In Our Time - Negative Numbers

Episode Date: March 9, 2006

Melvyn Bragg and guests discuss negative numbers, a history of mystery and suspicion. In 1759 the British mathematician Francis Maseres wrote that negative numbers "darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple". Because of their dark and mysterious nature, Maseres concluded that negative numbers did not exist, as did his contemporary, William Friend. However, other mathematicians were braver. They took a leap into the unknown and decided that negative numbers could be used during calculations, as long as they had disappeared upon reaching the solution. The history of negative numbers is one of stops and starts. The trailblazers were the Chinese who by 100 BC were able to solve simultaneous equations involving negative numbers. The Ancient Greeks rejected negative numbers as absurd, by 600 AD, the Indians had written the rules for the multiplication of negative numbers and 400 years later, Arabic mathematicians realised the importance of negative debt. But it wasn't until the Renaissance that European mathematicians finally began to accept and use these perplexing numbers. Why were negative numbers considered with such suspicion? Why were they such an abstract concept? And how did they finally get accepted? With Ian Stewart , Professor of Mathematics at the University of Warwick; Colva Roney-Dougal , Lecturer in Pure Mathematics at the University of St Andrews; Raymond Flood , Lecturer in Computing Studies and Mathematics at Kellogg College, Oxford.

This BBC podcast is supported by ads outside the UK. 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 programme. Hello. In 1759, the British mathematician Francis Masseres wrote that negative numbers, quote, darken the very whole doctrine of the equations and make dark of the things

which are in their nature excessively obvious and simple. Because of their dark and mysterious nature, Maser has concluded that negative numbers didn't exist, as did his contemporary William Friend. However, other mathematicians were braver. They took a leap into the unknown and decided that negative numbers could be used during calculations as long as they'd disappeared upon reaching the solution. The trailblazers were the Chinese, who by 100 BC were able to solve simultaneous equations involving negative numbers.

The ancient Greeks rejected negative numbers as absurd, but by 600 AD the Indians had written the rules for the multiplication of negative numbers. And 400 years later, Arab mathematicians realized the importance of negative debt. But it wasn't until the Renaissance that European mathematicians finally began to accept and use these perplexing numbers. Why were negative numbers considered with such suspicion? Why were they such an abstract concept? And how did they finally get accepted?

With me to discuss negative numbers are Ian Stewart, Professor of Mathematics at the University of Warwick. Colvaroni Dougal, lecture in pure mathematics at the University of St. Andrews, and Raymond Flood, lecturer in computing studies and mathematics at Keller College, Oxford. Ian Stewart, what are negative numbers? They're pretty straightforward from the mathematical point of view. They're just like positive numbers, but they point in the opposite direction. So as well as 1, 2, 3, 4, we've got minus 1, minus 2, minus 3, minus 4.

And when you add a negative number, it's like subtracting the positive number. So on that level, they're perfectly reasonable things. are various mathematical rules for how you add them, multiply them, subtract them, and it all fits together very, very prettily. It's when you start thinking about relations to the real world that it all starts to get a little bit pear-shaped, because sometimes negative numbers make sense, and sometimes they don't. For example, if I've got five cows and somebody takes away eight cows, I don't actually end up with minus three cows. It's just they can't take away eight. I haven't got them. So this is the sense in which a lot of cultures

said negative numbers are silly. But if I go into a bank and I've got five pounds in my account and the bank lets me take eight pounds out, which it might do, it'll probably charge me interest these days, but it might do that, then there's now minus three pounds in my account. It's not zero, it is possible, but I owe the bank rather than the bank owing me.

So the change of sign from plus to minus reflects a change in direction of the way in which the money moves and who owns, which, you know, whether it's a debt or whether it's an actual sum I can spend. I read it. It was the Chinese in 200, BC, who came, arrived at the notion, or as far as we know, the first arrived at the notion of negative numbers, was that to do with money as well?

It's quite astonishing what the Chinese were doing in 200 BC. Money was certainly part of it. They had a system for calculating which used coloured sticks. Think of it as a sort of, it's almost like an abacus. squared board and if you put a stick in a particular square that meant one and if you put that stick to the square next to it that meant 10 and so on and they had red sticks for positive numbers and black sticks for negative numbers this always confuses me because when I was growing up my father worked in a bank and of course it was red ink for debts and if you're overdrawn you

got a red number on your account and if you were in credit you got a black one so going into the red was the wrong thing to do. But with the Chinese going into the red was actually the right thing to do. But they did quite sophisticated calculations just by moving these sticks around. But the way the red and black sticks worked was basically that black sticks cancelled out the equivalent red sticks. If I had three red sticks and three black sticks, that was a plus three and a minus three, which was really zero. So one of their rules for moving sticks around was that if you had one of each colour, you could just remove them from the board completely. Were there any practical applications of this?

They weren't. They weren't. I don't know a lot about the practical uses, but they were using these things, among other things, for commercial calculations, tax collecting, any kind of sums. You know, if you went into the shop and bought something, the sticks might come out. Colva, Roni Doole, can I ask you about the Greeks? Unusually, the Greeks didn't take this up and run with it or hadn't invented it in the first place. What was difficulty with it. The idea of negative

numbers. In this story the Greeks are more interesting for what they didn't do almost than for what they did do. One of their problems with negative numbers was that Greek maths really was founded on geometry. Geometry was the key. So they were interested in lines, they were interested in areas, they wanted to know lengths and circles and this kind of thing. And when you're drawing

things, negative numbers literally don't arise. They also had problems because of the way they wrote down numbers. They didn't use a positional number system. like we do, where if you've got a one in the first column, it's one, and if you've got a one in a zero, one in the second column, it's ten. They use different letters to mean different numbers, which meant they never properly got to grips with zero,

and as such they could never properly get to grips with negative numbers. So Euclid, who's probably the most famous Greek mathematician, possibly the most famous mathematician of all time, wrote the elements about 300 years BC. And he's got some problems in there, which we would now see as equivalent to, quadratic equations. He's got problems which involve taking a line and then you're wanting to find the point where you need to bend it to make a rectangle of a particular area. Now, if you write that

down and draw it out, you'll find that it results in needing to solve a quadratic equation if you're trying to make some particular area rectangle. But you're only ever going to get the positive answer there because you're looking for a line length. So he's just hinting towards it. He's not writing it algebraically. He's writing it with pictures. Did the Greeks think the idea was absurd? I said in the introduction on good authority, i.e. the authority of something from your notes, one of you lots of notes. Did they think it's absurd? They did, but even the idea of it didn't occur to them until somewhat later than Euclid. We've got to go 600 years later than that to Diophantus in Alexandria,

kind of 300 years after the birth of Christ. And there he talks about an equation of the form 4 is equal to 4x plus 20. Now let's imagine what we'd do if we wanted to solve that. We'd take away 20 from both sides. one side would now become minus 16, the other side would still have the 4x there, so our final answer would be x is equal to minus 4. He doesn't even attempt to solve this equation.

He can see that it's going to lead to a negative answer. So he calls the whole equation absurd, let alone the number on its own account. How come that Hindu mathematicians in the 6th and 17th century took it up with such success, especially this man, Brahma Gupta? Did they get it directly from, I'm just interested in the way ideas travelled.

Did they get it directly from the Chinese? We have no evidence that they got it from the Chinese at all. It seems, I mean, the Hindu mathematicians had zero by then. They had quite a spiritual take on lots of their maths. They identified zero with the great god Brahman at one point and then Finacy with the god Brahma as well. But Hindu maths had always been very algebraic. They had the positional notation for numbers.

They had zero from very, very early days. They were solving quadratic equations when trying to build altars. and they wanted one layer of the altar to be half the area of the layer beneath it, so that forced them into solving quadratic equations. Brahmagups are defined zero as being what you get if you subtract any positive number from itself. And he then goes on, he talks about negative numbers as debts, positive numbers as fortunes, and zero as a cipher. And he describes what happens if you multiply or divide or add of subtract fortunes,

debts and ciphers. He gets division by zero wrong, but otherwise he's completely correct. Raymond Flood, what happens to negative numbers when we meet the great, a few centuries later when we meet the great Arab scholars? What do they do in negative numbers of zero? Well, part of that is picking up on your question about transmission, because one of the great ninth century Islamic scholars Al-Qurizmi did travel and did pick up on the Indian tradition. And what he did is he wrote two major textbooks, which have come down to us, one on arithmetic and one in what we now call algebra, and we call it algebra because of a word that was used in the title of his textbook.

But the one in arithmetic, as Colvo was saying, very much incorporates the Hindu notation. And when you think of it, it's really in a marvellous system where you've got ten characters, you've got ten digits, and you can represent any number with just ten digits. But what he's known to have done is to use the number zero as a place position, so that you can write 2106. and therefore be able to extend the range

and the simplicity of identifying numbers there. But he attributes the work that he has done to Bramogutra, so there is that transformational aspect of it going on there. And I think the other thing on your point coming back to the negative numbers, which is what you said, was this other book on algebra comes out of the word algebra in the title meaning of balancing which are a completion, which is where you add the same numbers to both sides.

of the equation. What happens there, and this relates to the point about suspicion, is that he wrote out the equation he was going to solve, but he only uses positive numbers. He was very suspicious of negative numbers because the way he went about solving these equations was to draw squares and to complete squares.

You may remember from your study of quadratic equations, which I'm sure you were keeping up. I'm struggling here, but you're talking very, Very well. And what you have to do there is that when you solve this equation, you look at it geometrically and you draw squares and you add bits onto the square and you complete the square. But if you're going to complete the square,

then having a negative answer is the solution to the equation is just ridiculous. So there was great suspicion of negative numbers. But what he did do in his book in algebra, the latter third off had to do with laws of inheritance in Islamic culture. And there it was very much his use and the development of the Hindu-Hindu numeric system that allowed him to make advances in that particular area. So as we're nudged along by business with the Chinese, we're nudged along by the laws of inheritance.

Can you just develop that a little bit around? Because it's interesting for me that the abstract side, it doesn't go step in step, it isn't as simple as that, but is accompanied by the practical side. Well, I think the practical side is crucially important because in these cultures they were civic actions that had to be taken related to governance, relating to tax collection, relating to business, relating to commerce. And in a similar way, the Colva had mentioned the idea of a debt and a fortune in Indian mathematics.

Albizani, about a century later, but again based at this House of Wisdom in Baghdad, which had been set up with a library and with a large astronomy to undertake scientific studies, what he did was to write a book on algebra where he introduced the idea of negative numbers and it was there as a debt. He had three minus five, so he had minus two as a debt. So it's picking up an Ains point earlier on

where you're coming through and using it for purposes of commerce and that there. In fact, I think the title of his book was something like book and what is necessary from the science of arithmetic for scribes and businessmen. So we've got enlightenment in mathematics, Ian Stewart, in China, in India, in the Islamic world, and in the 15th century Europe is still in the dark ages.

Europe's in the dark age, yes, the torch of mathematics is burning brightly in the east. But in Europe, it's almost flickering out. I mean, there's still sort of nucleid and various things, things the Greeks knew are. surviving there in text in monasteries there around. And it's a little unfair to say nothing was happening, but the great advances are being made elsewhere. So why is this? Well, I think it really is that Europe is,

attention is focused much more on the church and on various things to do with theology and also, I believe, on logic and rhetoric and argument, The Middle Ages were very strong on enumerating all the possible ways of arguing your case, but they were missing out on the new mathematics that was growing elsewhere. What kicked the study of mathematics into life then, Raymond Flood, in Europe? Because after the end of the 15th century, it grew massively and very intensively. What, as it were, gave it a kickstart?

A lot of it had to do with the recovery of the ancient texts. So it was, and that they, of course, had been transmissible. very much through the Islamic tradition. And there were lots of mathematicians who were trying to recover the knowledge of the ancients. So that was a very, very crucial motivating factor in the development of mathematics towards the sort of medieval age of times.

And that led on to a whole sort of variety of activities in bringing the mathematics along and in building upon really what it went before rather than sort of inventive and innovative study initially. Colver, Colver, Rone Dougal, something else kicked in, which was implied in what Ian Stewart said, the devotion to argument and rhetoric and discussions in the late Middle Ages coming into the Renaissance.

The public competition for place often depended on public debate. Now, that began to take place in the realm of mathematics, didn't it? And that, it seems to me, could be said as to one thing that spurred it forward. and there's a story which begins with a man called Del Farrow and his discoveries, and there's a good story there which you are going to tell, I hope. Sure. We're not quite sure why, but Delferro got interested in trying to solve cubic equations. Now, cubic equations are equations of the form some multiple of x cubed

plus some multiple of x squared, plus some multiple of x plus some number are equal to zero. Or that's how we'd write them now. That wasn't how they wrote them then, because they didn't have zero at that point. Cubics had been deemed to be incredibly hard things to solve. Somebody had written that with the present state of mathematics, they thought it was as hard to solve the cubic as it was to square the circle. Now, that wasn't correct. We now know that squaring the circle is impossible.

But they were out there as a kind of a holy grail of things to solve. And Delferro managed to solve a particular form of the cubic. He managed to solve cubics which looked like X cubes, so a number multiplied by itself three times, plus some positive multiple of X is equal to a whole positive number. So something like x cubed plus X is equal to 1. But he never told anyone how to do this

because of these public contests. People were very, very secretive in general about their mathematics. They wanted to guard ways of unlocking things that they could later use for public contests. He did write it down in a notebook, which he again didn't show anyone. On his deathbed, he told his student, Fior, and his family inherited the notebook.

Now, Fior was not nearly such a good mathematician, but he was itching to become famous. So he arranged for a public competition to be held with a much more well-renowned mathematician, Tartalia. Tartalia was from quite a poor background and was working as a schoolmaster in Venice at the time, getting by really on a pittance, but he had made quite a name for himself in these public competitions.

So he was a right person to challenge. Tartalia suspected that Fior knew how to solve the cubic, because Fior had been boasting about being able to solve this particular kind of cubic, called the depressed cubic. Now, Tartalia could already solve a slightly different form of cubic, but he knew that wasn't going to help him. Now, eight days before the contest was due to happen, he was sat up overnight, working and working away at this depressed cubic,

and he worked out how to do it. He saw the trick. So when the day of the contest happened, they'd each posed each other 30 problems, They'd arranged they were going to be on a whole broad spread of mathematics, and Tartalia had kept to his word, but Fior had set 30 problems entirely on this depressed cubic, because he thought Tartalia would not be able to solve it.

Tartalia solved all of Fior's problems in two hours flat, and didn't even bother to collect the price, he said it had been so easy. Fior didn't manage to solve a single one of Tartalia's problems and basically retired Hurt. We don't hear much from him again after that. Now Tartalia continued to keep secret how he'd managed to. solve this cubic. He wanted to win more competitions. And then Raymond Flood, enter Cardano, a mathematician and a gambler.

Yes, just one point that Culver brought up, I like to think of it that what Fior had done was to put all his eggs in the one cubic basket. You've been waiting to say that, aren't you? I could tell. So Cardano enters. It was a very, very famous Renaissance scholar and medical person and very very very a very strong work in mathematics and in medicine. In fact, I think was instrumental in curing a previous,

an archbishop in Scotland, of an allergy, of St Andrews, indeed, of Caldw's University, of an allergy by suggesting changing off the bedclothes. And he was also an inveterate gambler and wrote one of the first textbooks published after his death on probability, and also had hints in that as to heart. to go about cheating as well. So he's quite an interesting character,

knew about Tertaglia, wanted to get the result out about solving, how to solve these cubics, hard to get the algorithm, really, for solving the cubics. And what he did was to coax and cajole and to, well, bribe in a sense of introducing Tartaglia to the governor of the area

in which he was in Milan, because Tartaglia had a particular interest in military science, so he was offering him particular introductions in order to help his career along. and eventually Tartaglia told him the secret by way of a verse form in a poem but swore Cardano to secrecy

an oath of secrecy and because he said he was going to publish it himself and I think very reasonably so but he didn't publish itself Cardano published a book that he had in arithmetic and was really wanting to develop his work in algebra so he together with one of his pupils Ferrari works on the cubic and in fact works on the quart

equation, which is just one higher up where you've got the number you don't know is multiplied by itself four times. And they eventually make great progress in this year but felt bound by the oath that had been made. All very honourable. And then what happens

is exactly, as Colvert outlined, to do with the documentation that had come along and that still survived, they went along and to the son-in-law of Delferro and they found the notebooks and they examined the notebooks and they saw that Del Ferro had the secret, knew how to do it before Tartaglia.

Now Tartaglia derived it independently, as we know, the story had gone. But now he had a source, which was before the one against whom he'd given the oath. So I don't know what a lawyer would make of it. I'm sure they'd make lots of money on it. But when you think of it, what he then said was that he felt he had the right to go ahead and publish. And he did. And that's a terrific story. Ian Stewart, where did that take us in the Renaissance then? So they've burst through, really, haven't these Italian scholars,

through these competitions and through their own, fascination with mathematics, it's interesting that it's allied to gambling. We still have the outer world nudging away, whether it's business, gambling, inheritance, its complications in the real world and chance. Anyway, away you go. What's happening in the rest of the way. The great thing with mathematics, the outer world is always sort of there at the back of your mind, but the effects you get from it are rather strange.

And this whole story, this is actually directly related to our topic of negative numbers. But the root is highly, yes, the root is, is, indirect and nonetheless it was the work on the cubic equations that's really paved the way for mathematicians later to become convinced that negative numbers are very very sensible things and indeed they're so sensible that even once you've got if you accept negative numbers you start asking other questions about them and one of the questions was what about square roots of negative numbers and those are really quite curious perhaps let me come back to that so Colville was saying that these mathematicians in the Renaissance, there were different kinds of cubic equations as far as they were concerned. It depended whether certain numbers were positive or negative. And if, what we would now say positive or negative, they would say, suppose it's something cubed minus something squared equals five. They say that's a different kind of thing, that something cubed equals five plus something squared. They transfer all the unknowns around

until you have positive numbers of them, but some on the left-hand side of your equation, some on the right. And this meant lots and lots of different cases. And when they started expressing it in algebraic notation and thinking about it more generally, they began to realise that instead of having 14 or 15 different tricks for 14 or 15 different kinds of cubic, it was really all versions of the same trick.

And when you express that trick in mathematical formula, it's sort of complicated thing involving cube roots and square roots and you multiply this and you add that and you play around and you get this formula. The problem was when they applied this formula to cases where they knew perfectly well what the solutions were, it gave them a very, very complicated formula with square roots of negative numbers.

It did not give them something that obviously gave them the correct answer. So, for example, you can set up a cubic equation whose solutions are 1, 2 and 3. You usually get 3 solutions to a cubic. But if you apply their formula or their method to this equation, You don't get the numbers one, two and three popping out as the obvious answer. You get three very, very complicated algebraic expressions. Do you want to take that up, Colvin?

Yeah, there was an engineer called Bombelli, who we're not quite sure where his maths training came from. He'd come from up in the north of Italy. He'd been around Bologna at the time when Delpharo, I mean, he was born around the time that Delpharo died, and hence the solution to the cubic had been very much in the air. But he came from what we'd now call a fairly poor middle class family wound up being an engineer and he was busy draining a swamp he was draining a swamp in

Tuscany and he'd been doing this for about five years when work got held up and he decides hey I want to write a book on arithmetic I want to write a book on arithmetic that's self-contained such that people who haven't come from highly educated rich backgrounds are going to be able to read it and actually learn some mathematics so he sets out to do this and as part of his book on arithmetic, he wants to explain how to solve the cubic. There are these formulae around for solving the cubic, but the problem with them, as Ian was saying, is that you find you'd get these square roots of negative numbers cropping up, even when you can see by looking at the equation that there should be at least one whole, real, positive number that's going to work if you

stick it into the equation. You're going to get the zero out that you want. So, he says, well, let's imagine that these really are numbers. And he writes down the rules for multiplying the square roots of negative numbers. And he calls them plus of minus to mean plus the square root of minus 1 and minus of minus to mean minus the square root of minus 1. And he quite clearly lays down how to multiply them. He explains that you can't add and subtract them from numbers as we know it.

He comments to the reader that this initially seemed to him to be more a piece of sophistry than truth. But if you just plug through the equations, you will find out that you actually get a whole real number, answer in the end. He didn't care what it meant. He wanted the answer. What does the square root of minus one look like? Square root of minus one. The, if you just work with the ordinary numbers, including the negative one, just a second. Very, very, very briefly. Colvers explained it extremely lucidly, but still just to pin it home, what exactly is useful about the square

root of minus one? I'm sorry to be such a utilitarian bore, but there you go. What's useful about the square root of minus one? When it was first invented, it was just a gadget that let you get the correct answers out of a process that didn't seem to give the answers, although you were convinced it should. What it now is, is it's absolutely fundamental to quantum mechanics, to the mathematics of waves, to almost anything to do with things that happen in the plane. You can't do modern mathematics. From about 1700 onwards, you could not do mathematics sensibly without the square root of minus one. even though a physical picture of it at first sight is very difficult to understand because if you take a positive number and square it you get a positive number

and if you take a negative number and square it you also get a positive number and one way of approaching it is first to say well let's try and get some sort of representation of this and this is in terms of the the number line yes if you've got a ruler it's there are numbers associated with 0 1, 2, 3, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6. your ruler so these numbers go off to the right and then you say where would the negative numbers live and the answer is they go off to the left along the same, you put another ruler next to it

with negative numbers on. Yeah, so 5, 4, 3, 2, 1, minus 2, minus 3, minus 4, going to the left. Now the square root of minus 1 in some sense should be something which, when you do it twice, you get to minus 1. So it's sort of halfway between 1 and

minus 1. But if you stick to the ordinary number line, what's halfway in between is zero and that's clearly completely wrong. And it took an awful lot of faffing around trying to make sense of this before somebody said, I'll tell you what's halfway between one and minus one. You take your ruler and you rotate it through 90 degrees. Because 90 degrees is halfway between 0 and rotating at 180 degrees. And if I take my ruler and rotate 180 degrees, everything turns into it is negative.

The positive numbers and the negative numbers swap. square root and minus one should be halfway between those. Poking out at right angles is the place to put your so-called imaginary numbers. These square roots are negative numbers. And so instead of a number line, we get a number plane. And now it all starts to make sense. And all of the algebra of these numbers can be expressed very sensibly in terms of what happens on this plane.

But still is there not, Raymond Flood, suspicion about these negative numbers. They're still viewed with suspicion. in the 17th century still. Oh, yes. Definitely. I mean, Cardano, for example, that we were speaking about earlier on, who had come up with these negative numbers and then with the square root of negative numbers

said the negative numbers, for example, were false or fictitious. He said, and there's a nice quote I have here by the imaginary numbers, that so progresses arithmetic, arithmetic subtlety, the end of which, as is said, is as refined as it is useless. So what the stage that is at now, that we're at now, is that we have these negative numbers, we have these imaginary numbers, we know the rules for operating on them,

we know how to combine them together, to add them, to subtract them, to multiply them, to divide them, but we don't know what they are. Do you want to add anything, Colva, to what Ian said through the British mathematician John Wallace, about the number line? Yeah, John Wallace is the guy that we normally credit with having invented the number line in the first place.

He was very, very interested with giving geometrical interpretations to numbers. He wanted to get negative numbers into the story and finally came up with this idea that a negative number is the same as a positive number. You're just heading in the opposite direction. He attempted to come up

with a geometric interpretation for complex numbers, which in retrospect, we can see he's hinting towards the idea of going at right angles from his new number line that he'd invented. But he probably wouldn't have seen that. I think that's the benefit of hindsight there. But you can see the level of confusion at the time because whilst he's the guy that we say invented the number line,

he also thought that negative numbers were bigger than plus infinity. And he somehow held both of these ideas in his head at the same time. I'll explain to you why he thought this. If you imagine you've got a cake, if you divide it into two bits, then the bits you get are bigger than what you get if you divide it into three bits. Okay, so if you divide it by a bigger number, you get a smaller answer, you get smaller pieces.

Now, John Wallace believed that if you divide a number by zero, say divide one by zero, you get plus infinity. Now, his argument went, if I divide one by minus one, since minus one is less than naught, I must be getting bigger bits than I would get if I divided one by zero. Now, one over minus one is equal to negative one, so he concluded that negative one was in fact bigger than plus infinity. at exactly the same time as coming up with our beautiful picture of the number line. So much confusion, I would say, at that point. In the 19th century, Ian Stewart, we have the mathematician Carl Friedrich Gauss. What contribution did he make if we're moving this story forward?

Gouse, who was probably the greatest mathematician who ever lived, certainly is a pretty good contender for that, was interested in some actually very advanced mathematics of things which were related to square root of minus one and other such things, and which, from his point of view, he just had to have that kind of number available. And he was also a good enough mathematician to know that nobody had really pinned down firmly what these strange numbers were and how you should really use them. And so as part of preparing for his great work on much more interesting and sophisticated things,

he sorted out, at least in his own mind, a nice simple picture of how, square root a minus one and related numbers worked. And it's basically this plain again. But in Gauss it's all very, very neat and tidy and some algebraic formulas to show it really does the right job. And so he's probably the fourth mathematician to have the same ideas, but he's such a clear thinker that, and of course he has a very high reputation. So when Gauss says something is done in a certain way,

the rest of the world takes no... the rest of the mathematical world takes notice. And so he reduced the whole question of negative numbers and square roots of negative numbers to very simple pieces of algebra, a very simple picture of coordinates in the plane. And then he went on from there to do wonderful things in complex analysis,

which is doing calculus with square root of minus one, and algebraic curves and number theory and law of quadratic recipes, and all sorts of stuff, which actually makes a lot more sense if you can allow root minus one to come into your equations. Raymond Floyd, his contemporary Augustus de Morgan refocused terms of reference here. Can you tell him for negative numbers, as I understand it? Can I just say before that there in Ian's point about what Gauss was doing was that really a sort of culmination of a lot of work really on the theory of equations. and in a way, coming back to some of the initial points that were made,

what has been worked towards is working towards the connection between an equation and the coefficients of that equation and the roots of the equation. So that's what's underlying a lot of the development as it's going on. If you know the form of the equation in certain senses, what can you say about the roots, what can you say about the equation itself? But Augustus de Morgan said it's a point of departure from the measurement, didn't he, is I? And therefore you needn't look on negative never sometimes as negative. You can put a positive question to it.

What you can do is you can take attention away from the actual meaning of the symbol itself and just talk about the operations that you have upon the symbol. What advantage does that bring us? Well, the focus then is that you may be able to interpret the symbols in different ways. That you've got rules for combining them, rules for putting them together. Those rules will have certain properties. and then you may be able to get certain consequences coming from those. So you mentioned at the very start there, William Friend,

who was somebody who wrote a very polemical textbook, not believing in negative numbers, saying something like minus five should be understood as taking five away from a number that you haven't written down rather than minus five being a number in itself. What people like DeMorgan and Peacock before went on to do, was to say, look, we can think of these symbols like minus five. What we need to focus on is the interpretation that we give to how you combine them,

how you put them together, how you put them together. Yeah, I mean, very often with negative numbers, if the answer is a negative number, if you just change the question slightly, you discover it makes perfectly good sense. Can you give me an example? Yeah. I suppose I'm doing a calculation and I, basically, how far north do I have to go under certain conditions? and the answer is minus five,

then what that means is I go five miles south. I go the other way. And science is full of things which are measured in different directions. How many minutes before sunrise does something happen? Well, if it's negative answer, then it's minutes after sunrise and so on. So when we have these kind of directed quantities, quantities that can point one way or point the other way, the minus sign just means pointing in the opposite direction.

So you can always make sense of the negative answer. answer. And because the mathematical rules work perfectly well, this means you can do your calculation without bothering in which direction things point, and the sign of the answer will tell you that. So coming on from that, I mean, we started being interested in just the way the rules of combining numbers started to work. And there was a guy called Hamilton that was actually friends with De Morgan and was working much the same time as him. And he wanted to take both complex numbers away from the notion of geometry. He wanted a purely algebraic reason for their existence.

So what he did was rather than talking about the square root of minus one, he defined these pairs of positive real numbers called couples with rules for how to add them and multiply them and show that you could do away entirely with the notion of the square root of minus one and the notion of the complex plane, and you could just define a new kind of multiplication that was going to work. It's such an important development actually when you think of it I mean we're all very hung up with the notion of what is the square of minus 1

and very understandably. And what Hamilton did was to say that you can think of complex numbers as just pairs of real numbers, just couples of numbers. But what you have to do is to add, subtract, multiply and divide them in particular ways. Once you've got those rules of operation on them, then you can give them an interpretation. in the way that Ian was outlining earlier on, in the number plane, for example,

as being a representation of complex numbers. So what you've got now is, if you ask me the question at this point in the programme, what's a complex number? I say to you, it's just a pair of real numbers with certain ways of combining those pairs of real numbers. Then you want to go on and ask me the question, what's a real number?

And you can work your way down from that. But the complex numbers in themselves, with Hamilton's viewpoint, are no more complicated than just real numbers. There are pairs of real numbers that you put together and combine in different ways. Sorry, Calvert, can I just please come in, then I'll ask you in something.

And then Hamilton took it even further than that. He wanted to try and come up. So if complex numbers are pairs, he wanted to try and define a thing that would work with triples of numbers. And he spent years trying to find a way of multiplying triples, such that there would be a rule

that would correspond to this rotation by 90 degrees that you get with the square root of minus 1. And one day while walking along by a river in Dublin, he suddenly realised he could do it if he went to quadruples. He found a way of having I, J and K, such that each of them squares to minus one and rules for multiplying the I's and such that he got an internally consistent system. And he was so excited by this that he stopped

and carved the formulae for how to multiply these things into the stonework of a bridge by a river in Dublin. And there's a plaque there now to this day commemorating the point. So how is this to... Excuse me. How are these advances, and our listeners are all agog to me on this now, I so, yeah. How are these advances, seriously, how are these advances in thought by these brilliant men? And so far it's been all men. Yeah, well, never mind.

For the moment, it's all changing. We've got Culver. That's right. Right, fine. So how are these advances affecting the physical world? They're so built into the fabric of mathematics and engineering. and physics and science in general that we use them all the time.

Everyone practicing those subjects uses them all the time without really sitting there thinking, oh, I'm using a negative number. Oh gosh, I've got square root to minus one here. You just do it. And the reason is it lets you do the calculations without worrying about what's positive, what's negative. You get the same formulas, you do the same things,

you push the same buttons on your computer. I suppose I misphrase it then. I probably meant the material world, Raymond. I mean I mean the way I think about it is that you can think of numbers, the counting numbers

where you get and you're enumerating things you're just finding out how many there are of sheep, cows, goats, whatever then you've got negative quantities and with negative quantities

you can give the various sort of commercial or financial interpretations that you have there I mean things like then the square root of negative numbers or operate other operations performed in negative numbers

were quite useful, we're quite practical in finding the solutions of equations and that's essentially where they got their life from and there's now an underlying foundation for understanding those and as Ian says it sort of permeates not the direct world of experience that we bump into as we're going around but the things that allows us to go around

the things that are underlying all the technology all the science, all the engineering. Right. And it all kinds of back to money, as with the Chinese and their red sticks and black sticks, which were putting a value on a debt. Now we have the problem of putting a value on a derivative. We have all sorts of very sophisticated problems, which are mathematical problems. What is this thing worth? You know, I don't just buy things anymore. You buy options to buy things or options to sell things, or you can buy and sell options on the options, and so on, and so on, and so on. To put a mathematical

value that everyone can agree on on these financial instruments is actually taxing the ingenuity of the financial world. And this is where the rocket scientists come from, the people with fancy maths degrees, who actually try and tell them what these things are worth. Colber? Yeah, I mean, another way in which negative numbers are coming up throughout science at the moment is in the notion of frames of reference. We use coordinate systems just all the time now without thinking about it. We had negative numbers before as the plane, but if you're thinking about grids of the world and so forth. You want to be able to talk about going in the opposite direction

from the thing you were before. Complex numbers come up in all sorts of unexpected places. Electronic engineers need to use complex numbers when they're talking about how electricity flows through circuits because you wind up needing to talk about real-life things which should somehow square to a minus number. Einstein needed to use complex numbers when he was putting together his formulae for special relativity.

You find I times the speed of light cropping up all through because he needed things to square to minus numbers. Well, thank you very much for that education. I enjoyed it. Thank you very much to Ian Stewart, Colver Rony Dougal and Raymond Flood. And next week we'll be talking about Savanti's Don Quickshot, as we say, or Don Quixote, as I believe it should be said. And thanks very much for listening. 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. forward slash radio four