In Our Time - Imaginary Numbers

Episode Date: September 23, 2010

Melvyn Bragg and his guests discuss imaginary numbers. In the sixteenth century, a group of mathematicians in Bologna found a solution to a problem that had puzzled generations before them: a completely new kind of number. For more than a century this discovery was greeted with such scepticism that the great French thinker Rene Descartes dismissed it as an "imaginary" number.The name stuck - but so did the numbers. Long dismissed as useless or even fictitious, the imaginary number i and its properties were first explored seriously in the eighteenth century. Today the imaginary numbers are in daily use by engineers, and are vital to our understanding of phenomena including electricity and radio waves. With Marcus du SautoyProfessor of Mathematics at Oxford University Ian StewartEmeritus Professor of Mathematics at the University of WarwickCaroline SeriesProfessor of Mathematics at the University of WarwickProducer: Thomas Morris.

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terms of use, please go to BBC.co.com.uk forward slash radio four. I hope you enjoy the program. And sometime in the 1530s, a group of mathematicians in northern Italy made an alarming discovery. In solving a problem that had perplexed scholars for thousands of years, they stumbled across another, an entirely new type of number, mysteriously, it didn't fit into their conventional number system and was apparently useless for counting or measuring things. For more than 100 years, the new number attracted hostility from mathematicians who regarded it as fictitious, impossible or meaningless.

The French thinker René Descartes shared this opinion but also gave it a name. He called it the imaginary number. But since the 18th century, mathematicians have come to accept imaginary numbers as both genuine and very useful. They underpinned some of the modern world's most essential technologies from mains electricity to radio. Without imaginary numbers, you wouldn't be listening to Radio 4 now.

They led to complex number theory, which goes into the search of, for instance, complex molecules, such as insulin, viruses, even DNA. With me to discuss the discovery and uses of imaginary numbers are Marcus Usootoy, Professor of Mathematics at Oxford University, Ian Stewart, Emeritus Professor of Mathematics at the University of Warwick, and Caroline Sears, Professor of Mathematics, also at the University of Warwick. Mark Isot, before we go to imaginary knowledge,

Let's start with real ones. What is a real number and how did our number system come into being? Well, our real numbers are actually very rich and came out of the problem of solving equations, which is where this imaginary number is going to come from too. So I think it's very important to understand the buildup of our number system, which starts with just the numbers one, two, three, which we use to count things. But then you want to do a little bit more. We've got three apples on the table here and four people in the studio. want to divide this up, we need some new numbers to do that. So we invent fractions, which are, so

we each get a three quarters of the apple. So new numbers started to appear by trying to actually sort of solve sort of mathematical problems. Measurement, you mentioned earlier as well, well, if you want to measure, for example, the diagonal across a square where the square has one unit on each side, Pythagoras's theorem revealed that you couldn't use fractions to measure that length. It was actually quite a shock for the Pythagorean's. They thought that you, you know, you could measure everything using fractions. But Pythagoras' theorem says, well, the length of that diagonal across a square

is actually a number whose square is two. Well, what is a number whose square is two? They found there were no fractions that they could do that. You could get an approximate number, 1.4142, for example, when you square it almost gets you two, but not quite. So, in fact, there was an act of imagining a number in some sense which would solve this problem, number which when you square it gives you two.

So that was an irrational number. That's what we call irrational. So we talk about inventions. Often we're talking about just getting out there and inventing something to solve the problem and say, well, this solves one, we're going to move on and just keep solving problems and forget about proofs until we can get to them. Well, a lot of the, but you have to make sure that these new numbers are consistent, and that's always the problem to make sure that if you are going to add something in, that it doesn't collapse the whole of mathematics.

But there is still a feeling like you can see this number on a ruler, even though you may not be able to write down exactly what it is. If we want to write it down exactly, we have this little funny symbol, a square root sign over a two, and that represents a concept of something whose square is two. But then we get into really interesting numbers,

for example, those were quite early on, the sort of ancient Greeks, but what about negative numbers? They're there to solve another problem. What number when I add it to one, add it to three gives me one, for example? Well, I've got to add minus two to three to get me to one.

But what is minus two? You can't see minus two apples on this table. But again, you've got an act of imagining to bring alive these numbers, negative numbers, which can solve problems. I've got an equation there. What number, when I add it to three, gives me one? What problem would that solve?

I mean, the equation, yes, but what... Well, debt. Debt, exactly, the problem we all have. I mean, the government is happening. The government certainly know about negative numbers. But again, there is some sense in which you can give a reality to these numbers. numbers because we have this thing, the number line which all kids learn in school,

where the sort of whole, the positive numbers go off to the right. And then you can think of negative numbers going down to the left. So you can sort of visualise these numbers, or even if anyone who takes a lift who goes down to the basement is sort of visualising negative numbers. But I still think, you know, that's an incredible moment. It happens, you know, sort of 7th century AD in India. It takes Europe a heck of a long time to accept these negative numbers. And so you can see how the number system is building up.

And then at some point, they were struck with the equation, well, actually, I quite like to take the square root of a negative number. That came out of certain new equations called cubic equations. And, you know, is there a square root of minus one? And most people said, well, no, any of these numbers, where are the numbers that when you square them give you minus one? If you square a number, it becomes positive. And this became part of the great mathematical cluster in Volonia

and in northern Italy generally where people were... Well, they were starting to... ...competes with each other at courts and... Exactly. There was this... The O... And rushing forward a little bit, Marcus managed to do two and a half millennia.

Not quite as a big Australian period. Cardano, the mathematician called Cardano, found a way to solve the equations that Marcus led us to. Can you just repeat what the equation was that he was trying to? Was it the square root of minus one? And why was that so important? Yeah.

He said it tormented him, didn't it? The mathematical... The mathematical... mathematical gadget that comes up is the square root of minus 1, which comes from trying to solve the equation X squared equals minus 1 in school language. But actually what Cardano was interested in was the next level up, cubic equations. So this is where the unknown enters into your equation in terms of its cube. Okay, the cube of a number is what you get when you multiply the number by itself and then again.

So four cubed is four times four times four, that's 64. and Cardano who was a mathematician, he was a doctor, he was an inveterate gambler and a real scoundrel, he gambled away the family fortune, and then he made it back by curing the Bishop of St Andrews of some terrible disease. Gambling, family. I don't know. He was a fascinating character,

but he wrote one of the great mathematics textbooks, which was on algebra, how to solve equations. A lot of the work in it was not actually his work. It was from other people of the period, and there was some controversy about this. But with cubic equations, Kadano effectively had what in the language of the time

was a formula for solving the equation. And basically that formula boiled down to take a few square roots, take a few cube roots, throw them together in the right way, and that will give you the answer. However, there was a snag. sometimes when you did that the answer appeared in a completely impenetrable form

sometimes it was very nice you just did the sums and out-potter right no x equals 3 or whatever but an example cardano mentions is actually if x cubed is equal to 15x plus 4 now all of our listeners will instantly say I know the answer it's x is 4 because 4 cubed is 64 and 15 times 4 plus 4 is 60 plus 4

that's 64 okay we know the answer it's 4 Cardano's formula tells you that the answer is the cube root of 2 plus the square root of minus 121 plus the cube root of 2 minus the square root of minus 121. Now this is a bit of a bummer basically. And you can't just say let's do the sums. You know, get out your equivalence of a pocket calculator, press the square root button on minus 121. It's clearly 11 times the square root of minus 1, but it's a big puzzle. And so they kind of throw up their hands in horror and said, well, you know, sometimes it works, sometimes it doesn't.

Until 20 years later, when another Italian Renaissance mathematician, Rafael Bombelli, said, you know, I can see a way to handle this. Because if I take 2 plus root minus 1 and cube it, and stop asking myself what the square root of minus 1 is, and just pretend it's some sort of number which obeys all the usual rules, I get 2 plus the square root of minus 121. So Cardano's complicated formula boils down to 2 plus the square root of minus 1, plus 2 minus the square root of minus 1. And your horrible puzzling root minus 1 things cancel each other out, disappear completely. And what are you left with? 2 plus 2 equals 4, which is, of course, the answer. Phew. Well, I'm pleased about them.

It's a happy ending me. So the message that came over was, I don't know what. these imaginary number things are, but if you pretend that they work just like real numbers and stop worrying about taking what it means to take a square route, if you just do the sums, you get perfectly sensible answers in cases where you can check that it's perfectly sensible. Caroline's here is Ian mentioned Raphael Bombelli's treatise.

Could you tell us a bit more about that and why it didn't have much of an impact? Oh, actually, I think Bombelli's treatise actually did have reasonable amount of impacts because this was the point at which people actually saw that these numbers, these imaginary things, might conceivably lead them to a useful answer. You see, what Cardano did was he arrived at his formula, and he knew the answer to this equation was four, but there he had a formula, which he trusted.

He trusted his calculations and you have to give him credit that he didn't just sort of throw it out. But he absolutely didn't know what to do with it. He described the things as mental tortures. And in fact, he has this wonderful phrase, so ends arithmetic subtlety. It's refined as it is useless. So there we have the statement that these things are completely useless. And by the end of the program, I hope we'll have convinced you that they're not completely useless.

So what Bombelli did that was so important, was he showed that it was possible to manipulate these numbers in a form, according to reasonable rules of algebra, which he actually wrote down in rather a cumbersome way, but he wrote it down. You could manipulate them according to rules and actually come out with what everybody agreed was the correct answer. I'd like to just go back and say one thing that might be puzzling some people.

If you ever studied how to solve what's called a quadratic equation, which you may well have done if you did any amount of maths at school, these are equations which involve x squared and x and constants. And there's a magic formula for solving these equations, which was sort of the starting point of Cardano's work. And this formula involves a quantity which is normally just rattled off. You learn it when you do GCSE and so on. It's the square root of B squared minus 4AC.

Now, this B squared minus 4 AC could perfectly well be a negative number. So why weren't people, the Babylonians knew how to solve quadratic equations, ancient in the ancient world. Why weren't they worried about this? Well, the thing was that they just rejected these solutions as completely and utterly impossible. Who's they? Which they're talking about now? Well, all mathematicians up to the middle of the 16th. century, which is what we're talking about.

So, I mean, they used... They had a formula... They used it, but they didn't... No, they didn't use it. They had a formula, which they knew if the equation had a real number solution gave them the right answer. Sometimes the formula

involved square rooting a minus number, and they just said, well, then this equation is an impossible equation. It doesn't have an answer. It's true. It doesn't have a real number answer. So let's just ignore it. It's impossible. forget it.

But the real catch about Cardano's equation, this one that Ian just talked about, this 15x cubed equals 15 times X plus 4, is that people knew that it had a real number answer. You can check. You can go away and check the answer is 4. But Cardano's formula was a complicated formula involving all these square roots.

So the point was that Bombelli showed actually this complicated formula still had a sense. And from that point on really, gradually and kicking and screaming more or less, but mathematicians realized that these numbers, although they didn't understand them at all, they were completely mystified, nevertheless they had a point to them. And they better start paying attention to them. I mentioned in the introduction that Descartes had called them imaginary numbers, not in a complementary way.

Did that description in itself hold people back from doing much more with them? I think it was actually a rather unfortunate term, but actually the terms used before Descartes were also, they were people call them impossible numbers, improbable numbers, nonsense numbers, sophisticated numbers, all sorts of things. So you got away likely. So people didn't know what to call them. But by the time we come to Descartes,

people had realised something very interesting that if you have an equation of degree two, these quadratic equations that the Babylonians can solve, they have at most two solutions. Cubic equations have at most three solutions. Fourth power equations have at most

four solutions. And by the time of Descartes, it had been worked out that an equation of the nth's degree that's involving X to the n, say X to the hundreds and so on, would have a hundred solutions. But it might not have actually a full hundred solutions.

And what Descartes actually said, was that of these sort of potential 100 solutions that are out there, some will be perfectly good real numbers, but some of them will be just imaginaries, imaginary. Can I ask you if you can possibly answer this briefly, not because I'm hurrying you up, but because it'd be nice to encapsulated. Do imaginary numbers mean what you want them to mean?

Do they mean what we want them to mean? I think the thing that you have to hang on to, we're going to come on later to the geometry of imagination, numbers, but at this point, all you have to hang on to as your support in all of this is that whatever it means, if your algebra tells you to work out the square of it, then you get whatever is under the square root sign. If you have minus the square root of minus three, and your formula says square this number, the answer is minus three. So it's just at this point a calculating device. Marcus, broadly, over the next two or three hundred years,

until the 19th century, as I understand it, for goodness sake, correct me, all right, it was hovering around taken up by various mathematicians, but not in any sense in any mainstream. That would be more or less right? But in the 90th century, three mathematicians independently hit on a new way of representing a measuring numbers, and as it were, the thing got really started. Can you take us into that? Yes, I mean, I think the point is people could see that this,

sort of abstract concept something whose square is minus one was proving very useful in trying to solve these equations. But I think why people are having difficulty actually accepting it is, well, where is this number then? Because you could see, you had a picture of negative

numbers. They went off sort of to the left of your number line. The square root of two also quite an imaginary number whose square is two. You could see as a measurement on the number line. So we had these real numbers sort of lined up on a sort of a line running sort of east-west in the number line. Now, three mathematicians at the beginning of the 19th century,

Carl Friedrich Gauss, Vessel, who was a Norwegian, and Argan, who was an amateur mathematician in Paris, born in Switzerland, came up with a picture of these imaginary numbers, and suddenly we were able to see these imaginary numbers, which actually just helped to bring them an acceptance of them. And their picture is very clever. Well, if you haven't got room on the number line going sort of east-west, this one-dimensional line, we've got this new number, the square root of minus one, which was given the name I.

And we've got sort of combinations of these numbers. We've already talked about two plus the square root of minus one. Well, why not create a two-dimensional picture of numbers, where the real numbers run east-west? but if you want this imaginary number, well, why not take a step north one step and represent that on your two-dimensional picture of numbers as the imaginary number I? And if you want a combination of real and imaginary numbers, it's like coordinates on a map. Okay, I step three steps along the real line east, and then I step two steps north. And I've got a two-dimensional picture.

Let's just go into two-dimensional very carefully because we just so absolutely, you've got your ruler and you've got zero in the middle, let's say, and to the right you've got 1, 1,000, minus 2, minus 3, minus, minus, minus, minus, minus, minus, I did for 9,000, so that's it. What you're going to put, like a cross, is an up and down line north-south, with the zero's overlapping, and north is plus, and south is negative. So you've got these, you've got a cross. Exactly, so north is one step towards the imaginary direction, so it's the number

I, the square root of minus 1. Sorry? Is that technical imaginary, the eye? I think it is, yes. A good guess. It was oil. It was oiler who started choosing eyes. Which is useful.

It's good to have a symbol for this, and when you square it, it suddenly becomes minus one. Now, great to have a picture of this thing, but is it a useful picture? If it's just a picture which doesn't really tell you much about the arithmetic of these imaginary numbers, it doesn't really get you anywhere.

But the amazingly powerful thing that Argan and Vessel and Gauss discovered is that the geometry of this picture totally reflects the arithmetic of these numbers. So somehow if you multiply imaginary numbers in this picture, there's a geometry which helps you to understand what the answer is going to be. Ian, can you take us on without too many axes? Yeah. Okay, so Marcus has given us this beautiful picture of this complex plane Instead of the number line for the real numbers, we have a plane for the complex numbers with this cross giving you the reels and the imaginaries.

Now, suppose you want to multiply a number in that plane by three. What do you do with it? Well, the answer is you stretch the, you magnify the entire plane by a factor of three. You just stretch it uniformly in all directions so that everything gets three times as big. What happens if you want to multiply by I? This is the crucial point of the whole business. what does it mean to take a number and multiply it by I? Which is an imaginary number?

Once we know the multiplication rule, everything else is going to follow. This is at the core of the whole business. Well, how did we get from one to I? we take our horizontal real line and turn it vertically so that it's pointing north-south instead of east-west. We rotate it through a right angle. So multiplying by I takes the complex plane and just rotates the whole thing through a right angle.

And the reels go to the imaginaries. The imaginaries go to the reels. And if you do it twice, one goes from one to I, and then another right angle takes it to minus one. So we've done something twice and got minus one out of it. So you multiply by something, multiply again, you get minus one. That's i squared equals minus one. The algebra and the geometry all makes sense.

But the reason there isn't a number on the horizontal line, there isn't anything you do to the horizontal line if you do it twice you get things mapped to their negative values is because you have to go out into the complex plane It's another way to say you combine what we've called real numbers In an affectionate now seems rather a quaint way

And imagining numbers If you want three times three Three real three imagine Sorry about this You go tick tick tick three East on the horizontal line And then you go tick tick tick three On the imaginary line

You draw a line out there and you draw a line up there. That makes a... And so you've got the meeting in a place called whatever it is, called something or other. There they meet, and that starts to get a grid. That's right. And then the crucial thing is if I multiply that number

by I, which is the difficult thing to imagine, it just rotates that whole picture through a right end. It is difficult to imagine. So you're multiplying by I... So you're saying, I... Sorry about that. I think.

This is going to stand for that and that's that. But I is going to be the square root of minus one because it works. It's lateral thinking in a genuine sense. If I can't go along the real number line, let's move out sideways. Spinning through a right angle is going laterally compared to expanding things out along a radius. Okay, right. Caroline, can you, Gauss, one of the three independent thinkers, the three people who independently arrived at pushing on this theory very substantially, It also made a discovery about prime numbers, numbers which can be divided only by themselves or by one, numbers like five.

Well, until we've got to imagine numbers, that is, but let's leave that aside for the moment, five, seven, that sort of thing. Can you explain what Gouse was bringing by doing that? Yeah, before I say that, let me just say, for people who don't know, Gauss was really probably the greatest mathematician who ever lived. He described as the king of mathematics. He straddles the beginning of the 19th century his life. He left his mark on almost every branch of mathematics and a large part of physics and astronomy as well.

So he made enormous contributions that really still affect our modern world today in very crucial ways. So this was one thing he did. He, going on from this idea of Ian's, about representing complex numbers as points in a grid, he introduced the idea of what's called a complex integer, often nowadays it's called a Gaussian integer, which just means something like three plus three times I or two plus seven times I. It means one of the grid points where the coordinates are actually whole integer numbers.

And what he did was he extended the idea that you can factor ordinary numbers, like you can factor six as two times three, into the idea that you might be able to factor ordinary whole numbers into products of these complex numbers. And he then developed the whole theory of prime complex integers, prime numbers, which I think we're going to come on to. And this leads to the foundation of all sorts of men results just about, this is one of the miracles, you can prove results using these complex

integers about the ordinary integers. So let me give you one example. We can take the number five, which you know you can't factor as an ordinary integer, and you can work out that if you take the complex integer 2 plus i and multiply it by the complex integer 2 minus i, if you do the sums and you remember that i squared is minus one, you come out with the answer that two plus i times two minus i is equal to five. And it is no coincidence that the number five is actually two squared plus one squared, four plus one, which is five. And what Gauss was able to show, using his theory of factoring into complex integers, was a fact about, which has nothing to do with complex numbers, which says that how do you tell when a prime number, that's a number, an ordinary

whole number that can't be factored, how can you tell when you can write it in terms of the sum of two squares? And the answer is that you can do it exactly when that number has remained a one when you divide it by four. So it's a wonderful theorem and you prove it by going into this realm of complex number factorization. And at this stage, I'm going to give you a norm of our experience, just as sort of where we are. At this stage, we're in the realm, and we have been for centuries, of people doing things to solve problems,

people doing things for the mental exercise, the mental enjoyment, or just because they're their problems to solve. They're not looking for things to make from it at this stage, except to solve more problems and more problems. It's what we used to call pure science, pure mathematics. And out of this is going to come an explosion of practical applications, but it has taken centuries to do.

So, as it were, to listen to just hold it, Hold on, things are going to happen in about 10 minutes. Right. Marcus. Right. Is it worthwhile spending time on another of Gauss's discovery, is doctoral thesis 1799, the fundamental theorem of algebra? Absolutely, because I think this shows the power of these ematuary numbers,

because you say, well, we've added the square root of minus one, and we've got a load of new numbers out of this. But what about if we wanted the fourth root of minus one? Oh, my gosh, you've got a number who's, when you raise it to the power, power four gives you minus one. Are we going to have to imagine some new numbers which solve this? So we've added I. Are we going to have to add a J which is going to solve that one? And if, you know, there are millions, infinitely many equations? So we're going to have to add more and more numbers. The amazing thing that Gauss discovered is that this imaginary numbers,

this two-dimensional picture, this plane of numbers, is enough to solve all of these problems. and that's what he proved in his doctoral thesis. And it's in a way why he produced this picture. He was a bit nervous of pictures, and he didn't tell anybody about his picture. We call this picture the organ diagram that he was actually the third person to discover it. But Gals used this picture to prove

that if I want to solve X to the 4 equals minus 1, there is actually a point in this two-dimensional picture, which, when you raise it the power 4, gets you to minus 1. And it is, in fact, 1 over the square root of 2, plus 1 over the square root of 2 times I. Actually, a point if you go 45 degrees to the horizontal line with the real numbers, it's a place on the sort of unit circle, the circle with radius 1, which is at that point. And now if you go back to Ian's description of how you multiply things in here,

well, you just sort of, when I multiply these things, I sort of add the angles together. So I go 45 degrees, 90 degrees, 90 plus 45, add another 45, I'm at 180 degrees, which is minus 1. I'm gone down in the other direction. So, well, okay, that might have been a bit confusing. Sorry. But the essential thing here that Gow's discovered, it's called the fundamental theorem of algebra, is that this picture of these imaginary numbers

is enough to solve any equation. If you give me an equation, X to the 100 plus 4x equals minus 6, there'll be a number, several, in this complex plane, this imaginary 100, which will solve that equation. It was an absolutely staggering discovery that we, and quite good actually, because we've got a nice two-dimensional picture of these numbers. If we had to add new numbers to solve all of these equations, it'd be very difficult to see them. But that's the power of these imaginary numbers, is that now we can actually solve any equation we want to.

So, Karen, briefly before me, can you, the complex numbers help the range of conventional numbers as well, didn't they? Help these develop and this development and helped conventional mathematics. Oh, absolutely. I mean, the thing I was just explaining about gas factoring, there you have a great example. So we have something about ordinary integers about the fact that the sum, he's telling you the answer to the question which whole numbers can be represented as the sum of two perfect squares. Nothing to do with square roots of minus one.

But actually to find the solution, find this incredibly elegant solution, in fact, you go into the realm of these imaginary numbers, and there you come back with a solution, which explains why. We're now steering towards practical applications that turn there because there's nowhere near the minds of all the people we've been talking about. And those of you who come so far, thank you, and here we go. Towards the end of the 90s, in the 90th century, Ian,

practical solutions began to heave upon what we might call the horizon. Yeah, let's up the ante a bit. We've sorted out of algebra now. Let's go for calculus. Complex calculus. You thought this was a difficult program. Calculus is about solving differential equations, things about rates of change, things about...

It comes out of Isaac Newton's Law of Motion that the acceleration of a body is proportional to the force that acts. What's the acceleration? It's how fast the velocity is changing. What's the velocity, how fast the position is changing? So, suppose we want to do that kind of... of thing with complex numbers. It turns out that there's a wonderful theory.

It all works beautifully. And just as Caroline has said, you can apply complex arithmetic to ordinary real arithmetic prime numbers. So it turns out you can apply complex calculus to solving the kinds of real differential equations that were coming up in mathematical physics. So things about the flow of a fluid

or a magnetic field. or heat or sound or waves or gravity. All of these things, you could use complex calculus to find real solutions. So, for example, you know, bar magnet, we all did this experiment at school, you take a piece of paper, you stick a bar magnet underneath, you shake out some iron filings onto the piece of paper, and they arrange themselves in these wonderful curves, which are the lines of force of the magnetic field.

you can use complex calculus to give formulas for that kind of geometry. And Marcus, when we come to Raymond, only one or two in one hours, what is he pointing us in the direction of? I want to get to the adaptations of this. So I'm trying to, I know we never have enough time. Yeah, yeah.

We do, actually, mostly. Right, let's get on, Ryan. The amazing thing is that these imaginary numbers started being incredibly useful for lots of different people. In fact, there was one mathematician said that the shortest way to get to the real answer is to go via the imagine. And very often this proves to be incredibly powerful.

So actually, Reiman was looking at trying to put, we have things called functions, like sign functions, cosine functions, the exponential function, where you feed in a real number. It calculates away this function, gives you an answer. When these imaginary numbers came along, mathematicians said, okay, well, what happens that we start feeding imaginary numbers into these functions

that we've been using for centuries? Is there a sort of mathematics of these? And amazing discovery. started to pop up, connections between things that didn't seem to be connected at all. So, for example, if you take the exponential function, which is raising things to certain powers, put imaginary numbers in there, suddenly you get waves coming out of these. You get cosines and signs coming out. And there's something, a discovery that oiler made. It actually leads to one of the most beautiful formulas in the whole of mathematics, which is E to the I pi equals minus one.

I mean, somehow beautiful, I mean, it's called the most beautiful formula of mathematics because it combines the most basic numbers, pi, e, square to minus one and minus one in some beautiful formula. But that form, hiding behind that formula is a connection between putting in imaginary numbers into an exponential function and getting out waves. This is actually, you mentioned why we can hear ourselves coming down the airwaves and you can hear us in your living room or in your car or wherever you are. It's all based on the fact that imaginary numbers somehow give rise to wave functions. Now, you mentioned Reamon.

Reamon started putting these complex numbers into an interesting function, something called the Zeta function, and suddenly out popped the answer to prime numbers out of this. Suddenly he found a new way to understand. Prime numbers are a totally chaotic-looking set of numbers with no pattern,

but he suddenly identified some imaginary numbers who held the secret, the sort of DNA, of these prime numbers. And now we're actually trying to understand not prime numbers, the subject has totally shifted in the middle of the 19th century and actually all we're obsessed with, I mean I'm obsessed with trying to calculate some imaginary numbers

and where they are, because I know that that will tell me about prime numbers. Can you, Caroline, can you just take this line on, what is it leading to that we have, the world we live in now, the people say, oh, that's because of imaginary numbers. We are at this point in our technology because of the pure mathematics of the last several hundred years. This is what, just can you give us a few specific examples? Yeah, let me first of all say,

something quite different that Reiman did. Remen's great thing was that he explored this geometry of complex numbers in a very original way. And one of the things he did is who proved a great theorem called the Remen Mapping theorem, which set very sort of abstract theorem, but it says you can take any region enclosed by a curve you like on the plane, and you can transform it into a circular region by using these complex functions. And now the point of that is that, If you have a problem like about heat flow in this funny shaped region or fluid flow in this funny shaped region or a gravitational field in this funny shape region and you're trying to find out how the flow lines go, you use Riemann's theorem to change this funny shape region into a circle. And in the circle, these problems, the solution is known, was worked out during the 19th century.

You solve the problem there, you change it back again. And they, there, Bob's your uncle. You have the solution. So all sorts of problems about how does heat flow, how does magnetism work, how does electricity work, are all solved by Remens' theorem. Can we keep on this tick line? Yeah. If you want a real example, in your home, imaginary numbers are basically flowing around all the time. The electrical wiring in your home uses alternating current.

That is, the current doesn't just flow one way, it goes from positive to negative and back again. and there are all sorts of good engineering reasons for this. And this is basically back to what Marcus was saying, the mathematics of waves. And engineers who work with alternating current use a mathematical trick to simplify all of their calculations, and that is to represent the current in terms of imaginary complex numbers.

And if you do this, all the calculations are much, much simpler. And you can turn... You can think of the equations for... alternating current as being just like the normal equations for a direct current except that things like the resistances in the wires should be considered to be complex numbers. So it's as if imaginary numbers are flowing around your house through the wiring and into your television set. None of this technology would work without the engineers being able to do the calculations.

Do you want to take this on a bit, Marcus, as to what else is going on? I mean, perhaps it would work, but it would be so much more. difficult, I think. I mean, the point is that I think it shows the positive attitude of mathematicians. We could have said this number doesn't exist. Squared to minus one. What's that? There is no number. And then let's just get on with the world in our real numbers.

But we're a positive lot of people and we say, no, come on, let's imagine something that would work because it's going to make our mathematics so much more powerful. And by that act of imagining, introducing this thing which didn't, which made was consistent, it made absolute sense, suddenly are able to

make problems in physics by using Remens Mapping theorem, so much simpler that you can actually solve them. The understanding the electricity flows suddenly becomes so much more straightforward and you can actually make progress that you couldn't have done without the facility of these new numbers. I said in the beginning of the programme, but in research into complex molecules in viruses and insulin and that is because of this, isn't it? Sorry, Caroline. Yeah, and there's something called a Fourier transform, a magic thing called a Fourier Transformer and a just

incredible amount of our modern technological society really comes back to the power of this device. So this device is something What is the incredible amount for instance? Well, for example, the BBC probably wouldn't exist without it, certainly not in its

present form. We wouldn't be talking here down this, certainly not down these digital ways. We're always using the radio as an example. Are there others? Oh, television. When you go to hospital and you have a cat scan, NMR scans, the technology that enabled us to first discover the structure of molecular structure, atomic structure of crystals. And then as you say, we went on to discover the structure of viruses, insulin, chlorophyll. Eventually we came to the structure of DNA.

All the technology that does that and the physics behind it, this freer transform which intrinsically involves the square root of minus one in this formula of oilers. It really, you know, you might be able to do it without it, but it would be so impossibly complicated, and the calculations would take such an enormous amount of time. It's very hard to think that you could do it. All the big models that people run, so people do climate modelling. They do sampling of lots and lots of information

about what happened in lots of places at lots of times, and then they put it into a huge computer model. Well, if you tried to do that, and you didn't have the power of these imaginary numbers to do the calculations, it would take so impossibly long that even with modern computers you couldn't do it. Do you want to carry that forward, Ian? Yeah, another one, digital photography.

When you take a snap with your digital camera, I went on holiday recently and took 1,400 photographs and still had some spare room on a 2 gigabyte card. How does that much information get into such a small space? And the answer is that this Fourier transform is actually built into the electronics of the camera, and it's used to compress the information in the picture. So if you're carrying a digital camera around with you,

such as in your mobile phone, you actually have a chip with a phosphorya transform on it that does the maths for you. And that absolutely, without complex numbers, as we're saying, you could perhaps have invented this kind of thing, but in practice it would be so terribly hard, it would be a silly way to go about it. So what mathematicians and engineers and everybody are looking,

for are the simplest ways to solve problems and that's what the imaginary numbers do for you. I think another area that we have to talk about is quantum physics that you really can't understand. I know it's not about 58 seconds. I think we'll do another program. Give us a few headlines. Exactly. Quantum physics is all about you're not quite sure where a particle is until you observe it. That's because actually the particle is existing in this imaginary world is described by imaginary numbers. you observe it, it sort of has to collapse onto the sort of measuring line into the real world, this one-dimensional world. So without quantum, without imaginary numbers, you really couldn't

describe the world of the quantum. And since the quantum physics is really what describes reality, these shouldn't be called imaginary numbers. They're actually as real as anything else. And you see this, it's extraordinary, as long as it were, long, slow run from the edge of the boundary until they get to the wicket and they deliver a ball and it explodes into all this, isn't it? I mean, that's the power of mathematics that you. you'd never quite know how much it's going to unleash. And these imaginary numbers have unleashed just fantastic bits of mathematics that would be such a shame if we just said that that number didn't exist.

Well, I'm glad we get repeated and all the rest of it, so people can really study this, including me, and find out what you were saying. I was holding on now. Anyway, thank you very much to Caroline's series, Marcus Chesotoy, and Ian Seward next week. Where's next week? Yes, we're going to talk about the Delphic Oracle, which ran from 600 BC to 500 AD,

great powers on the side of Mounted of Panassas. Thank you for listening. If you've enjoyed this Radio 4 podcast, why not try others, such as Thinking Aloud, where Laurie Taylor discusses the latest social science research. To find out more, visit bbc.co.com.uk forward slash radio 4.