In Our Time - e

Episode Date: September 25, 2014

Melvyn Bragg and his guests discuss Euler's number, also known as e. First discovered in the seventeenth century by the Swiss mathematician Jacob Bernoulli when he was studying compound interest, e is now recognised as one of the most important and interesting numbers in mathematics. Roughly equal to 2.718, e is useful in studying many everyday situations, from personal savings to epidemics. It also features in Euler's Identity, sometimes described as the most beautiful equation ever written. With:Colva Roney-Dougal Reader in Pure Mathematics at the University of St AndrewsJune Barrow-Green Senior Lecturer in the History of Maths at the Open UniversityVicky Neale Whitehead Lecturer at the Mathematical Institute and Balliol College at the University of OxfordProducer: Thomas Morris.

This BBC podcast is supported by ads outside the UK. Every Sunday, we talk about the week's tech news on this week in tech. Hi, this is Leo Leport. inviting you to join me this week with Lisa Schmeiser, Dan Patterson, and Yanko Rekkers. We're going to talk about the new 49 megabyte web page. It's the standard, you know. We'll also talk about Elon Musk. You've got some spleenin to do and the Yassify filter, new from Nvidia.

That's this week on this week in tech. You'll find it at Twitter. or wherever you get your podcasts. Thank you for downloading this episode of In Our Time. For more details about in our time, and for our terms of use, please go to BBC.co.com.uk slash radio 4. I hope you enjoy the program. Hello, centuries ago, when thinkers started to look at the world around them

using the language of mathematics, they found that a few very important numbers seem to underpin everything. Perhaps the best known of these is pi, the ratio of a circle circumference to its diameter, which is roughly equal to 3.141. Less famous, but arguably every bit as important, is a number studied in the 18th century by the great Swiss mathematician and an Euler.

He called it E, and in honour of him it's sometimes referred to as Euler's number. E begins 2.71828 and continues for an infinite number of decimal places. It's a number that can be found in all sorts of surprising places. It crops up in the study of interest rates, electronics, and radioactive decay. A mathematician would also tell you that it's irrational, transcendental, and part of the most beautiful equation ever written.

With me to discuss the number known as E, are Colbert Roney-Dougall, reader in pure mathematics at the University of St. Andrews. Vicki Neal, Whitehead Lecturer at the Mathematical Institute and Balliol College at the University of Oxford, and June Barrow-Green, senior lecturer in the history of mathematics at the Open University. Colver and Dougal, most people have come across pi in their maths classes. If you're familiar with E, would you give us a quick summary of what it is? Well, as you mentioned in your introduction, E is a number.

It's a number between 2 and 3 that begins with 2.7. It goes on forever. But what's important about E, there's two main things. The first is that it's a number that just pops up again and again and again in a huge range of both pure maths and applications. So you mentioned some of them, but already we can have applications featuring things like the way that a hot coal will cool if you put it in a cool environment, the way in which populations of rabbits and foxes grow and shrink as the foxes catch the rabbits and the rabbits breed. And in pure maths, we're going to see some applications during the show, including things like the number of prime numbers less than a given number.

E appears everywhere. And the second thing that's really important about the number E, and in terms of the way it was discovered, is that this very small number, is infinitely bound up with infinity. All of the ways that we define E involve infinite processes. And so to understand what the number E is, mathematicians had to get to grips with infinity. Pi was a number known to the ancient Greeks,

but E wasn't discovered in about the 17th century. Why did it take so long and how was it discovered? Why it took so long is because the maths involved was considerably more advanced them was available to the Greeks. The maths involved was infinitely bound up with rates of change and of movement and of shifts in direction and we just didn't have the maths to talk about those until quite a bit later. How it was discovered... When you say he didn't have the maths, what does that actually mean? What maths didn't you have? The Greeks were afraid of infinity in many ways.

They were afraid by concepts such as Zeno's paradox, which says that to get from here to there, I first of all have to go half the distance from here to there. Then I have to go half the distance again. That makes me three quarters. Then I have to go half again. I'm now at 7.8s and then half again. And reasoning like that, I'm never going to get to the other side of this table. So they effectively banished all notion of infinity from their mathematics

and would only consider finite processes which could be big but couldn't go on forever. The way in which E was first discovered was, in fact, thinking about quite, compound interest, which might disturb our listeners to know that infinity's bound up with compound interest. But the Swiss mathematician, Jacob Bernoulli, was interested in the problem of suppose I borrow one pound and my credit rating is poor, so there's 100% interest rate. If the interest is simply added on at the end of the year, then at the end of that year, I owe one pound plus one more pound, so I owe two pounds. But imagine instead that the interest is credited twice. So 50% gets added on after six months and 50% gets added on again at the end of the

year. Well, after six months, I owe three over two pounds or one pound fifty. At the end of the year, I owe three over two times three over two or nine over four pounds, which is two pounds 25. That's what happens if it's credited twice. If it's credited three times, then after four months, I owe one and a third pounds. After eight months, I owe four thirds squared pounds. And after a year, I now, now owe four cubed over three cube pounds, which is a little bit more than two pounds, 25. He wondered what would happen if interest was added constantly, if we allowed the number of times that the interest was added to go off to infinity. So rather than twice or three times, it's added infinitely many times.

And what he was able to show, so there's two ways you can think about it. We've got a number that looks like one plus one over N. So we started with one plus a half when it was 50%. and I'm raising it to the power n, that's the number of times I'm adding it. You could think that 1 plus 1 over N is very close to 1, and so it doesn't matter how many times I multiply it by itself. It's still going to be 1. Or you could think that it's a bit bigger than 1,

and I'm multiplying it by itself infinitely often, so it'll be infinity. What Bernoulli was able to show is that, in fact, it's equal to a number between 2 and 3, and that number is E. Thank you very much. Vicki Neal, on the way you're getting there, it sort of began to crop up in various other departments,

if I can use that word, of mathematics. In logarithms, for instance, I'm told, solely invented by the Scot John Napier took him 20 years. What happened there? That's right. So Napier was working in the late 16th century, early 17th century.

I'm sure that Colver would like us to mention that he studied us at St Andrews for a while. He did all sorts of things. things. So he spent quite a lot of time managing the family estates and in between those, as you do, he thought about maths and he thought about trying to solve problems that he could see around him with the help of mathematics. So one of the problems that he was trying to address was that arithmetic is hard. We have to remember that there were no computers, there were no pocket calculators to help. So anybody needing to do big calculations got a very large piece of paper and sat down for a

long time and did them. The astronomers of the day needed to do these big calculations. They tried to predict the motion of the planets, what they could see around them. They're needing to do big calculations. And that's sort of hard and time-consuming. And Napier thought, wouldn't it be a good idea if we could make this process easier and quicker and also more accurate? So the book he published, he wrote about this problem of arithmetic being difficult.

He also said that the slippery errors creep in. I love this phrase slippery errors, because I think we've all experienced slippery errors creeping into our calculations. I have. The thing is, multiplication's difficult. If I give you two five-digit numbers to multiply, it's going to take a long time. If I give you two five-digit numbers to add, that's going to be much quicker. Addition is much easier than multiplication. So, NAPIA's fantastic idea was, wouldn't it be a good idea if instead of multiplying numbers, we could add instead?

So he came up with this kind of idea that he called logarithms that enabled us to transform multiplication problems into addition problems, which then become much easier. What was the basic notion in constructing these logarithms? It's an extraordinary idea at the time. And first of all, what was the basis of it? Yeah, okay. So I don't really want to tell you about the finer details of exactly what Napier did.

No, please don't. No. Let me tell you about how we sort of think about it nowadays. So his idea was, multiplying is difficult, addition is easy. So one way that you can turn multiplication into addition is to write numbers as powers. So let me give you an example. If I want to do two to the power six times two to the power seven, that's quite straightforward because there are these laws of indices.

There are these rules that tell me when I've got a power of two multiplied by another power of two, I add the exponents, the numbers in the powers. So two to the six times two to the seven. The little numbers at the top, exactly. Got it. So two to the power six times two to the power seven. I add the six and the seven. The little numbers at the top, I get two to the power 13.

That was incredibly painless. I mean, that's 64 times 128. The answer is 8,192. I'm looking at the piece of paper on which I write this down because I can't do this in my head. I can't do 64 times 1228 in my head. I can do two to the power six times two to the power seven. That's precisely the point.

You have drawn attention to it beautifully, Melvin. So the plan was to write numbers as powers of a fixed number. So two to the six and two to the seven are easy, because the little numbers, the exponents are whole numbers. You have to worry about the numbers in between, but that can be done. Can you summarise why this was so useful in the age before calculators? Because Napier wanted this to be useful and obviously it was why and how. Absolutely.

So the reason that he was doing this was to address a very practical problem, which is that all sorts of people, you know, him looking after the grain on his family estates, but also astronomers, all the rest of it, are needing to do these huge calculations. And instead of having to multiply these numbers, you say, I'm going to get hold of Napier's book that he spent 20 years putting together these tables,

and I look up some numbers there and add them, and then look up the answer rather than having to multiply. So Napier had to do a phenomenal amount of work to produce the tables, but thereafter everybody was able to. to use those. June, Barry Green, how was his invention refined in the years immediately after it came up?

Because he was 20 years on it, and then a chap came, took four days to get Briggs, four days to get, Gresham Professor, four days from London to Edinburgh, and then a miracle happened, really? Well, exactly. Well, Napier's logarithms, in fact,

were logarithms, not of integers, not of whole numbers, or real numbers, but they were logarithms of signs, to begin with. And that was one of the reasons for that, The main reason for that was because, of course, the people who needed these calculations, as Vicky has mentioned, were the astronomers, people doing long distance navigation, people doing Jodacy and things like that.

And those were exactly, they were using signs. That's the angles. They were working with angles and the sign function was something that they were adding with. So Napier's logarithms were logarithms of signs. So that was one thing about them. Another thing about them was that they were not logarithms to the base 10, which is the base that people are probably most familiar with if you've ever seen a log table school. Some of us are old enough to have used them. But they were logarithms to actually very closely, approximately, to the base of 1 over E. So this number E comes up again, but NAPIA didn't know E. He didn't actually think of them in those ways. and that's perhaps one of the reasons why sometimes Napier and Napierian logarithms

are confused with natural logarithms which are logarithms. What did Briggs do? Gresh and Professor Briggs went to Edinburgh and we were told they sat opposite each other for 15 minutes without talking because they're so in awe of each other. I hope that's true. It sounds so about it.

I think it's a great story and then Briggs sort of took control. Napier said he was tired. Yes. And so Napier's logarithms and also Napeer's logarithms were rather conversational them to calculate the method that he had devised to calculate them. And included in his calculations, when you had to, when you multiplied two numbers together and the logarithms were adding them, he also had to have an additional factor, which was the

logarithm of one. Now in Napier's system, the logarithm of one was a nine-digit number. So this really complicated the computation. I'll say. And so one of the things that came out of Briggs's meeting with Napier, was the fact that they decided that it was much better if you had the logarithm of 1 being 0. So then you get rid of it in terms of the calculations. And then also you have logarithms to base 10 because you have logarithms,

the logarithm of 10 becomes 1, the logarithm of 100 becomes 2 and so on. So that really facilitated the calculations. Now Napier by this time, it was Briggs meets him, I think it's in 1615, meets him again in 1616. Napier dies in 1617, and Napier's not terribly well. So Briggs goes back, and he calculates the logarithms and produces a table himself of, actually of integers, not of signs, in 1617. And then in 1624, these get published, and they become the basis of the logarithms,

the logarithm tables that we all, you know, well, as I say, those of us who were educated in the 60s are familiar with. And so this was a huge kind of breakthrough. And you copied it all out by hand, and it's lasted 300 years. There's two summers. Yeah, I mean, it's amazing. And the number of computations were phenomenal to produce these tables. And also another thing that one has to think about, not only just writing out these numbers, then they get printed. We're getting carried away by logarithms.

I mean, you can easily get carried away from it. Or you can easily look at the logarithms and think. Right, never mind what you think when you're... Anyway, I look. Can you, where does E crop up? So E crops up because, as I've said, the Napierian logarithms were to the base E, which means that if you raise E to the power of a number,

then that is the logarithm of that number. So E to the power two, for example, equals something. Did they know, did Briggs and Napier know, about E? No. So what were they, were you calling it E? Yes. What did they call it? They didn't, they didn't call it E. So what did they call it? I don't know actually. Oh, well. So, so, so, yeah, so it didn't, it, although it, it, it actually appears for the first time, there's a table of, uh, of, uh, of algorithms that are produced in 60, a couple of years after Napiers, which are tables of logarithm to the base E, but again,

It doesn't appear as a name. So the first time it's actually named is in about 16, 68, I think, in a publication then, and it actually appears. Well, let's go on with this journey towards E. And Colver Rony Dougal again. One of the most important uses to be in a branch of mathematics called calculus. So can we talk briefly? I mean, when you're talking about Newton and Leibniz and calculus, never mind. Rates of change, calculus, very important for cannonballs and planets and everything else, right.

Where does he crop up? Okay, so yes, calculus is the maths of rates of change and of their opposite of rates of change, which is accumulation. So, for example, my rate of change of distance is my speed. My rate of change of speed is my acceleration. and conversely, my accumulation of acceleration is my speed and my accumulation of my speed is the distance that I've travelled. So that's what calculus is about. On the one hand rates of change and on the other hand, accumulation.

The way in which E first appears in this context, or the way in which we now think of E is appearing in this context, it wasn't quite seen as that initially, is that we've been talking lots about E to the power this and E to the power that. So slowly there was appearing the idea of a function called E to the power. the x and what that means is i pick a value of x and then i multiply e by itself that many times and that's what e to the x could be 11 x could be 11 and the answer then would be e to the 11 which is a

horrible messy number but anyway um it's a football too the magic thing about e to the x is if i look at what that function's doing at any point its value is the same as its rate of change at that point so if i Can you just say that again slowly? Absolutely. So if I look at say E to the naught is one, if I look at the slope of the curve of E to the X at the value naught, then it's going up one for every one that it's going along. Right. E to the one is E. If I look at the slope of the curve at one, it's going up E for every one that it's going along. So it's a magic value, Ian, that it's equal to its own rate of change when I look at the function E to the X. Now we have Leibniz and Newton arguing, not arguing, they did dispute who was the true

original idea of calculus. We'll have to leave that to one side at the moment. Leibniz was beguiled by infinite numbers. Newton was beguiled by everything to do with mathematics. Where were they? How did they take calculus to get towards E? The way in which they took calculus to get towards E was they properly understood the way in which calculus could be applied to infinite sums. So they weren't working with E to the X, the way I just explained it to you. They were working with E to the X as a big complicated infinite sum,

which I'm hoping somebody's going to get right for this show in a minute. And what they started realizing was that if you wanted to find the slope of these infinite sums, then they could do so by working term by term. So each term looks something like a little quadratic, say, X squared on two. And they knew how to find the slope of X squared on two. So what they brought in was the ability to find these slopes of things that, look complicated like E to the X. I'm going to go to Vicki Neal.

What's fascinating with these scientists? It's curiosity, isn't it? Curiosity for the sake of curiosity. It's like ours and gradi, art for the sake of art. It's just maths for the sake of maths. They aren't saying this is going to make electronics or tell us about population growth. They're just going on doing it.

Getting past that, Vicki. Let's continue about calculus. We've got two main operations, that integration and differentiation. Right. Yes. Okay, so how have I got the job of explaining differentiation integration in two minutes flat? Well, COVA's set this up. So differentiation is about finding rates of change. So that's thinking about my train journey coming to do this program today, thinking about finding my speed. So if I know how far I've travelled and I know the time it took, how far, what was my speed? I can find my average speed, but what I really want is my speed. at a particular time.

So I travelled about 60 miles in about an hour. So my average speed was about 60 miles an hour, but I wasn't doing 60 miles an hour the whole time. I got on the train, it wasn't moving for a start, that it speeded up, went faster than 60 miles an hour for a bit, slowed down for another station and so on. So this bit about finding rates of change,

finding the speed at this moment. So I can find the average speed over the course of an hour, or I can find the average speed of five minutes. I could think, well, in this five minutes, section of my journey, how far have I travelled, and then do that distance divided by five minutes, find my average speed for that. I could keep doing that in shorter and shorter sections of time. What I really want to find my speed right now is to do that in a sort of infinitely short period of time, whatever that might mean. Well, whatever that might mean is where Newton and Leibniz came in,

is making sense of that rather difficult, rather subtle notion. So they were building on work of others, but they were able to kind of pin down what that was. So in terms of a graph, that's finding the slope of the graph, the gradient of a graph at a particular point. So if I've got a straight line, it's easy to find the gradient. That's how far I've gone up, divided by how far I've gone across.

If I've got a curve, it's a little bit more subtle. That's where differentiation comes in. Then integration is kind of the reverse of that. So this is a phenomenon that crops up all over mathematics. You have a process that you use in one direction, and then you ask, can I undo that process? And actually, I suspect that we're going to touch on this

in the context of natural algorithms and the exponential functions because they're inverses of each other. So integration is the process of undoing differentiation. If instead of telling you the distance at each time and ask you the speed, that would be where differentiation comes in, if instead I tell you my speed at each point and say, can you tell how far I've travelled, that would be where integration comes in.

That's a rather brief introduction to quite subtle ideas. Very clear. I have to concentrate hard if you're me, but it's very clear. Very clear. June, Barrow Green. So the number E turns out a particular significance in calculus. Did Leibniz or Newton feel that there was this? No, your head shaking.

Right. Well, what's the particular function of E in calculus? Well, I mean, I think as Colver has already explained, really, it's the fact that when you differentiate it, you get the same function back again. So E to the X is differentiated, exponential function, differentiated is itself again. And of course, and it's not just E to the X,

any exponential function, so a natural, because E is just a number. And so any number, say like A to the X, if you differentiate that, you get itself back again times a constant. And so this number, E, when raised to the power X, and you get it back again and you differentiate it,

means that it tells, us about rates of change that are proportional. So the rate of change of E to the X is you get a proportion. So

and this is very important in particularly as some of the things that Vicky has mentioned particularly in things like population growth and so on. Can I bring Culver in here? There's a lot of exponentials of what's going on here. Can we? I'm not asking you to simplify, but I have asking you to clarify.

One of the early ways that E actually appears in calculus wasn't as E to the X. It was as the opposite, as Vicki just mentioned. It was as the natural logarithm. So people had been interested in integration. The natural logarithm. So therefore it's differentiating from another sort of logarithm. The natural logarithm means what number do I need to raise E2 to get the number I'm thinking of? So the natural log of 2 is the power that I have to raise E2 to get 2.

So that's 0.69-ish. The way in which this pops up in calculus is that people were interested in the areas under curves. This goes back to precalculus and firma. And they knew how to find the area under various curves like x squared and x and all these other different powers of x. But x to the minus one had been left as this unsolvable problem. Nobody could work out what the area was under the curve.

Y equals 1 over x. So if you picture it, it shoots up to infinity at x equals nought because we're not allowed to divide by nought, and then it slowly comes down and disappears off to nought as X goes off to infinity. So we want to find the area of the curve from x equals 1 to some arbitrary point. And what was slowly realized

was that the area under that curve is described by the natural logarithm function. People took a while to realize that it was logged to the base C. What they had was one of these infinite series expansions describing the area under the curve, and sometime after the fact, they realized that infinite series expansion

was in fact the natural log. Before we come to ELA and the E in plain view, can you tell us where we might encounter the exponential function, which has been referred to in the real world? Well, I mean mathematics is the real world, because you three, mathematics is the real world, but in the shadow world in which the rest of us live? It's everywhere.

One of the ways we encounter it might describe quite well one thing that might have puzzled some liftness. So if we think about radioactive substances, they're always described as their radioactivity in terms of a concept called a half-life, which is the amount of time it takes for half of the substance to decay. So you'll hear people talking about the half-life of uranium or whatever, and hence how long we have to look after nuclear waste for. Well, the reason why we have to talk about it in that slightly strange way

is that individual atoms decay randomly. All I can say is that they've got a certain probability of decaying in any given, say, minute. So that means when I want to describe what's going on with the decay of that substance, I need to say that the rate of change of the amount of the substance as it decays is proportional to how much of it there is. In fact, it's negatively proportional to how much of it there is because it's decaying. So when you solve an equation like the rate of change of this is proportional to itself, that means you exactly get an answer out looking like E to the something.

Fine, June, back to you. Let's now move to Euler, the mathematician I mentioned at the beginning. Can you tell us a little about him before we go into what he did? Yes. I mean, Euler is the mathematician of the 18th century, without doubt. His life spans the 18th century. He was born in 1707. He dies in 1783. And he's probably, and we always say probably because one can never be sure, but he's probably the most prolific mathematician that has ever lived.

because I think right at this moment his collected works reach something like 75 volumes, 25, and they're still publishing. And that's well over 200 years after his death. And there's just not a subject he didn't touch really in mathematics and physics. So you just name any subject, whether it's ballistics, astronomy, number theory, cartography, shipbuilding, algebra, you know, I was just extraordinary his output. And one of the things is really important about him is that he published. And he published like mad. And before him, people like Newton, for example,

is well known for being very secretive about his work and not publishing. For Euler, you know, publishing was really, you know, he wanted to get it out there. And he published in different languages. He also corresponded with lots of people. and his correspondence, we've got something like 3,000 letters in the Euler correspondence. Some of the letters are written. He changed his language in the middle of the letters.

I mean, he was just phenomenal in, I had a phenomenal memory. I mean, there's just so much about him. And he spent a lot of his time as the court mathematician in Russia. Yes, he did. He started off. He had 13 children. He did. And he had two.

He was blind halfway through his life. And his children helped him to get on with a job. And he had two wives. His house burnt down at some point. I mean, he spent his first bit of his working life in St. Petersburg, then he goes to Berlin, then he goes back to St. Petersburg. And of the things that he wrote, apart from all the masses of mathematical papers, there are a number of books, and one of them is particularly important in the context of E.

And that's the text that he wrote in 1748. Can we move a... Sorry, can we move to Vicky now? Just to share it up it. How did he investigate? the properties of E? He did all sorts of things. And as you mentioned, some of his work was really very practical, kind of dealing with very specific real world problems that were there. Some of it was playing around with mathematics for the sake of playing around with mathematics.

And my sort of hunch is that that's a rather artificial distinction that he thought it was just investigating interesting things. So he was working in this period just after people were really kind of understanding what calculus was all about, starting to explore things infinite series. So adding together infinitely many things. So if I've got 10 things on my piece of paper, I can add them together that might take me a while if I'm not very good at addition, but it's going to be a manageable thing.

If I've got infinitely many things I want to add together, that's a more daunting prospect. And it's also quite a subtle prospect because I have to be very careful to make sure that I understand what kind of answer I'm going to get out at the end. And sometimes when you add together infinitely many things, it doesn't converge. You sort of get something that that doesn't behave in a meaningful way. Sometimes you get a nice sensible answer. So one of the things that Euler did was play around with these infinite series relating to E

and to the exponential function. So the exponential function is this function E to the X that Culver has talked about already. So very closely bound up with properties with E. So it turns out, for example, that you can write E as an infinite series. So you can write this decimal that we sort of find slightly hard in this two point. seven something and I can never remember what comes after the seven because it's kind of complicated and unpredictable. But there's this beautiful series, this sum of infinitely many things I can add together to get E and it's beautiful because it's very predictable. So it's one plus one over one,

plus one over two, plus one over six, plus one over 24 and so on. And the denominators there, the numbers on the bottom of those fractions form a very nice series. It's very, sequence, it's very easy to write down. So Euler played around with a number of things related to this, just kind of testing out. What can I find here? So I've just told you about a series for E. There's also a series for E to the X. And one of the things that he did was play around with that

and see what he could come up with there. Colver, he also known for something known as the Euler identity. It's been described as the most beautiful equation in mathematics. Can you tell us what that equation is? And this is where E comes into its own. is now on the map. Absolutely. So this is a justly famous equation.

I'll say it slowly because it features five of the most important numbers in mathematics. So the equation is E to the power of I times pi plus one is zero. So I should tell our listeners what the various bits of that mean. And I'll also, if we've got time, briefly say where it came from. So I is the square root of negative 1. Now, obviously there is no number that if you multiply it by itself, you get minus 1 because if you multiply a number by itself,

you get a positive number. But a little bit before Euler, mathematicians had said, well, let's just pretend. Let's just call I the square root of negative 1. Let's imagine. It's imagine, hence imaginary number. And they discovered that maths works when you do that. The equations you write down behave themselves

and you can solve real-life problems that you couldn't solve without putting this imaginary number, the square root of minus 1 around. and they realized that you could represent numbers like the square root of minus 1 by instead of thinking about the number line, let's imagine the number plane. So everything's gone two-dimensional. So I've got I-2-I-3-I at right angles to the real numbers 1-2-3-4. We think of 1-2-3-4 going off to infinity along the x-axis of our graph

and the complex numbers i-2-I-3-I going off to infinity on the y-axis of our graph. that's what the i bit is. Pi we've mentioned already, that's coming from circles. So in this context, it's a measure of an angle. Rather than 360 degrees for a circle, we're going to say how much of the radius of the circumference of a circle it is, so pi's a half turn. So what e to the i pi plus one equals nought is telling us is that if you take e to the power i pi, then you do 180 degree twist and you're now at minus one so minus one on the x axis.

June Baragin, what makes the the Euler identity is such an intriguing piece of mathematics besides the fact that it uses the five most important numbers? Well, it also uses three of the basic operations. So it uses the multiplication, addition

and an exponentiation. And of course it's involving these important numbers which are not, we haven't just got numbers coming in here, as Colver's shown as we've got angles coming in as well. So from a mathematical point of view, it contains all the things that mathematicians use in various different fields all the time.

And it's just encapsulated in this really short equation that you can just write down. And as you've mentioned, I think probably any mathematician you ask, it will come up in their list of one of their sort of favorite equations for this reason because of including these five numbers and these three operations. Vicky then, I mentioned at the beginning of this programme that E is an irrational number. Can you tell us what your wrath that means?

Yes, yes. It doesn't mean that it behaves erratically and unpredictably and goes out on the town, except it does behave erratically and unpredictably in some ways. but it's sort of it's the opposite of rational and rational a rational number is one that can be written as the ratio of two whole numbers so it's a fraction so half and two thirds and minus four-fifths are all rational numbers

they're the ratios of two integers and we're somehow go through primary school and we learn about whole numbers and then we learn about negative whole numbers and then we learn about fractions and we learn how to add these and things on our experience kind of mirrors the way in which mathematicians have come across these things.

So these are all quite familiar, quite safe numbers? But then you might wonder, are there any numbers out there that can't be written as the ratio of two whole numbers that aren't a fraction? And it's sort of quite hard to imagine them. But then it turns out that there are. In fact, it turns out there are inconceivably more of these things than you can possibly imagine. Somehow our experience at primary school is entirely, entirely unrepresentative. almost every number cannot be written as a fraction is irrational. The rational ones are quite unusual.

So one very famous example of an irrational number is the square root of two. And that goes right back to the ancient Greeks. The ancient Greeks knew that square root of two is irrational. Some of them were a bit unhappy about it being irrational, which is possibly why this kind of atmosphere has arisen about these things being slightly strange things. So it turns out the E is another example of an irrational number. And that's somehow not terribly surprising because if you pick any old number, the chances are it's irrational, because there are so many more of them than there are rational numbers. The difficult bit is proving that it's irrational.

You might guess this it is because you haven't got any other information, but actually proving that it's irrational, requires quite a bit of work. So one of the properties that the irrational numbers have is that their decimal expansions are difficult to understand. So a rational number, a fraction, when you look at its decimal expansion, it's period. You get these repeating blocks. So a third is 0.333333 recurring. You might get a number that has a block of six digits that recur, but it's still recurring. A number like the squares of two or E or pi, these irrational numbers aren't like that. There is no recurring block.

They keep behaving in kind of so. So if you want to know the millionth digits of the decimal expansion, you kind of have to do some work to find that you can't just guess based on the fact that it's this repeating block. during Barrow Green it's also transcendental really getting a lot of adjectives to it a lot of baggage this year

hasn't it right okay well away you go okay well so it's part of as Vicki says it's irrational now there's a bunch of irrational numbers that have a particular

property and E is one of those numbers and what that means is that it's not the solution of an algebraic what we call an algebraic equation with rational coefficients. Now let me explain what that means. It's an equation, so let's take something like x squared minus 5x plus 6 equals 0. So we have that's an... That's the easier way around,

is it? So that's an equation. It's got rational coefficients. Vickis explained what rational means. And so if you've solved that equation, I think it should be something like x equals 2, x equals three, other solutions. Now, you can have an equation that includes irrational numbers as the solution. So, for example, x squared minus two equals zero. We get root two as a solution. So that's fine. We've picked up all these numbers like root two and so on. But are there numbers that don't fit into that scheme? And in fact, it turns out that E is one of those numbers. You cannot write down an equation as I say, an algebraic equation with rational coefficients

and which E, and for that matter, pie, is going to be a solution. Why does that make it transcendental? Is that just a word to grab your attention? Well, I mean, it's a way of describing these numbers. In fact, the word was first used by Leibniz in the 17th century, but it wasn't until the 19th century

that a French mathematician Joseph Lovil proved that these numbers existed. As Vicky has mentioned, I mean, we can think that these numbers might exist, but it's a completely different thing to prove that they exist. And Louville did that, and he didn't only do that. He found some transcendental numbers. So he didn't only prove that they existed, he actually found some. But E wasn't one of the ones he found.

It was believed that E was transcendental, but it took until 1873, I think, when Charles Hermit, a French mathematician, proved that E was actually one of these transcendental numbers. Colver, can we talk about it? There are many applications, and I like a lot of lists that are quite interested in the applications because the pure science is fascinating,

but the applications are right, we haven't got as much time as I thought. We've had a signal from through the window. Right, let's talk about the application and statistics. Very good. Well, incredibly briefly, lots of our listeners will have heard about the bell curve, the normal distribution. This will have been heard about in the context of, say, IQ. Famously, Francis Galton, who was Charles Darwin's cousin, discovered this.

This is the distribution of the chest circumferences of Scottish soldiers. It pops up all over the place when we're measuring things. If you picture that bell curve, it turns out that the equation describing that curve is a multiple of E to the mind. minus x squared on two. So this number E is popping up in the very nature of randomness itself. What other applications can we think of as we... Can I tell you about prime numbers? Please do. But my goodness, you've got to do it quickly. I'm awfully sorry. I seem to lay the sprints on you. Okay, there are loads of prime numbers in the world. There are infinitely many prime numbers. But if I want to count how many prime numbers there are up to some point,

if I want to know how many prime numbers there are up to a million or a billion or a squillion, the way to do that is the prime number theorem. One of the most celebrated theorems in number theory proved right at the end of the 19th century and the number of prime numbers up to X is X divided by the natural logarithm of X approximately. So the natural logarithm is cropping up even when we're counting prime numbers, these numbers who only factors one in themselves. It's all about whole numbers, but to count those,

we're using this natural logarithm function. Right. Now, do you have an application? I don't, but what I do want to mention, because I don't think it's been mentioned so far, is why E is called E. Oh, God, what a good way to do I mean. It's quite shy, that question. Because it is often referred to as Euler's number, and some people have thought that it was because Euler was the first

to actually publish it in his book, The Mechanica, in 1736. But in fact, OILA was a very modest man. Oliver being EU, L.A.R. Oh, yes, of course. That could be E of his name. Yes, the first letter of his name. And so, in fact, it's probably thought that he chose E because it was just the next available letter.

A, B, C and D had already been, he'd already used them up. So E was the obvious choice. So that's the answer. That's the answer for E. Any other, any variations on this? Well, have we got the whole morning? I just wondered if you're a complete, accepted June's explanation, which I do

wholly, and I think the three of you very much indeed for taking me through this, and I hope I can retain it for as long as possible. I really enjoyed that, and I hope a lot of other people did. It's good to be back. Next week we were talking about Julius Caesar, Roman general, statesman, and writer. Thank you very much for listening.

And the In Our Time podcast gets some extra time now, with a few minutes of bonus material from Melvin and his guests. It's, um, do you find, when of course you don't, because you're mathematicians, you move along mathematicians, that when you say X squared plus

thing equals zero, that people think they go. Algebra. Is that true? I'm hoping it's becoming less true. I think there is a point in school when things become a little bit more abstract and some people sort of, for whatever reason,

find it difficult to keep up at that point. Once you've had that experience of being terrified, it's very difficult to get beyond that. I think some people have that experience of being terrified by the abstracts a little bit later in that mathematical career. Yeah, I mean, if you're going to freeze when you hear the phrase x squared, then you're not going to remember that the next bit was minus 5x plus 6, so holding it all in your head is going to be a little bit more difficult.

What are you finding at the universe is that people come into mathematics? Are they daunted by the complications of the city? Except at the same time, the Euler's identity, that equation is looks. Beautiful and square. Very simple, doesn't it? Yes. I mean, I do think that there is this fear factor, which is quite, quite difficult to dislodge with, and sometimes it comes up with sort of certain words. So before they even start, you know, you say, well, we're going to be doing, you know,

some algebra or something, and just the word algebra. And, you know, that is something that's somehow in the culture a bit. I mean, we do hear it, unfortunately, too much. So sometimes it's almost the words themselves, I think, kind of, and they haven't even, they don't even know what it is, but somehow they've heard that algebra is frightening. Yeah, and if you focus on the concepts of maths and you sort of think about the intuition and unpicking what's going on,

that's what mathematicians do all the time. In a way, one of the things that I think maybe people who don't feel their mathematicians, the reason that they're struggling is that they find it very difficult to get to that stage because they're so inhibitive by the kind of baggage of the notation and the way it's phrased and so on. So some of my, you know, I've come across undergraduates

who have this where they reach university maths and it's getting beyond those kind of barriers of the language in which to articulate it, just understanding what's going on. I was finding conversations with taxi drivers, so they ask what I do and I tell them that I'm a mathematician, and there's two standard responses. One is that they always loved maths,

but life just didn't pan out correctly for them to continue and become a mathematician themselves. And the other is always to point to some incredibly painful incident that happened to them in a school math class when they were about seven or eight, and Mrs. Whatever told them off for not understanding, and then ever since they've had a block. So it's amazing,

how actually for such an abstract subject people relate quite emotionally to their own feelings of how good they are at maths and almost that sense of oh I'm terrible is stronger than it would be for almost anything else The clever people knew maths didn't they maths in physics yes those are the tough subjects

Well and also I think it's because as Colbert says I mean it's this moment you have a bad moment and if you're not rescued very very quickly then you're lost because everything else is bekeiling on top You lost another subject so just in a month Oh, here's Tom. Therefore, we stop this chat and we perhaps get a cup of tea. Yes, tea.

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