one and the same. To him? That suggests that there is something else happening.

And I'm not saying that's God or whatever it is. It's just there's something else here. And I can't, by definition, I cannot on this side of the door articulate what that reality would look like.

But... I'm middle-aged. Now that Kareem is rolling into his mid-40s... I don't have children, a spouse. He finds himself unable to stop wondering about what that something else could really look like. I look at my life and I think, well, after 44 years, you're still not content with this. That must be a sign that either you're doomed to be discontented or...

That's a sign that like you're not going to find it here. You need to go through the door because honestly, what's your alternative? But how do you actually do it? Like, how do you I don't get how you how do you actually divide by zero and go through the door? I don't know. Whatever that means. I have no idea what it would mean practically to divide by zero.

But he says he does know it would have to start with some pretty major changes. Like, he would definitely need to quit his job. He would need to leave behind his house in the D.C. burbs. Look, I'm Arab. I feel this weird, like, attraction to the desert. Like, I would probably go take camping gear and go find a desert and sit in a desert. And then...

Well, he's not entirely sure. All he knows is that he would need to connect with that mathy part of his brain he has been using for decades, thinking about numbers as these discrete and different things, and then try to turn it off. That is the thing that I will need to put down. And then maybe if he listened really close, he could begin to hear things.

or feel the something else behind all of this. Now, okay, so what's my personal reaction to that? By the way, there's a guy named Steve Strogatz. Yeah, sure. We talked to him about you. We were behind your back and we talked to him about you, yep. And we told him about how you were thinking about

Trying to access a world where there are no differences in numbers? I would say you can do that. If you want to do that, you can do it. You can make a universe in your mind where all numbers are the same number. Let me describe that universe. Yeah. There's a universe I'm going to call Zero World. Welcome to Zero World. Okay. Where, in fact, there's only one number. Zero.

And here are the properties of the mathematical zero world. Zero plus zero. Equals zero. And that's true no matter how many times you add zero. You can't get any new numbers in this world because there are no additional numbers. There's only zero. Zero plus zero plus zero plus zero. As far as the eye can see. Yes. And that's it.

That's your universe. It's the universe of zero. All numbers are the same because they're all zero.

Are you happy now? He keeps going. He says, to me. That's like such a solipsistic, pathetic little universe. That is the ultimate in navel gazing that does nothing for anybody. But it's self-consistent. You can live in that universe if you want to pretend there's nothing but zero. Oh, see. OK. And let me respond to that then. Yeah. Because it's a beautiful. Steven Strogatz is a really smart dude.

But the question, that first question of, are you happy now? I would say, well, Stephen, if you live in one world or where every number is distinct from one another, like if you're happy in that world, great. I'm not because I have this question in the back of my mind. This question of...

What is actually on the other side of that door? To me, it is zero world and I just find it incredibly stultifying. It's a very impoverished little self-contained logical place. Stultifying but mathematically sound? I think it is. It's defensible. You can have it. There's nothing wrong with it. It's just as minimal as a thing can be. It has no potential for anything beyond itself, but it's just a fine little solipsist looking at its own belly button.

But the inside your belly button is everyone and everything. It's like, I don't know. I'm just trying to defend him because he's not here. I don't know if I want to go there. You can try. I'm not buying it. No, but it's like division. He kept saying like division goes away. Political division. Spiritual division. Duality goes away. Let me try to make the case for it.

Right, the case for it, I guess, is this is a noble impulse to see the unity and it's also a productive impulse, scientifically looking for unified theories.

has historically been the way to great progress in physics. So to recognize that electricity and magnetism are actually two sides of the same coin that we now call electromagnetism, that was a great invention, a great breakthrough of the middle 1800s that gave us modern things like wireless technology

and telegraphs and telephone. And then Einstein unifying space and time, matter and energy. This is a trend. We've been doing this unification program in physics for the past 150 years and it's very, very successful and it reveals these underlying deep commonalities among things that are superficially different. So the idea that there's great insight to be had by realizing that things that look different are actually deep down the same,

That's a good move. That is historically a very good move much of the time. But there's also the move that along with the unifying impulse, you also have to have the diversifying impulse. You have to realize that not all things are the same, that there is great abundance in the world, all kinds of diversity, whether of people or biological species or

or phenomena. There are two kinds of scientists, or more than two, but there are unifiers and diversifiers, and there's a need for both. I guess I want to argue for the happy middle that if you're all about diversity, you won't see patterns. If you're all about unity, you won't see richness. I think both are blinkered visions of the world. I just don't believe in either extreme.

Talking to Steve and talking to Kareem, I think the question we were really kicking around is, does your experience of the world feel fulfilling and complete, even true? And I think for Steve, there is a deep pleasure and joy and a benefit, like a real tangible benefit to accepting math exactly as it is and reveling in how it describes reality.

And for Kareem? Every day I sit at my computer. There isn't. Kind of rewriting our lessons to tighten things up. The one I was working on yesterday was about concert tickets and about all the fees involved.

And like our secondary ticket brokers discourage or are they actually like correcting kind of a market failure? That sounds interesting. Oh, yeah. All of our lessons are interesting. I mean, I think. But that is so based on math. And it sounds like you're every day you're staring at these things that you believe are confining you.

these numbers and you're, you're literally not just staring at them. You're like working with them even more intimately than most people, because you're trying to like fit them around the universe and explain that back to kids. Like you're, you're playing with these tools that sound like they have, you feel like are failing you or, or maybe not failing you, but they aren't all that's there. I sort of feel like I'm spinning my wheels needlessly. I feel like I'm ready for something. I feel like I'm ready for whatever is the next thing. But,

But what's crazy to me is like, but to do that because of the nature of what you do and what your passion has been, you have to turn your back on math. It sort of sounds like. I mean, I think, look, I think we live our lives in phases and that isn't, I'm not going to put it down and then stomp all over it. Yeah. Right. It's like, it's a gentle putting down. It's not throwing it on the ground, but I feel like I've,

I've sucked all the juice out of that orange for me. Okay. One last question. When you think about the world, when you think about zero world mathematically, where one equals two equals zero equals infinity. Everything gets sucked into the black hole of zero. Yeah. This place that you, it sounds like you yearn for, that you want to go experience and understand and feel, right? I mean, is that, okay. What? What?

Has thinking about it and spending time there theoretically, has it changed your understanding of numbers or math at all? Has it expanded math for you at all? I respect math more by virtue of it writing the sign. Writing the sign? Yeah. What does that mean? Mathematics saying there's something we can account for.

I admire that. Why? Why? Why? Because everybody, I am Christian. This is the truth. There is no truth but for this. I am Muslim. This is the truth. There is no truth but for this. Mathematics is an incredibly powerful tool. And for the institution or for mathematics personified to say, I'm an exceptionally powerful tool. If you master me and if you use me, you're going to be able to do so much. But

I'm not complete. There is something I can't account for. I think that humility, I really, I think that is enviable. When I first wrote that paper about division by zero, I was like, I'm really going to stick it to math. And now it's more like, what a wonderful gift for this powerful tool that we use to do so much to say, but if you want to go further, you need to put me down now.

This episode was produced by Matthew Kilty with help from Akati Foster-Keys and Alyssa Jean-Perry. Mixing help from Arianne Wack. Fact-checking by Diane Kelly. It was edited by Pat Walters. Steve Strogatz, by the way, also hosts a podcast all about math where he zips and zazzles through different...

and questions with all kinds of fun guests. It is called The Joy of Why. W-H-Y, The Joy of Why. And Kareem wrote a book all about how to get kids talking about how math interplays with real world puzzles. It's called Dear Citizen Math. And you can check out citizenmath.com to see all sorts of neat lessons he and his team have dreamed up over the years for middle school and high school classrooms.

That'll do it for today. That'll do it for this year. Thank you so much for listening to Radiolab. I hope you all get a little bit of Zero World over the break, like where nothing is happening. Just low stress, low thought. Rest. Dare we say rest? Welcome back to Zero World. Where there are no phones. Yes, your precious little phone is gone.

Oh no, oh no, oh no. Going somewhere? I don't think so. There are no cars, there's no planes, motorcycles, bicycles. None of it. No money, oh how good freedom, no money. You can't even count here. There's nothing, nothing but zero. As far as the eye can see. Are you happy now? Hi, I'm Hazel, and I'm from Silver Spring.

Radio Lab was created by Chad Abomac and is edited by Soren Wheeler. Lulu Miller and Latif Nasser are our co-hosts. Dylan Keith is our director of sound design. Our staff includes Simon Adler, Jeremy Bloom, Becca Bressner,

Our fact checkers are Diane Kelly, Emily Krieger, and Natalie Middleton. Thank you.

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