Numbers Numbers Everywhere

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Hello, I'm Robin Ins.

And I'm Brian Cox.

And welcome to the podcast version of the Infinite Monkey Cage, which contains extra material that wasn't considered good enough for the radio.

Enjoy it.

Welcome to the E to the IPi Plus 11 series of the Infinite Monkey Cage.

I'm Brian Cox, and here is someone who has no idea what I just said.

And I'm Robin Inks, and he's quite correct.

I have no idea what he just said, but hopefully by the end of today's show about mathematics, you will understand exactly what is going on.

So today we are talking about numbers.

Are numbers merely a human invention or are they so fundamental to reality that our behaviour is influenced by unconscious responses to them?

Is the universe inherently mathematical or is mathematics merely the best language we've yet found to describe reality?

So to investigate why there are more to numbers than meets the eye.

Aye.

Aye.

Square root of minus one.

Aye.

It's a pun.

Good, there we go.

So that's the aye.

One minute into the first episode of the new series, and we've already had our first visual pun for the radio there.

How this has got to ten series, I've got absolutely no idea.

Anyway, so one of our panelists has been on University Challenge.

In fact, at least one of our panelists has been on University Challenge.

So for once, we're actually going to hand over the introductions to them.

So here are our panel.

Hi, I'm Vicki Neal.

I'm a senior teaching associate in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge and Fellow of Murray Edwards College.

And disappointingly, I didn't arrive in a taxi with number 1729 this evening, but I'm hoping that's not a bad omen.

I'm Alex Bellos.

I'm the author of Alex Through the Looking Glass: How Life Reflects Numbers and Numbers Reflect Life.

And my chosen number is 224,

which is the lowest whole number not to have its own Wikipedia page.

So boring it doesn't have one, which makes it interesting.

My name is Dave Gorman.

I am a university dropout,

and the number of significance to me is two

simply because I'm a twin, and throughout my entire childhood, people go, Oh, you're a twin.

What's it like being a twin?

And you spend your entire childhood thinking, I don't know, I've got nothing else to compare it to.

We haven't had an aggressive twin on this show for quite a while.

I'm looking forward to it.

So, that is our panel.

Now, Vicky, I thought we should get the simple, easy-to-answer stuff out of the way first.

So, this is a show about mathematics.

So, what is a number?

Yeah, well, so if we take the ZF model for set theory, so you start with the empty set, that's the set that's got nothing in it, and then you take the set containing the empty set, which isn't the empty set because it's got the empty set in it, and so on, you keep building these up, and then you can use those to behave as the whole numbers, positive whole numbers, and then from that, according to the axioms of Zimeno-Frenkel set theory, you build up the whole of number theory.

Isn't that an old Abbott and Costello routine?

Are you happy with the answer, Brian, or would you like more?

I think more would make it simpler.

Well,

not more complicated.

There is a natural follow-up question, which is why is it not easy enough just to describe them as things like one and two and three?

What's the complication?

Well, the trouble is that, you know, what do you mean by two?

I don't know whether you and I mean the same thing by two.

And if we're going to prove theorems about two, we'd better agree what two is.

Dave, it's really aggressive.

That was like a spaghetti western.

This is really important stuff, right?

I really care what two is.

Okay, well, I really care what a set is.

Yes.

So your theory, you start in defining it with sets that contain nothing and then other sets that contain that.

Yes.

Well, okay, I'm going to insist that you define set first.

And then we just keep going around for whatever you ever do.

I think modern mathematics is unraveling in front of our eyes.

Yes.

So now you have to do that first answer, but changing set for a sort of collection of things.

It's not quite so snappy, is it?

It isn't, but I'm not sure.

But there is not a branch of mathematics called a sort of collection of things theory.

And maybe that's where science and mathematics have been going wrong.

The concept of phenomenology,

basically, it's the study of stuff involving stuff, and sometimes there's more stuff and less stuff in a stuff situation.

But it's very precise stuff.

Yeah, precise.

Well, we don't know.

Precise stuff.

Precise stuff dealing with precise stuff as opposed to biology, which is broader stuff.

And then kind of psychology, which is stuff all over the shop.

Well, I think we've covered everything.

We don't need to do five more parts of this series, do we?

Maybe we don't need to know what it is, we need to know what it does.

And that's important.

When you look at something, there's a distance from you, which is easy to see, and closer and closer becomes really complicated and difficult.

And that's what set theory is: it's trying to get closer and closer and closer.

And what we want, we want to start, we actually want to get further away because we want to sort of use numbers to do things.

So, mathematics is trying to make simple things very difficult.

Trying.

I'm so glad that our producer said, make sure you don't go off on a tangent within the first three minutes.

Well, can we just define what a tangent is?

Dave, because I mean, you studied mathematics up to you started a degree course in mathematics.

Yeah.

And we were talking earlier about the fact that this is exactly the thing where you get to a point for some people, it's quite early on in their school age, where you just feel that mathematics is not for you, that you don't have a mathematical mind.

And I wonder, for instance, for you, why did you not complete your degree course?

What was it that you suddenly went, there seemed to be stumbling blocks?

Well, I discovered when I went to university that the real reason I'd gone to university was as my easiest way of leaving home

and I'd achieved it by turning up.

Did you go to university thinking that I want to be a mathematician?

What was it about mathematics that interested you?

I can tell you a moment at which I decided I really liked it, and it's to do with a really good teacher.

I was in high school, and there was a problem on the blackboard that was completely incomprehensible to everyone in the class.

And there were three potential answers, and I knew that it had to be an even number.

That's all I knew, but I didn't know anything else.

And two of the answers were even, and one of them was odd.

And the teacher said, Does anyone know?

And a kid in the class put his hand up and he said, What is it then, Simpson?

And he answered it, and he said, No, that's wrong.

And he'd eliminated one of the even numbers.

And I put my hand up and he said, What is it?

And I said, It's twelve, sir.

And he said, Very good, very good.

How did you work it out?

And I said, Well, it was either twelve or thirty-six, and you've just told Simpson it wasn't thirty-six.

And instead of giving me a clip round the ear and telling me I was being cheeky, he said, Brilliant.

That is exactly how a mathematician thinks.

Never do more work than you have to.

And that is, that is what maths is.

Maths is about trying to make life easier for yourself by finding patterns and shortcuts and working out a rule that applies in all situations instead of having to always add it up from the start and always do all the work.

And that really appealed to me, which is why when it got to university and it was actually hard work, it sort of lost some of its charm.

And there's an element of playfulness in mathematics.

Right, I think one of the reasons that people switch off is when they stop being allowed to play around with mathematical ideas.

I think playing around with mathematical ideas is doing mathematics.

Doing mathematics isn't about routinely solving quadratic equations or differentiating stuff, it's about playing around and seeing what you can discover.

And the longer people have the opportunity to do that at school, the longer they're doing real mathematics and the longer they're excited.

I mean, I sort of work on the assumption that everybody can do mathematics, can think mathematically.

It's just sometimes people lose sight of that because they're being made to do quadratic equations or long division with big numbers or something.

And you mentioned it in your introduction, actually, the taxi number.

What was it that you did?

That's right, 1729.

So why?

So the story is that this was back in the 20s or 30s.

The great Cambridge mathematician Hardy was visiting a colleague in hospital.

His colleague was this Indian mathematician, Ramanujan, who was a kind of prodigy who was largely self-taught, who'd managed to get himself invited over to Cambridge by sending Hardy a manuscript with all these kind of fantastic calculations, observations he came up with, some of which were known to mathematics and some of which weren't, and some of which he said had been given to him by a goddess in a dream, all of this.

He became very ill, he was still quite young, late 20s, early 30s, became very ill.

Hardy was visiting him in hospital, and it's not clear that Hardy was great on small talk.

So he arrived and wanted to talk about maths.

And he said, Oh, well, I came in taxi number 1729.

I hope that's not a bad omen because it seems like a very boring number.

And the story is that quick as a flash, your manager said, No, no, Hardy, it's a very interesting number.

It's the smallest number that can be written as the sum of two cubes in two different ways.

And he had some kind of understanding of the whole numbers in a way that very few people do.

He had some kind of intuitive feeling for them, I think, somehow.

Well, actually, Alex, you emphasise in your book actually that people do have a feeling for numbers, people have favorite numbers,

superstitions about numbers.

They do.

I mean, numbers essentially, you know, if we're excluding set theory, are abstract ideas that signify quantity and order, and that's what we use them for.

Incredibly objective things that don't have personality, but they

we're human, we understand them through culture, through words, through language, and we do have these feelings and emotional reactions to them.

And for example, there are certain numbers that people prefer than other numbers.

The most popular reason for having a favorite number is it's the day that you're born.

Yet, if you're born on the 30th of a month, you're not going to choose 30.

But if you're born on the 7th of the month, you're probably going to choose 7.

So some numbers just feel a bit more interesting.

And actually, if you think about it, what are the numbers just throughout history that are more mystical, more spiritual, that seem to have more meaning?

They're all low primes.

3, 5, 7, 13.

And, you know, you think, well, this is silly, maybe this is not science.

But actually, there is science there.

For example, seven is the world's favorite number.

It's the one that most people come up with.

Seven is also the answer.

And this is something which has been done in lab conditions.

Think of a number between one and ten off the top of your head.

Most people say seven.

So it's both the number that when you're, what's the number you love the most, and the number just off the top of your head, it's the same number.

And why is that?

It must be to do with the arithmetical properties of seven.

So when we're thinking of a number at the top of our head, we're thinking, well, it can't be one, that's just too obvious.

Well, it can't be ten.

Well, it can't be five, that's halfway.

We're like doing the five times table, the two times table.

It can't be two, that's just too boring.

You eliminate two, four, six, eight.

And basically, what you're left with, you're left with seven, because seven is the only number that you can't multiply or divide within the first ten numbers.

So actually,

without realizing it, that spontaneous thought of the number at the top of your head, you're actually doing all the math you learn at school, basically.

But but what happens if you ask people to choose a number between 200 and 300, where they maybe don't know which numbers are prime and which aren't?

When they do experiments like that, there's usually a seven there.

So between one and twenty, it will be seventeen.

And actually, there's a famous magic trick which has also been tested in the lab, so to speak.

And this is, I can read your mind.

I know you're thinking of a number.

You're thinking of a number.

The number is between one and fifty.

You're thinking of thirty-seven.

Most people think of thirty-seven.

And so many people think of it as thirty-seven that magicians can actually use it to read people's minds.

So there's some theorist psychology, and the psychology is linked to the properties of numbers.

I love that idea of this musical earth, the songs sound great, everything, but it's just not working.

It's six brides for six brothers.

I've got an idea, really.

And then it's just kind of

true.

But numbers have tell these stories that we don't talk about very much, but they really do.

And brands use it also.

WD-40, would it be as popular or as successful?

WD-41?

I mean, it might have done, you don't know, because there weren't weren't two products and you couldn't choose.

But there's something about 40 that feels much better.

41 feels, you know, it's prime,

it feels different.

You know, the answer to the life of the US and everything being 40.

I tried to test that theory by launching a product called WD41, and they sued me.

So

is that true?

I don't think they're prepared to take the risk, Alex.

I really don't.

But there is some interesting psychology research on branding, and it turns out that people are more likely to buy and spend more money on household products with an even number in them.

So, one of the experiments was: would you prefer the shampoo zinc 24 or zinc 31?

Zinc 24 every time.

It's just a hypothesis.

Shampoo, yeah.

Called zinc.

And the zinc bits pop me off a lot.

I'm not buying either of them.

Well, zinc may make this some special something as zinc.

Do we have a chemist?

Oh, it's one of those ones where they go, if we put in something from the periodic table, people think it has properties that are magic.

Exactly, but it's not that.

you're right about chemistry.

The periodic table is not the table that we should be talking about, it's the times tables.

Because the argument is we are very familiar with numbers that appear in the times tables because we spend years and years learning them, and we just say them all the time.

We never say, we never actually process prime numbers because they're not obviously in any of the times tables.

Prime numbers, numbers that are only divisible by themselves, and one, by definition, not in the times tables.

So, what we do is when we see a product with a number that is in a times table, it's easy to process.

We remember it and and we misattribute that familiarity with a liking and their feeling of I like this product.

Actually, we're talking about prime numbers there.

You mentioned them several times.

It's worth, Vicky, perhaps, just giving us a brief introduction to prime numbers, which are so important in mathematics.

Yeah, they are really important.

So they're partly important because they're really interesting in and of themselves, and it turns out to be surprisingly easy to ask attractive kind of questions about prime numbers that turn out to be really hard, but they're also the building blocks from which all the other numbers are made.

So if you pick any number, you can write it as a product of prime numbers.

So, you know, 24 you can write as two times two times two times three.

But what's really important about that is not just that I can do that, but that there's only one way of doing it.

So it turns out one of the things that's really fundamental about prime numbers is that there's only one prime factorization for each number.

That uniqueness is what makes all sorts of properties of whole numbers tick.

It's somehow really fundamental to mathematics.

It's so fundamental.

It's called the fundamental theorem of arithmetic.

It's what makes number theory work, in a sense.

So partly they're important because if you understand primes, then you understand other things.

And partly they're important just because they're really intriguing and quite difficult and quite complicated to understand themselves.

And they have, of course, a long history, as you say.

So

Euclid's famous proof that there are an infinite number of primes is one of the first and beautifully easy proofs to state, isn't it?

Yeah.

I challenge you to state it on the radio.

So Euclid said, let's do a thought experiment.

He said it in Greek.

Let's do a thought experiment.

I think you should do it.

This is radio four.

I think you should do it in the original Greek.

Fortunately, I am fluent in ancient Greek.

Well, he also did it using geometry.

Yeah.

So M.

Gotta was a very good idea.

You know, there's a small part of me that's very impressed at him being bilingual.

He knew if he wanted to get his book published, he needed the English translation ready.

So he said, let's do this thought experiment.

We think there are infinitely many primes.

So let's imagine we're in some terrible kind of other universe, which Brian obviously understands in a way I don't, where there are only finitely many primes.

So there are only finitely many of them, so I can write all the primes in a list.

So here are all the prime numbers in the world.

Because there are only finitely many of them, I can multiply them all together and get some very big number.

Who's knowing what it is?

This is just a thought experiment.

And then Euclid's fantastic idea was: not just let's multiply together all the primes, but now let's add one.

So this is some very big number, who knows what it is.

It's a very big number, so it must have a prime factor.

Either because it's prime itself or because it's divisible by a smaller prime.

So could this number be divisible by 2?

Well, no, because 2 is in my list of primes.

So, my number is 2 times some stuff plus 1.

So, it needs remain to 1 when I divide by 2.

So, 2 is its prime factor.

Could its prime factor be 3?

Well, no, because it's 3 times some stuff plus 1.

It needs to remain to 1.

So, could it be divisible by any of the primes on the list?

No.

But at the same time, it has a prime factor.

So, then we all feel slightly ill,

which officially we say is a contradiction.

So, that kind of awful, ghastly alternative universe can't exist.

So, there must be infinitely many primes.

Beautiful.

Belated applause for Euclid.

Yeah, exactly.

The guy deserves some credit.

And the primes, to this day, many theorems that are yet to be proved about primes.

In particular, the theorem about pairs of primes.

Well, it's not a theorem because we haven't proved it, it's a conjecture.

A conjecture.

Yeah, yeah.

So Fermat's Last Theorem, yeah, it wasn't a theorem for a long time, despite it being called Fermat's Last Theorem.

Mathematician's very bad at naming things.

So yeah, the twin primes conjecture says that there are infinitely many pairs of primes that differ by two.

So 11 and 13, or 41 and 43, or 107 and 109.

So pairs of primes, the gap is just two.

We know there are infinitely many primes.

That was Euclid's thing just now.

So the conjecture is that there are infinitely many pairs of primes that differ by two.

Euclid proved there are infinitely many primes 2,000 years ago.

How hard can it be to show there are infinitely many pairs of primes that differ by two?

Well, very nobody's done it yet.

What's the practicality of that?

I mean, this is right.

This is

why I know that this is the intriguing thing is we

make that sound as if you're thinking, well, I'll give it a go, but I'd like to know

if it's going to be useful

at the end.

You know,

will it help me mend my lawnmower?

You'll have contributed to human knowledge.

So, is there a practical application to proving the twin primes conjecture?

Not immediately, but who cares?

That's not the point.

Nobody who's working on it is is hoping that by doing so they're going to cure cancer or change the world in some way.

They're doing it because it's such a simple question to state.

It's such a natural thing to wonder.

It's somehow so fundamental, and yet we don't understand it, and wouldn't it be great if we did?

But there are applications of prime numbers.

So when you're using your credit card shopping online,

you kind of don't want somebody else to be able to read your credit card details.

The cryptography that's keeping that secure is based on number theory.

It's based based on some fairly fundamental properties of prime numbers.

It's work that goes back to Fermat, who was working in the 17th century, and Euler working a little bit later.

They weren't studying it because they were hoping to keep your credit card details secure when you were shopping online.

They were doing it because they just thought this is fantastically interesting.

Fermat wasn't even a professional mathematician.

He was a lawyer.

He did this as his spare time.

They were just excited by prime numbers and trying to understand what's going on.

What are the structures here?

And what can we prove?

300 years later, it found an application.

So, you know, I'm excited by this just because, in and of itself, who knows which bits might have an application in 50 years' time or 300 years' time.

Some of it might not, but at least we'll have understood it a bit better.

Why the impatience and wanting the application right now?

There are so many examples of inventions or mathematical discoveries that hundreds, if not thousands of years later, have been crucial, pivotal for massive discoveries.

For example, Apollonius, who wrote about the conic sections, and he made a point of saying, I'm writing about these and studying these purely for their own beauty.

If it wasn't for Apollonius and the conic sections, Kepler wouldn't have worked out that planets orbit in ellipsis, and Galileo wouldn't have worked out that projectiles fall in parabolas.

And this is,

you know, 2,000 years before.

And we mentioned complex numbers, or i, in the introduction, which is absolutely fundamental to quantum theory.

It'd be very difficult, actually, to do quantum theory without complex numbers.

It might be worth exploring a little bit, actually, if you'd explain briefly what complex numbers are.

A complex number, well, first we need to say what an imaginary number is.

An imaginary number is a square root of a negative number, which is quite hard to get your head around.

And i is the square root of minus one.

And

a complex number is a number that has two parts.

One is a normal number, and one is a multiple of i.

That is a kind of layman's explanation.

And what makes the complex numbers so interesting is that in the same way that when we understand, now when we learn positive and negative numbers,

negative numbers are really quite recent, only a few hundred years old that they were completely accepted.

But what the reason why we understand effortlessly negative numbers is that we know the number line, which can be a line, which in one direction can go positive and the other negative.

The way we can imagine visually complex numbers on a complex plane, and the complex numbers give a wonderful language for expressing rotation.

And I think that I'm not the particle physicist, but I believe that that is what is really useful in quantum mechanics: in explaining, basically, giving a language to explain

waves and rotations.

And if it wasn't for your complex numbers, we wouldn't be able to do it.

What worries me is that there was a point in history where no one really could deal with the concept of negative numbers, and then everyone kind of got comfortable with negative numbers, and that gave us debt.

It was the other way around.

I'm just worried that complex numbers are just just going to give us another form of debt.

It was the other way around.

One of the reasons why negative numbers weren't invented was because there was no application.

So the Greeks didn't have negative numbers because they saw all maths as geometrical, as visual.

It was the Indians who realized, oh, we need to have a language to talk about debts and assets.

And, oh, yeah.

Things can actually exist and be negative.

And it was that application which really drove right at the beginning.

And the very first person, Brahma Gupta, one and a half thousand years ago, who wrote the kind of the laws of arithmetic, wrote it in terms of debts and assets.

So, yeah, it's right.

It's right.

Yes, well, I don't want to owe anyone 3i.

That's not helping.

This idea that mathematics is developed because we're interested, perhaps out of playfulness or generalizing certain contexts, negative numbers, imaginary numbers, complex numbers.

But then they find an application in physics many centuries later, perhaps.

It seems to suggest that mathematics is out there in a sense.

You almost get the sense it's waiting to be discovered.

And this dates back to Plato, this idea that mathematics is a thing to be discovered rather than a human invention.

Yeah, there's a kind of abstract imaginary realm, but this realm really does exist.

And I think that mathematicians, the kind of romantics mathematicians, like to think that it's really out there.

So that when you're doing your math, you're somehow kind of exploring this world rather than just like creating the stuff in your own brain and it doesn't exist anywhere else.

Do you think you can imagine a reasonably advanced civilization that didn't have numbers and mathematics?

No.

Because that's what I the you know, we see in certain kind of parts of humanity where there is a you know certain groups that have a limited

number, you know, numbers, there's basically kind of one, two, three, and then more.

And that's but you believe that underpinning a civilization, if it gets to a a certain level of complexity, mathematics, the nature of numbers, is required.

Yeah, I'd thought that all scientific advances have come from mathematical advances right from the beginning.

You know, numbers in themselves are a massive advance, and then you have the concept of zero, which gives us a number system which we can actually do science with, negative numbers, you know, the idea of the curvature of space, which gives us relativity.

Yeah, you really need this math to, you know, if it's going to be any kind of advanced civilization, they're going to need to do.

I mean, they may be doing math in a different way, they may have different base systems, so they may, you know,

they've got four fingers, not five fingers, so they count in base eight.

It's still the same mathematics, it's still the same platonic realm.

And there does seem to be,

it seems to be the case that mathematics is the language of science.

Certainly, when you talk about theoretical physics, you mentioned Einstein's theory of general relativity there.

It's the language of curvature, the language of geometry.

So, how would you speculate about the deep reasons for that?

Is it possible to think about why there may be deep reasons that our universe appears to be mathematical?

I think it was Galileo who said, the language of nature is written in mathematics.

So, the question, the simple question, is why, Alex?

And I would say, why do we need to know?

As long as we can use it.

Mathematicians are very cagey, aren't you?

I like this.

So, I'm a pure mathematician.

Mathematics is the kind of pure, clean, fundamental thing to be discovered.

The universe is kind of complicated and fiddly and hard to understand.

Mathematics is the kind of thing that's really out there.

I'm not sure the universe exists, I'm sure mathematics exists.

It is interesting with them.

You have to, in your next series, go, the universe is fiddly.

That has to be said at some point.

If you explain that, because I also, in a programme I made, actually, the great mathematician Richard Borchardt, very famous, won't the Fields Medal, I think, the Nobel Prize of Mathematics, said that he thinks that mathematics is more real than the universe because you can imagine many different universes but only one mathematics.

The sense in which you mean it, there's really you feel there's one fundamental mathematical truth there.

There's one mathematical truth.

When something is true in mathematics, it's true for all time everywhere.

So I think that gives this idea that it's kind of eternal.

Right, so Einstein came along and

said, well, this guy Newton, what did he know?

Nobody has come along and said, this guy Euclid, what did he know about primes?

2,000 years later, we are still completely sure that there are infinitely many primes.

Yeah.

So it's a very static.

I find that very exciting.

Very static, exactly.

That's why we've all finished mathematics, we've packed our bags and gone home.

Yeah, that's the annoying thing.

That's the thing you did so well a few thousand years ago, and now you're still dragging it out.

And it's just.

And we still don't understand the twin primes conjecture yet.

Well, it's interesting just to backtrack a moment, because I think it sounds like it's perhaps nonsensical and not much progress was made, but there's very rapid progress being made in the twin primes conjecture, isn't it?

Yeah, exactly.

So this question that's been around for, actually, I don't know how long, I haven't been able to find out how long this question has been around for, but it's such a natural question that you can imagine this having been around potentially back to Euclid.

In the last twelve months, there's been dramatic kind of progress on understanding this problem.

So, this is the problem of showing there are infinitely many pairs of primes that differ by two.

And

almost exactly a year ago, almost to the day,

a mathematician called Zhang, working in the States, put out a paper showing that there are infinitely many pairs pairs of primes that differ by at most

70 million.

You should have done.

He's hit the crossbars.

I have done a theory that gets it down to

noted numbers, 69 million.

Well, it's funny you should say that.

because this paper came out,

and 70 million is a really big number, right, when your target is two.

But on the other hand, it's a whole lot smaller than infinity, which was the best we knew before that.

So he put out his paper online.

Lots of mathematicians poured over it, checking it.

But then, via blogs and wikis, there was this very kind of public project where mathematicians were getting together trying to say, well, can we do better than 70 million?

Because there are all sorts of points in the argument where if you worked a little bit harder, you could get a better number.

So last summer, from sort of May through to about July, there was was this kind of project with an online league table or a wiki.

It's all online, you can go and look, it's completely public.

So you kind of have to keep checking back every day because this number's going down and down, and everybody's thinking, well, is it going to get to two or not?

And then it's sort of progress dried up a bit in the summer when mathematicians had shown there are infinitely many pairs of primes that differ by at most 4,680, which is one of those numbers that seems small when you compare it with 70 million rather than two.

And then progress sort of stopped for a bit until a young postdoc called Maynard came along along, and a few months later in the autumn showed there are infinitely many pairs of primes that differ by at most 600.

It was going down and down, and then this internet project resumes, and

news keeps changing.

So I checked earlier today to make sure that I was up to date.

The best known at the moment is that there are infinitely many pairs of primes that differ by 246.

It's almost that uninteresting number you have.

24.

224, was it?

We get to 224.

It gets its own Wikipedia page, though, right?

So I'll move up to 225.

Well, what I find interesting about this, it sounds joyful.

It sounds like it's absolutely a cultural pursuit.

There's often a discussion about the two cultures and how there are scientists, mathematicians, and artists who do something for aesthetic reasons.

But mathematics is surely in that sense.

Mathematics for aesthetic reasons too.

Yes, yes.

Well, that's what I wonder, Dave.

When we've had mathematicians on in the past, and as someone who, you know, there was a point where I just found mathematics very hard, you get this sense of a tremendous adventure, which is not necessarily instilled in you through the education system, of saying, you know, when listening to you there, going through it, you think that sounds really exciting.

You're getting these things in from all around the world.

You know, there's actually, you know, the best maths film we've had so far is probably Goodwill Hunting.

Oh, look, the cleaner's good at maths, it turns out.

You know, one of those things.

Pie, actually, by Darren Aronofsky is probably more interesting, but that adventure.

Now, do you think if you'd had that sense of the adventure in numbers, you might have, you know, made it all the way to the third year and not wasted all that grant?

No.

I think there is

really elegant things.

And I remember having this conversation when I was 18, 19 with friends who were studying other things, and they would try to understand why you like mathematics.

And

I forget who this is, it's probably Euler, but I might be completely wrong.

The thing about adding up one to a hundred.

Gauss.

Gauss, thank you.

Okay.

So most people, if you're given the task of adding up the numbers from one to a hundred, most people would start by going one plus two is three, plus three is six, plus four is ten, plus five.

And that's a really hard work way of solving the problem.

But if you put them in two rows of uh one to fifty

and uh and then count backwards the other way with a hundred to fifty-one on top of each other, well one and a hundred add up to one hundred and one, and two and ninety-nine add up to a hundred and one, and three and uh ninety eight add up to one hundred and one, and so on all the way down, so you've got fifty and fifty-one on top of each other, and they're up to a hundred and one as well.

So actually, you've got fifty times a 101.

That's a really beautiful thing for someone to have thought, oh, I can make this easier.

That's really elegant.

That's really exciting thought.

That's the, that's easy.

I didn't have to do all that work.

That's what it always was for me.

That was always the thing that was exciting.

And hearing those little stories behind things like that are the things that turn you on to it rather than just being taught by rote, rather than just being taught, this is a secret someone else discovered, don't worry about where it came from, just learn it.

That's what makes it feel like hard work.

And actually, I think the kind of idea that savants exist kind of put people off.

Maths and music are the two things where you'll hear about sort of genius 12-year-olds, and there's the guy who can take that taxi number and say it's this many primes multiplied and whatever.

Those people, you go, oh, somebody can do it without doing any work.

I'm not one of them.

Oh, I won't bother then.

See, that's what I love: is the difference between Vicky and you.

So, Vicky, when you speak, you talk about the adventure of maths, how you were drawn into numbers, and you speak about the idea of maths meaning you can be in a hammock earlier having a snooze.

That's the great divide between Vicky.

But the great thing is that, you know, once Gauss or Dave or whoever has, you know, come up with this clever way of adding the numbers, then you can generalise it.

Then you can do that for all sorts.

If I want to add the numbers from one to a million instead, now I've got a plan for doing that.

So, all of those things all at once.

It's all about looking for opportunities to generalise and understand the structure of what's going on.

If I just add them on my calculator, I don't learn anything about what's going on.

If I do it that way, then I can kind of see how it fits into a bigger picture and understand what's going on.

That's really exciting.

Yes, absolutely.

Dave raised an important point there, I think, because I think that, and Robin had raised earlier, that at some point through our education, many people get turned off mathematics.

And you hear many people say, I just can't do it.

It was trigonometry that did it for me, or whatever it is.

But do you think that there really are, there obviously are people who are absolutely brilliant, as there are in music, like Beethoven or something you can't really understand.

But I want to ask you both, actually, did you always find mathematics easy, or did you find you had to work at it but you were interested?

What's the balance for you?

I start with Vicky, perhaps, between practice and just natural talent?

I've always liked maths, but because I've always liked maths, I've always liked doing maths, and that's quite a good way to get better at doing something.

So I'm sure that some people are fantastic at the piano because they're fantastic at the piano, but I think a lot of people are fantastic at the piano because they spend a lot of time playing the piano.

So, I've spent a lot of time thinking about hard maths problems, and no, I don't always find maths easy because I don't want to go and solve easy maths problems, because what's the point of that?

I want to find problems that are just a little bit outside what I can do at the moment, because then I can try to understand those.

I'm not interested in the things that I already know how to do.

I don't want to repetitively do things I can already do.

I want to build on that and find hard problems.

Not too hard, because that's terrifying, but a bit harder.

I think it's really exciting being on on sort of the boundary of what's understandable and just sort of playing with that.

And once you kind of break through and you understand something, I mean, it's so satisfying.

I mean, it really you get a kick out of it.

Well, one of the things you talk about in your book, I mean, there's lots of things

in your book which are fascinating about why we're drawn towards things that are £7.99 more than £8,

all of these different kinds of ways that we've used.

But Benford's Law was the thing that I found particularly intriguing, which is this fascinating thing.

I'll ask you first of all what Benford's Law is, and then the fact that it also can be used by kind of, you know, maths cops, basically.

That it's a way of finding scurrilous individuals who are involved in financial skullduggery.

Benford's Law is the law that there are more ones in the world than twos, and more twos than threes, and there are less nines than anything else.

And in its simplest form, let's just talk about the first digit of numbers.

So in a million, that would be a one, in twenty-three, that would be two.

In 0.005, it would be 5.

The first digit,

the leftmost digit.

If you were to look in most random data sets, so all the numbers that are in a newspaper, all the numbers that are in your bank account, all the numbers that are in an atlas with populations or areas, you will find that

about

30.1% of all numbers begin with a one, about 17.6% of all numbers begin with a two all the way down, and only about four and a half percent of numbers begin with a nine.

And that is that's about a sixth as much numbers beginning with a nine as beginning with one.

That's such a huge discrepancy.

And you see this everywhere, and because it is the case, if you look at data sets that don't obey it, then you think, well, it raises a red flag.

The way that I came across this is in financial investigation, and

it is

an important tool now within financial investigation to just check that in anyone's account you have 30.1% of what's in the ledger begins with a one.

But it's not only in financial cases, in polling.

They discovered that the Iranian elections a few years ago were most probably fraudulent because they ran the Benford's test in terms of all the different ballot boxes and it wasn't the precise Benford's curve.

And so you know that

something is up.

This is quite sure about this, because if you look at this audience, for example, and you take the the first digit of their age, I reckon you get more sixes.

That's because

I reckon everyone here is over 10 and under 100.

So it doesn't work.

I'm sure it's not 70 million.

The nice thing about you explaining that is one of the hardest demographics we've found to get a break is career criminals and hopeful dictators.

And now, 20 minutes into the show, they're going, hang on, there's something for me here.

Good.

There we go.

But actually,

the the person who was the expert on Benford's law said that he was approached by people, probably criminals, just saying, Do you know where I can get some like random Benford's data?

So there is a kind of marketing, you know, proper data that will pass the Benford's test.

There is an interesting

final question we had written here.

It's absolutely clear now that the answer will be interesting.

I'll start with Dave, actually.

Given what you've just said,

the question is: the final question is: would the world be a better place if there were more mathematicians?

This is going to sound really pious, and I don't mean it to, but I think there will be a

better place with more good maths teachers.

I think

it's one thing for the people who are doing at the sort of further reaches of what we understand,

as many of them as possible, please.

Let's, you know, obviously that's really exciting and interesting, but it's but actually having more people leave school equipped to understand what the interest rate is on that debt, on that payday lender, on like that level of maths, that should be improved by teaching.

That's the thing that would be the most advantageous to most people.

Yeah, the day when it comes where you can't say, Well, I'm rubbish at sums, that's as embarrassing as saying, Well, I can't read.

That would be great to get there.

So my definition of a mathematician is somebody who does mathematics, not somebody who's paid by a university or company or something to sit somewhere proving theorems or writing paper.

I think

everybody can be a mathematician.

I sort of start from the assumption that everybody is sort of interested in maths, even if they don't know it yet.

And that the world would be a better place if more people did more mathematics more than the time.

They don't have to be academic mathematicians, you know, they're allowed to do other stuff as well.

But I want people to understand what mathematics really is, as opposed to kind of boring stuff.

And I want people to have the opportunity to play around with mathematical ideas.

I think it is tremendously interesting.

I think the world would surely be a better place of just ideas, like the prime numbers, and the fact that no matter how big the number is you write down, there's a bigger one that can't be divided by anything except itself and one.

I find it a wonderful thing, actually.

I think if more children knew that, then the world would be a better place.

Yeah, we need to tell people that math is the most creative of all disciplines because it's the one that's always creating new concepts out of nothing, and it's not just about learning a times table.

What's that moment when a child thinks of the biggest number and that excitement when they're told add one to it and they go, What?

Oh, well, then that's no, hang on, and that's you know, that's a fantastic and beautiful cage.

Is there a number so big that you can't fit it in an infinite cage?

How big is your infinite cage?

What kind of infinity do you have?

No, we're not doing that again.

No, no, no, no, no.

I refer you to series nine.

What about the smallest infinity?

The smallest infinity is the number of real numbers.

Is that the smallest infinity?

No, no, no, no, no, no, no, no, no, no, no, no, whole numbers.

Real numbers.

Whole numbers.

There are masses there are loads of whole numbers, but there are loads and loads of real numbers.

Like loads of of numbers.

So if I had a cage that was just big enough to fit the infinity of real numbers in it.

That would be huge.

That would be huge.

Yeah.

But whole numbers, then I could, it wouldn't fit the infinity.

That would just be quite big.

It wouldn't fit the infinity.

Whole numbers are not that huge because you can quite easily, there are an infinite number of infinities that are even bigger than that.

Yeah.

So the infinite cage is really

problematic.

If we were small, treating the numbers humanely, could we make it smaller?

Right, so we were, as usual, we always ask the audience a question slightly related to the show.

And today's question was: What is your least favourite number?

And can you tell us why?

And the first one I've got is 3.141592653589793238462643383279,

because it reminds me of food and I am on a diet.

I just want to check, Alex.

Is that pie?

Is that right, though?

Wait, can't be a good idea.

Did it disfrane once?

Yeah, exactly.

It says dot, dot, dot.

Yeah, so it's, yeah.

What's your least favourite number and why?

745, thought for the day.

I think Alex

might take issue.

It says one because it's not as interesting as the other numbers.

Is that really fair to one?

Well, it's true.

In the favourite number survey, certain numbers performed performed incredibly badly, and they were number one, but also all round numbers: ten, a hundred, a thousand.

No one liked them.

I think that's because they octopuses do.

Octopuses count on their tentacles, so ten is different for them.

Oh, and we had a bit of a debate about this earlier.

There's still some of the octopus experts are out on this one.

Well, sadly, octopuses

didn't enter my survey, so this is just short.

We asked eight octopuses

many, many years ago.

Um, uh, my bank sent they knew I'd moved house because my statements were an ox.

You are overdrawn.

It's an ox.

They knew I'd moved house because my statements were arriving at my new address.

But they sent my checkbook, a new checkbook, to my old address.

And luckily for me, it was a nice person who knew where I'd gone and forwarded it on.

And I wrote a letter of complaint to my bank saying, How dare you do this?

And it could have been, could have caused considerable damage.

And I ended my letter by saying, I'm charging you £15.6 for this letter.

And they paid it.

I thought, well, that's what they do.

They charge us for writing a letter.

I'm going to charge them for writing a letter.

And I chose £15.06 because I thought they'll think I've accounted for every penny.

I'm not going to say £15, because that feels arbitrary.

That feels like it's just been plucked from someone going, I was going to a nominal amount, I was like, £15.

So £15.06.

And that way, someone somewhere is going, he knows what all that's for.

And then they did it again, six months later, with something else.

And I wrote them, I charged them £15.12,

and they paid it again.

And I had a third go about a year later.

I'm just adding on six pence every time.

And

I'm charging you £15 and £18, please credit my account.

And I got a letter back saying, We no longer charge people for writing letters, so we're not going to honor this.

And I think I brought them to their knees.

When did the banking crisis begin?

A practical joke some years ago.

24, because when I say it's my age, no one believes me.

That's from Rosemary.

24, because when I say it's my age, no one believes me.

Rosemary's dad.

So,

there we go.

Thank you very much to our panelists, Dave Gorman, Vicki Neal, and Alex Belos.

And I should say that during the last series, we received some listener complaints that we were too boisterous for Radio 4 and overly excitable about particle accelerators and genome sequences.

So, we would like to apologise to anyone who's affected by this majestic vision of reality as revealed by the scientific imagination, moderated by experiment and experience.

And as a way of saying sorry, we'd just like to say it's taken 13.8 billion years of cosmic evolution for small groups of atoms to assemble themselves into conscious beings who can look up in wonder at the billions of stars in the night sky and dream about their own origins in the heart of long-dead stars.

So, wake up!

This is no time to sleep!

Goodbye.

That was the Infinite Monkey Cage podcast.

Hope you enjoyed it.

Did you spot the 15 minutes that was cut out for radio?

Hmm.

Anyway, there's a competition in itself.

What do you think?

It should be more than 15 minutes.

Shut up.

It's your fault.

You downloaded it.

Anyway, there's other scientific programmes also that you can listen to.

Yeah, there's that one with Jimmy Alkaceltzer.

Life Scientific.

His dad discovered the atomic nucleus.

That's Inside Science.

All in the Mind with Claudia Hammond.

Richard Hammers' sister.

Richard Hammond's sister.

Thank you very much, Brian.

And also Frontiers, a selection of science documents on many, many different subjects.

These are some of the science programmes that you can listen to.

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