In Our Time - Pi

Episode Date: September 2, 2004

Melvyn Bragg and guests discuss the history of the most detailed number in nature. In the Bible's description of Solomon's temple it comes out as three, Archimedes calculated it to the equivalent of 14 decimal places and today's super computers have defined it with an extraordinary degree of accuracy to its first 1.4 trillion digits. It is the longest number in nature and we only need its first 32 figures to calculate the size of the known universe within the accuracy of one proton. We are talking about Pi, 3.14159 etc, the number which describes the ratio of a circle's diameter to its circumference. How has something so commonplace in nature been such a challenge for maths? And what does the oddly ubiquitous nature of Pi tell us about the hidden complexities of our world? With Robert Kaplan, co-founder of the Maths Circle at Harvard University, Eleanor Robson, Lecturer in the Department of History and Philosophy of Science at Cambridge University; and Ian Stewart, Professor of Mathematics at the University of Warwick.

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or wherever you get your pods. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio 4. I hope you enjoy the program. Hello, in the Bible's description of Solomon's Temple, it comes out as three. Archimedes in the 3rd century,

calculated it to the 3rd century BC, that is, calculated it to the equivalent of 14 decimal places. And today's supercomputers have defined it with an extraordinary degree of accuracy to its first 1.4 trillion digits, but there are more to come. It's the longest number in nature, probably the most potent, and we need only its first 32 figures to calculate the size of the known universe within the accuracy of one proton.

I'm talking about pi, 3.14159, etc., etc. The number which describes the ratio of a circle's diameter to its circumference. How has something so commonplace in nature been such a challenge for mathematics? and what does the ubiquitous nature of Pi tell us about the hidden complexities of our world today? With me to discuss Pi are Eleanor Robson, lecturer in the Department of the History and Philosophy of Science at Cambridge University,

Ian Stewart, Professor of Mathematics at Warwick University, and Professor Robert Kaplan, co-founder of the Math Circle at Harvard University and co-author of the Art of the Infinite. Robert Kaplan, to start with you, why is Pi so important? I think of it as historically the first of the slippery numbers. It means so much to us both in practical terms, finding out the area, the circumference

of a circle, but also in trying to come to grips with what numbers are. And yet, like a virus, it keeps slipping through our finest filters. It isn't commensurable with our ways of being precise. And how was it first come upon as an idea rather than a number? Well, our first bit of evidence for Pi comes up in the rind papyrus from 1600 BC, though the problem in which it appears, maybe dated back to 2000 BC, where one is asked to find the area of a circle whose diameter is 9 Ket. And by wonderful manipulations involving imposing a grid of squares on the circle, they come up with pie being something. like three and a seventh. Before we move on to Archimedes, who brought a new light to shine on it, and could be in a way they start of modern studies of it,

there's the idea that was in Vida culture, the idea of squaring the circle, which has run alongside this. Now, can you just briefly tell us why that was important? Came up as more of a religious than a mathematical problem, didn't it at first? Yes, there's a text, the Sulbosutras, which date from 800 to 20. 200 BC, in which it's necessary to make pits or altars of a circular cross-section and to have their area be precise. And one thinks of precise areas in terms of squares. One makes one's areas by little squares built up. And so they had to know precisely how to make these circular pits or

altars to have the right area. They had to know how to square the circle. So out of the same hole, you could have a square and a circle. And this was a great trial and squaring the circle. Alan Robson was one of the great problems that the Greeks set themselves, and one of the problems that they couldn't solve. In fact, centuries and centuries and centuries later, it was proved to be insoluble. But why were they so taken with this squaring of the circle?

Well, there's one historical argument that goes that the Greeks were fascinated by the idea of commensurability. That is the idea that you could draw a line and measure it against another line you drew, so that, for instance, three is commensurable with one because the line of length three goes, can be divided into three equal parts by a line of length one. So they were interested in whether you could measure the circumference of a circle

against the square drawn outside the circle, so that the circumference of the circle was touching it, and they were also concerned about whether you could transform a circle into a square of the same area. How could you do this simply by manipulating it geometrically? And this goes together with two other very famous Greek problems about whether you could double the size of a cube, just using geometrical methods,

and whether you can divide an angle into three equal parts by the same methods as well. So by this time it had become an intellectual project. Yes, exactly. It wasn't a way to describe things, so much as a way to try to arrive at things. And the man who pushed that forward most of all, I know I'm rushing. but it was Archimedes, a man from Syracuse,

who became, as it seems, obsessed by circles and cylinders and had the symbol of the circle in the cylinder inscribed on his tomb. How far did he take us forward? How far did he revolutionise the study of pyre, the understanding of this number, which seems to sort of creep through ancient civilizations and to be kind of there, but not as a way of an area of intellectual inquiry

until he comes along? Well, our community is very interesting because he takes a very different approach to Pi. He tries rather than to define it absolutely, but to pin it down with almost to sort of cage it inside and outside. So he draws a circle and then inside the circle he draws a hexagon. So if you imagine that is made up of six triangles of equal side. So the diameter of the circle and the hexagon are both, let's say, one. then the diameter of the hexagon inside the circle will be three very simply. And you can do the same with a hexagon outside the circle.

So you've got this circle trapped inside two hexagons, which are touching it. And then you can take each side of those hexagons and break them in. And let's tell people that you're drawing on the table as fast as you come with your fourth and you take each side of the hexagons inside and outside and break them in half. So you have a polygon of 12 sides inside and outside the circle still touch. And then you can calculate its perimeter. And then he did the same again and again until he had two polygons of 96 sides each, inside and outside the circle.

So they almost themselves circles. And this is as far as he decides to get. And he's doing this without algebra, without trigonometry, and with a terrible number system as well. So it's an incredible feat of mathematics. And by this means he narrows down the perimeter of the circle to between three and a sevenths

and 3 and 10.71st. That's an extraordinary feat. And he's the first to realise that Pi can't be expressed exactly, but can be narrowed down. He's almost sort of trapped it between these two big polygons. But the interesting thing really for us today about Archimede is say that he did it, and people use that for the next 14, 1600 years. You can Ptolemy used it, and they're just to floged a bit and refined it a bit,

but that he saw it as something that demanded intellectual inquiry because it was so elusive and kept being elusive and kept being lucid, this apparently simple measurement. Everybody listening to the programme just draw a circle, put a line through it and say, oh, what's the fuss? But he did drive that on. And how important do you think, Ian Stewart, that Archimedes' view was to later mathematicians think

because of philosophical breakthrough? Because he broke, the Greeks wanted straight lines and fixed points. And he was saying, this is infinite, this is fudge, This is not like we're brought up to believe that numbers should be at all. This is against the face, wasn't it? I think so. I mean, Archimedes was one of the all-time greats. He's very high on every mathematician's list.

And, in fact, on engineers and physicists list, so we all like to sort of claim him. That's right. And he did a lot of things which pushed Greek mathematics further than was normal at the time. and as Eleanor said, he not only gets some approximation to pi, but he can tell you that it very definitely is bigger than one number and less than... He tells you how accurate the approximation is,

instead of just saying, well, I think it's about 3 and 1-7th, you know, because I made a circle and wrapped a piece of string round it, measured how long it was. Much more than that, he wanted to really pin this thing, whatever it was down. And he used a technique which had been invented a little bit earlier, called exhaustion. And that is precisely, as Eleanor said,

this is the idea that you can sandwich the length or area or we would say number that you're interested in. If you can't calculate it exactly, you can often sandwich it between some set of numbers that are smaller and some set of numbers that are bigger and such that the gap between them is getting smaller and smaller and smaller and smaller.

In fact, as small as you like. And that process lies at the heart. of what later became the calculus. And nowadays often introduced in terms of slicing things up into very, very thin slices, infinitely many infinitely thin slices and adding them all together and seeing what you get. Now Archimedes is bright enough to know you can't do that. But he was also bright enough to use precisely that method himself. His great work, the one that had the diagram inscribed on his tomb,

is the sphere and cylinder. And roughly, it's, think of a tennis ball inside a tin can, so that the can fits the tennis ball exactly, and of course do this with complete mathematical precision. And then Archimedes proved, using this exhaustion method, that the area of the sphere, the area of the tennis ball, is the same as the area of the curved surface of the cylinder. And he also related the volumes of those.

And all of these are formulas these days with pi. but the crucial thing here is in order to make exhaustion work you actually have to know what the answer is you have to have some idea of what it is your sandwiching and he had to know that the pie he was talking about so to speak for the sphere was the same pie as the one you get for the circle and he discovered this by

conceptually slicing a sphere into infinitely many slices like sliced bread or like putting it through a bacon slicer and hanging all the little bits of sphere from a balance in his mind and they're saying what shape hung on the other end of the balance would exactly make this match. And we know this because an ancient manuscript was discovered, rediscovered in the Vatican Library early in the 1900s

and it contains a complete description of this non-rigorous but essential piece of Archimedes' work. So we've got an insight into the great man's mind. So it was an active imagination really as we're talking about. and Einstein said the most important thing was imagination. But the interesting thing is that already there's a divergence. People are getting on making wheels. And Ptolemy is getting on with reckoning up the universe,

even though the distances are different. It's a wonderful model that he's got with the sphere. It's working, as just like it works for all of us today. But Archimedes is obsessed. He was taught to be obsessed by Circles. He wants to know what's behind it. And it's that obsession that's driven through the study of Pi. But he took a long time, Robert Kaplan.

We've got Archimedes. Ptolemy takes it up. People keep using it. But we wait to the 15th century in India before the thing moves forward again. In 403, this Madhava, the Indian mathematician, introduced the idea of infinite series. Now, you're going to have to talk about that. Wonderful man, Madhava. He, as you said before, Melbourne, it's imagination that drives all this on, imagination and this compulsive desire to know, to pin things down precisely.

And what Madhalla does, something which James Gregory in Scotland at St. Andrews does 200 years later and then Leibniz, one of the co-inventors of calculus, is to rethink Pi in particular and numbers in general as an infinite series, if you can imagine that. He finds that pi can be given by this two times the following sum. 1 minus a third plus a fifth minus a seventh plus a ninth plus one over the next odd number

and then one over the odd number after that with the signs alternating forever if you do that forever and multiply by 4 you will get pie precisely

what an amazing act of the imagination and once that infinite series means whenever you cut it off at whatever stage, after 50 numbers, a million number, it's very, very nearly accurate, but never quite. That's right, but more and more accurate, the higher off.

But it's never been proved, yeah, even with supercomputers, macho maths, as you call them, pounding through, he tarches and that walloping the thing through, it still hasn't got there. And it never will, because pie isn't a precise number. Well, it's a precise number, but not one that our puny methods of reckoning will give us precisely. It's an infinite series.

Its decimal places go on forever without any pattern. Alan, there's the fact of the non-precision of this number. Did that become, after matter, did that become intoxicating for mathematicians rather than something to be feared? Yes, I think so, because what the discovery of the infinite series does

was completely changed the way people thought about Pi. Before it was a geometrical ratio, the ratio between the diameter and the secretion. conference of a circle. Now here it is not at all a geometrical thing but purely a numerical and arithmetical thing. It's the sum of an infinite number of numbers. It's completely changed the way you think about pie. Ian's sure,

how did that move in? Leibniz and Newton have been mentioned. How did that move into modern maths? I think it's the realisation that having observed there is no pattern in the digits of pie. having observed that it doesn't look like a fraction in whole numbers, that it's not rational, the next step for mathematicians is to prove that that's right, to prove that pi is irrational.

Right, here we go again. And that's right. And that was done, and it was done using infinite series. It was done by showing that certain interesting infinite expressions related to pi contradicted any possibility of pi being an exact fraction. Right, can you unroll that a bit? So, okay, so we, at school you're taught that pi is 22 over 7, 3 in the 7th. Yeah.

But actually that's not quite right. And it's a lot closer is 355 divided by 113. I'm glad we didn't have that at school. That's nine decimal places, and that's not quite right either. And so what a mathematician would like to do is, is there some enormous fraction which is exactly the same as pi? Or is it the case that no such fractions exist? Now the Greeks knew that there were numbers that could not be expressed as fractions.

The square root of two is the standard one, the diagonal of a square of side one. Absolutely basic geometrical object, it is not an exact fraction and they could prove it geometrically. They also knew that if you took a pentagon and looked at the star that is inscribed in... If you draw in every alternate vertex of a pentagon,

you get a star, that the lengths of the sides of that star are irrational compared to the sides of the pentagon. We now know that the cube root of two, which is what comes up in Ireland as duplicating the cube problem, is another irrational number. So these were floating around. So what's pi? Is pi rational? Is pi irrational? And it was, I think, it was a mathematician called Lambert, who proved that pi is not an exact fraction.

So if it's irrational, if it's not an exact fraction, can you just say precisely what not an exact fraction means? And then the consequences of it being discovered to be an irrational number. And is it the only one? I'm sorry to ask you three questions, but how much will join in the complication? I'll give you four answers. What's marvelous about fractions is that when you turn them into decimals,

the decimal pattern will repeat. There will be waves in it. one-seventh, for example, is 0.142-857, 142-857, and so on forever, looping in patterns of six. Now, what if you have a decimal that doesn't repeat, where there is no such pattern? It won't correspond to a fraction. And what Lambert proved in 1761 was that pi's decimal pattern

won't repeat. It is not a rational number. It is irrational. And that's very interesting. It's not fatal news for circle-squarers. It might still be possible to square a circle, even if pi is

irrational. The bad news was to come a century later. But it means that pie is, as I said, a slippery number. It's slipping between our sieve of rational numbers. Rational numbers are whole numbers that can be brought together and

there's a ratio between them that does not exist with this three point forever. Rational, i.e. a ratio, like one, seven. Was it the only irrational number? Was it the first? Was it the greatest? Was it one alone? As Ian pointed out, the first irrational that the

Greeks had come up with was the square root of two devastating discovery to the Pythagorean who thought that everything in the universe was a ratio of whole numbers or a whole number itself. No, it's not only not the first and not the only most numbers are irrational.

Okay, back. Anna Mutant, if it wasn't bad enough being irrational, a man called Linderman pronounced it by to be transcendental. Yes, now. Now, this, that is liftoff. So that's over to you now. Yes, transcendental numbers

are even more slippery than irrational. Can I just go, sorry, go back to slippery. You use slippery right at the beginning of the program. When you just say again what slippery is. I mean, it's very hard to pin them down. So you can't find nice mathematical expressions for them in terms of whole numbers. So Bob and Ian have both mentioned the square root of 2, which the Greeks knew was irrational.

But you can express the square root of 2 in terms of a very nice little equation that says X squared equals 2. and if only you solve that, that will give you the square root of two. But you can't solve it mathematically. Well, numerically, you can't get an absolute answer, but you can write a nice little neat equation that just uses X's and whole numbers to do it.

And the trouble with pi is that you can't find a nice expression that just uses powers of X and whole numbers to capture it. So you can only have infinite series that go on forever and ever. You can't write a nice little short equation that says this, using just X's and powers of X's and whole numbers that say this defines pi. Now that's what a transcendental number is, the one that does not, is not expressable in a finite, what we call a polynomial equation. Is this creeping towards, is this obsession for individual persons developing into something that the mathematical community,

if one can use that, overused word, I'm sorry about that, is beginning to take into itself and say, look, there's a lot we've, a lot we can find, out about mathematics and about the world using this? And is there a gathering of interest and force in this in this pipe? I think it's... I think mathematicians are exploring the nature of numbers and they're beginning to realise that it's a real Pandora's box and that what you think of as entirely straightforward things

that everybody must know, and you know that the area of a rectangle you multiply the two sides together and that kind of thing. And then it's that, well, what if the sides are root two and root three, that means the area should be root 6. Do we know if that's true? And it turns out to be incredibly difficult to prove this. It's way beyond anything.

To do it properly is way beyond anything you can do at school. If you thought you understood the area of rectangles, you are wrong. You know what the formula is, but you don't know why it's true. And the mathematicians of the 18th and early 19th centuries didn't know why it was true. And Pi is sitting there as a kind of peak in the mathematical landscape. It's a Mount Everest of numbers. It's sitting there as something.

extraordinarily special and unusual and you suspect all sorts of interesting things about it and the challenge to the mathematician is to climb the mountain it's to prove that you're right and so Lindemann's work is the culmination of a long series is the end of the 19th century that's right 1882 yeah 1882 and it starts with people not knowing whether

these transcendental numbers exist at all perhaps it's also a question do certain mathematical tools give you a complete grasp of things. Can you grasp numbers using rational numbers, fractions? No, Route 2 says you can't. Can you grasp numbers using solutions of polynomial equations, like Eleanor was saying? Well, root 2 satisfies a polynomial equation. But maybe there are things that don't.

And pi is prime candidate for this. And eventually, after an enormously complicated and long struggle, it turned out this is the case, that pi is transcendental. soon after it turned out that almost all numbers are transcendental. It's just very hard to pin specific ones down. The reason this is such bad news, fatal news, for circle-squarers, is this. To square a circle means to make a square that has the same area as a circle, to make it, as Eleanor said, with Euclidean tools that is a straight-edge and a compass.

Any construction with straight-edge and compass can be expressed by a polynomial equation, an equation which has X's and numbers and the numbers are whole numbers to show that pi was transcendental that it couldn't be the root of any such equation meant the circle could not be squared with Euclidean tools that didn't stop people from going on trying

and they still do they still say things like well yeah but that's an algebraic proof and my construction is geometric but let's get a hold on let's just get a hold on as we come into the last third of the program let's get a hold on where we are at the end of the 19th century We have this number which emerged because of the intellectual demands made on it by Archimedes and then by Madhavar and then the use it gave to people like Leibniz and Newton with calculus and then Lambert comes and says it's irrational and then Lemberin comes and says it's transcendental.

But it's gathering force as a method of inquiry into the deeper nature of mathematics, therefore the deeper nature of the description of the world we live in as far as you see. I'm moving towards the idea that numbers are abstract approximations. to a particular sort of truth rather than exact delineations of an outer reality? As Bob said, I think it's a question of our notation system for numbers, that the decimal system, the whole numbers and the decimal system that we're used to describe a small patch of the world of numbers.

There's a whole lot of numbers out there that we don't in our everyday world use and don't have very good notation for. So what this does is suggest that there's the big universe of other numbers out there. That old story of the drunk looking for his keys under the lamplight, because that's where the light is. But the keys are out there in the darkness. That's where most of the numbers are out there in the darkness.

Our feeble little tools of whole numbers and rationales are in the light. But pie, other numbers like E, all the crucial numbers that are so important to us that describe how the universe works, how organic things grow. these things are beyond our rational grasp, but mathematics sets itself to coming to grips with approximating two, better and better approximations to these numbers, bringing them and us with them into the light. It's very interesting that approximations too can lead you so very, very far. The approximations to pie, the 32 can describe the universe,

the approximations to, Ian, and so on, can get you very far, the intellectual determination of the problem is still elusive. In a sense, what's happening is there is a mismatch between our number notation, which is what Alan has just been saying, and the real essence of pie. The essence of pie is right back to the Greeks, it's circles, its rotations, it's spinning things round. Anything in the universe that spins around

that has rotational symmetry, we're getting a sort of deep Einstein-like structure of the physical laws and so forth. The laws of physics are symmetric. space is symmetric, and one of the important symmetors of space is you can rotate things. Whenever rotations come up, implicitly there are circles, whenever there are circles, you get pi. So as soon as you start thinking about rotations, you find pi. All over the laws of quantum mechanics, the equations of quantum mechanics, you find little pi is all over the place. So are we talking about pi as being a key way to explain the universe there, Robert?

It comes up in the things that Ian has described, it comes up. in probability all over the place. I can give you an example in probability. Let's say you have an epiphany cake with a ring in it, and you want to know what your chances are of getting the ring if the cake is sliced into 20 pieces. Say the cake has a radius of one. Well, your chances are a 20th of the area of the cake,

a 20th of pie, pie over 20. So probabilities are going to come up all over the place with pie in them, as Ian said, whenever you have rotation. And this number, this slippery number which keeps evading us, is one which we keep pursuing. Why do we keep trying to get further and further decimal approximations to it? Newton wrote to a friend of his in 1666, I am ashamed to tell you to how many figures, namely 15, I carry these calculations having no other business at this time.

It's as if we kept doing what we could do, although we want to do something else. we want to grasp its nature. This reminds me so much of the beginning of Proust's great remembrance of things past where Swan sees his beloved Odette in the crowd, disappearing in the crowd, and he follows her, he pursues her,

and she continually evades him and leads him on and leads Proust on to writing his great novel. That's Pi. Pye is our Odette, leading us on this chase through the crowd of numbers, through the crowd of phenomena, hoping to capture that elusive ratio,

that elusive number which describes to us what rotation, what circles mean. Well, I'm going to, I mean, we've got all the 30 seconds left. Usually, I would spend that thriftily, we would race through it, but if you think I'm going to interrupt an ending that goes from Pi to Proust, on the first program if I'm back to you, you're mistaken. Thank you all very much indeed, to Robert Kaplan, the insured, and Eleanor Robson, I hope you got as much out of that as I did.

Next week we're going to discuss another Odyssey, the Odyssey. So we'll see what happens there. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programmes about history, science and philosophy at BBC.com.com.com.com.