How to Prove 1=0, And Other Maths Illusions - Sarah Hart

In this lecture I will show you some mathematical illusions: “proofs” that 1=0, that fractions don’t exist, and more. There are curious and important implications behind what’s going on.

These “proofs” reveal some very common logical slips that can go unnoticed when we are trying to prove more plausible statements. And the stakes are high. As I’ll show you, once you have “proved” one false claim, you can prove absolutely any statement at all.

This lecture was recorded by Sarah Hart on 14th May 2024 at Barnard's Inn Hall, London

The transcript of the lecture is available from the Gresham College website:
https://www.gresham.ac.uk/watch-now/maths-illusions

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Transcript

0:05

Today I'm going to share with you some mathematical illusions, seeming proofs that one equals zero, or that fractions don't exist. Many others like that. But behind the scenes what's going on, uh, with some of the seeming paradoxes that we are going to see has actually caused real difficulty for mathematicians over the years. And sorting it all out has led to some fascinating new mathematical developments. We're going to begin with three truly startling proofs, but I'm not gonna reveal the tricks of those straight away because it's fun to try and work out what's what's gone wrong. However, I do promise you that by the end of the lecture, we will have resolved all of the problems and everything's going to be fine. So try and watch out and see where you think my little mathematical slight of hand is happening. Will we ever be able to see beyond the current observable universe? Oh, it's much worse than that. The observable universe is shrinking. If you want to know what are my odds of winning the lottery, you come straight to probability. Yeah, because probability is all about how likely or not events are to happen. I think the chance of there being an undiscovered second species, very like humans out there in the world today is pretty slender. However, And these are the pictures if you haven't seen them. I mean, New York was orange. The air was orange. They said one day out in that air was like smoking a pack of cigarettes. It had the same effect on the lungs as smoking a number of cigarettes. So people who'd never smoked in their life were suddenly going to suffer some of the same health effects. Any further questions is a brand new podcast from Gresham College, A place where we ask our speakers all of your questions that went unanswered following their lecture guests have included Ronald Hutton, Robin May, crystalline, TOTT, Sarah Hart and Maggie snowing. Any further questions? All episodes are available wherever you listen to your podcasts. And don't worry if it feels difficult to believe these impossible things. Just follow the advice of the Red Queen. She says to Alice, it's just all about practice. When I was your age, she says, I always did it for half an hour every day. Sometimes I've believed as many as six impossible things before breakfast. So, you know, we've all had our breakfast, so this should be much more achievable for us. So we're gonna begin with the first three impossible things. Let's see. First, impossible fact one equals zero. So this might come as a surprise, but you know, we are gonna see a proof. And I'm a mathematician, so you know, I'm gonna be very convincing, I hope. So here's, here's the argument. Why does one equals zero? Well, we're just gonna consider and think about this innocuous little, little series of numbers. One minus one plus one minus one plus one minus one, and so on. Now, it's pretty obvious what this'll all finish with because we, it, it all comes in pairs, right? One minus one, another one minus another one. We're just adding, but then taking away repeatedly. So this thing is just a bunch of one takeaway once, so we're adding a load of zeros together. So bullet, it was zero. Now there's another way of looking at it, the sort of bird in a hand argument, which is to say we start off with one, right? That we've got one. And then what do we do? We take away one, but we add it again. We take away one, but we add it again, minus one, plus one, minus one plus one. So really what we have here is one plus a bunch of zeros. So one, there we go. So we've, we've worked out the same thing two different ways and we've got zero and we've got one. So those must be equal. QED one equals zero. So I'm sure you're all happily believing. Now that one equals zero, just in case you are not entirely convinced. I will give you actually a couple more proofs of this as we go along. As I say, I'm not gonna tell you exactly why this doesn't work right now, but we might suspect something murky is going on with these.dot dots, <laugh> that they had a lot of, uh, problems. So we'll come back and think more about infinite series later on, but let's have our second impossible fact. The smallest positive number is one. Now, this is actually very good pedagogically, right?'cause teachers tell us, research has shown that one of the major stumbling blocks for youngsters learning mathematics, and some of them never get beyond it, it's fractions. So once we've proved the smallest positive number is one, there'll be no need for fractions anymore and everyone can be happy. So let's prove that the smallest positive number is one. So currently we don't know what it is, right? So let's investigate it a bit. So here comes the proof. So we dunno what this smallest positive number is. Let's call it X for the moment. Obviously X is at most one 'cause one is a positive number and X is the smallest one. So X is certainly less than or equal to one, okay? But X is a positive number. If you take the square of a positive number, you still have a positive number, right? So x squared is, is also a positive. Number X is the smallest positive number. So X squared can't be any smaller than X. So that tells us that X is less than or equal to x squared. Okay? Now x less than equal to x squared, X is positive. It's not zero. So we can divide through by X without, I'm not dividing by zero or anything weird. This is okay, I can divide that, that little equation through by X on both sides. And I get one is less than wrinkle to X. And now we've got a number sandwich, best kind of sandwich, uh, because we see, we've showed that one is less than equal to x, but X is also less than or equal to one. So X is trapped, it has to equal one. So we've shown that the smallest positive number is in fact one, which is great. No more fractions, mass has become suddenly a lot easier. So again, have a thi what do you think has gone wrong there? Well, we'll come back to that later. Final of these three, uh, impossible factors coming up. And I thought as I'm the G regression, progressive geometry, we should do some geometry. So we're gonna do some geometry around congruent triangles. Um, I'll show you before then a couple of warnings because I don't want to like, you know, lead you astray here. We just have to say ahead of this, watch out. Um, Saint Augustine does tell us to be aware of mathematicians and all those who make empty prophecies. But okay, well, like, you know, well, I dunno. See, see what you like Saint Augustine, you know, be honest with us. Er, on the other hand, I dunno if this is more insulting to mathematicians or to Frenchmen, um, but he says, whatever you say to either of them, uh, they translate it into their own language and forth with it means something entirely different. So those are the warnings. Uh, but now let's proceed and see what we can see about triangles. Uh, okay, so there's the basic setup. I've got any old triangle, A, B, C, right? Take a triangle, your favorite triangle. Now what I've, I've drawn in two lines. We're gonna have some more lines. I thought I'd do it step by step. So we've bisected that angle. A. So bisecting an angle is something we've probably all learned to do at school with a ruler and compass. You can, you can do that construction. So bisect the angle A and that line is coming down. And then we're gonna cut the bottom, the opposite side in half. So the perpendicular bisector of that site. So we've cut it into two halves. Um, then we point is it D and then we've got that perpendicular coming up and they're meeting at O and we're gonna add some lines from O we're gonna add in, we're just gonna join O to the other two sides, O to B, O to C. I'm not claiming those angles are being bisector or anything. We're just joining those lines. And we're gonna drop perpendiculars to the remaining sides. Again, I'm not claiming that that chops those sides in half, but we can still drop those perpendiculars. Okay, that's a bit of complicated setup. So now watch me like a hawk, 'cause I'm going to do some things with congruent triangles. And you've gotta see where I may or may not be lying to. Okay? So to, we're gonna take them in pairs. We've divided this triangle into six little smaller triangles. Let's look at the two blue ones at the bottom. That pair. So I claim they're congruent. How do I know? Well, we've got a common side OD and BD and CD are equal because we, we, what we've got od there is the, the bisector, right? So BD equals cd. So those two sides are equal. And between them, we've got the same angle. Happens to be a right angle. So one of the rules for congruent triangles is side, angle side, okay? So by that rule, these two blue triangles are congruent to each other. Alright, next pair of triangles, the red ones, what have we got here? The angles at a are, are equal by construction. We made the angle by sector, uh, at p and at Q we've got right angles. So we've got two matching angles. And then of course the third angles must be the same. So all the angles are the same in these triangles. So they're definitely similar triangles, but also they have a common side, uh, ao. So that makes them congruent. Okay? So it's another pair of congruent triangles. And now finally, think about the two yellow triangles there. They're the ones that are left. Uh, and so these ones, what can we say about them? So they're right angled triangles. P and A Q got RightAngle. The hypo uses of those are the same. OB equals OC because the blue triangles are congruent and we've got another pair of matching sides. P, o and QO are equal because the red triangles are congruent. So RightAngle, side, side, another of the criteria for being congruent, the yellow triangles are congruent. So we've now got these three pairs of congruent triangles. So what can we say from this? Well think about the side AB now from, because the red triangles are congruent, AP is equal to aq, and because the yellow triangles are congruent, PB is equal to qc. So AP equals aq, PB equals qc. Now you might see where we're going. AB is AP plus pb, but AP equals aq and PB uh, equals qc. So we've got the AB is equal, equal to in length AQ plus qc, which is just ac. So what we've shown is that AB equals ac, so it's an is OES triangle, but this was any old triangle. So we have proved the amazing theorem that all triangles are isosceles. Fantastic. So if you are puzzled by that, good, we'll come back and think more about it later. Okay, well by this point you might be starting to ask yourself many things about your life choices that led you to this room today. But you might say, well, okay, what actually then is a proof?'cause these proofs perhaps are, seem to prove something that isn't true and that's not really what we want in a proof. Uh, so what is a proof? And the idea of proof goes back thousands of years. It's one of the things that makes mathematics, mathematics. So we can sort of loosely say that it, what are we doing when we prove something? Well, we, it's, it's a sequence of logical deductions starting from some things that we all believe are true. You know, we might call them axioms, some, some initial assumptions. Then there's a sequence of logical deductions, everything following from what's gone before that establishes the truth of, of a given statement. Um, so we, we, we started with three impossible facts, three proofs, and we've had two quotes. So now we need to have one terrible joke because that has to be the ratio. Um, there's another meaning of the word proof, half a percent of alcohol. And occasionally combining those things can make things go terribly wrong. So here's the advice for today. Don't drink and derive there. We carry that. Thank you. Yeah, I know I, you know, I thought I, I'm gonna, I I've only got one more lecture after this as ion professor and I don't think I've told that joke yet. So, you know, it had to happen. Anyway, let's carry on and think about the first definition of proof. Not involving any percent of our goal at all, if you're doing it properly. Um, where we make deductions following on from our initial assumptions and we go through the proof. So there's a direction of travel we start here, everything follows from what's gone over, gone before. Now getting that the wrong way round is something that can cause big issues. And I'll give you an example. And this is something that, a mistake that is a really common pitfall in fact. So you might get asked to show that some two algebraic expressions are equal to each other. It's very tempting to make that the first line of your, of your proof and say, right, let's start here and then mess around with the algebra and multiply things out and cancel things and do some stuff. And then we end up with zero equals zero. Well, that's true. Great. So the original thing was true. Now this is very dangerous. It's dangerous thing in this case. I mean the, the original statement does happen to be true, but deducing something true, um, is possible even starting from something false. So you've gotta make sure your direction of travel is, is, is in the right direction, because if you don't do that, you risk proving things like this. So here I'm proving that one equals minus one. Um, so I start with one equals minus one, and then I square both sides. One squared will equal minus one squared. Well then on the left, you get one on the right, you get one, so one equals one. Well, I know that's true. So the original thing must have been true. Now <laugh>, of course, that's nonsense. We can see that it's nonsense because I've proved, proved, uh, something false here. But it doesn't look like nonsense necessarily in the thing above because it just so happened that what we're aiming to do is it happens to be a true statement. But in mathematics we don't only want to be able to trust proofs of things we already know are true, rather destroys the purpose of this. So it's so, so important to make sure that yeah, we, our, our, our proofs are going from beginning assumptions to the end and, uh, not the other way around. Okay? Uh, let's have our fourth impossible fact, and it's a good one. It's gonna help all of our, all of our financial, uh, calculations there. One penny equals one pound. Okay? Here's the proof. It's a very nice short proof. So one penny, well what is that in pounds? That's N point N one, okay? Now N point N one is n 0.1 squared. So we just make that change. Okay? So now I've got N 0.1 squared, but n 0.1 pounds, put that back again. That's 10 P yes, 10 pence. Um, but now of course we've gotta square it now. So 10 squared is a hundred and a hundred pence is one pound. Hooray. So I, I'm pretty sure this is how the Bank of England does quantitative easing<laugh>. It's at least as good as any other explanation I've heard. Um, but okay, what's going on here is there's something up with the units. Something's going wrong when we trying to change between, uh, pennies and pounds. And actually problems with units have caused real world issues on many, many occasions. I wanted to just mention my top three of these 'cause it, it does matter. I mean, we can see that this is wrong, but sometimes there are, there are issues that arise when people don't notice problems with units. So the first of these, what are the most famous? Yeah, the Mars, the Mars Climate Orbiter, which, uh, on very unfortunately that mission failed, uh, at a cost of gosh, something like $120 million. Because, because, um, some bits of the collection of, uh, you know, companies and of people who were doing this, some people were expecting to receive some numbers that are needed to calculate various, you know, thrust in the, in the, uh, craft wanted to receive the numbers and were thinking they were receiving the numbers in Newton's of thrust and they were actually being sent the numbers in pounds of thrust. And those two do not give you the same answer. So unfortunately, because of that me mess up between metric and imperial, uh, the, the, the mission sadly failed. Um, we can go back. This is not a new phenomenon. So it's not all the fault of metric. Uh, this is a much older example. So this is the Swedish warship vasa, uh, which sank, you know, mere moments after launch in 1628. And then they raised it in the 1980s and they had a look at what might have gone wrong and what they found, or one of the, one of the problems was they found on the, on the ship, you know, it just literally just been launched. So there was still a few bits and bobs of their construction, uh, materials and they found some rulers, they found four rulers and they were able to sort of deduce that there seemed to have been two teams of carpenters or ship builders working on this ship. One on the port side, one on the starboard side. And on one side the rulers were measuring Swedish feet. And on the other side they were measuring Amsterdam feet. And one of those is about 28 centimeters and the other one is about 29 centimeters. And that discrepancy was enough to make the, the weight of the ship unbalanced like one half that the, the things were thicker than others. And so it was unbalanced and catastrophe. So even if you are both using feet, you need to make sure you're both using the same feet. Now the final example, uh, is a slightly different one. So this is the High Rhine Bridge. Uh, it was built to connect the city, the Swiss city of Laufenberg to the German city of Laufenberg. So both are the same name, but they're on different sides of the Rhine. One's in Germany, one's in Switzerland. Okay? So this, uh, was a problem because in Switzerland the um, altitude is measured from a base point, which is related to essentially a big rock in Lake Geneva. And on the German side altitude, um, sort of mean altitude is measured, um, based on a measurement that's sort of mean water level somewhere in Amsterdam. Okay? Those are different from each other. Now they knew this, they did know this, you know, they got complete idiots. Um, the difference is about 27 centimeters. So they said, okay, well we just have to take account of this in our calculations. And at this point my heart goes out to them because I too, uh, often miss out minus signs and then have to go back and find 'em again.<laugh>, right? Rogue minus signs of the bans of many people's lives including mine. Um, a minus sign went wrong. And so instead of getting rid of the 27 degree, uh, centimeter discrepancy, they doubled it. And when the halves at the bridge were kind of coming together and closer to meet, they found there were 54 centimeters apart from each other and sort of emergency changes had to be made. So in that case it was, yeah, we, we using the same measurements, but we are starting from different levels and the rogue minus sign messed everything up, which uh, they seem often to do. So if we think back to, so units mattering, if we think back to our, our impossible proof that one penny equals one pound, it's easier to spot what's gone wrong if you hadn't already, if we were to try and do this same thing, but with centimeters and meters because we straightaway can see that we, some of the things that we are saying are equal to each other aren't even measuring the same thing because we've got meters not 0.1 meters and we are claiming that that is equal to some number of meters squared, but meters are a unit of length and meters squared are a unit of area. So there's no way those can be the same thing. Um, it's harder to spot that with pounds and pennies 'cause we don't often think about square roots of a pound or square of a penny. But so what went wrong here was that we just sort of glibly put these units inside squares and square roots and, and didn't pay attention to what was happening. So yes, units important, uh, and that, that is a warning to us all. Okay, so let's think now again about one equals zero, just in case you didn't believe me. Um, I'm gonna give you two more proofs. That one equals zero 'cause I really want you to believe it by the end of the day. And they're, they're increasing in mathematical, uh, sophistication. So that, so we just use ones and minus ones before. Now we're gonna bring exponents into it. So just a couple of observations. We all know the laws of powers. Yeah, anything to the power zero is one, right? So X to the nor is one for any value of X. But on the other hand, if we take zero and raise it to any power, you just get zero, multiply some zeros together, still zero. So zero raised to the power Y equals zero for any Y. So I simply set X and Y both to be zero. And you'll see that from the first expression I've got that, uh, zero to zero is one, and from the second one it clearly equals zero. So it clearly is equal to both one and zero, which are therefore equal to each other. So one equals zero, okay? Marvelous, easy. Just in case you're still not convinced, I'm gonna, I'm gonna bring calculus into it, okay?'cause calculus can do anything. So this is where, this is the our only glimpse of calculus in this, in this talk, but it's a good one. So this isn't calculus yet, but look, I'm just showing you here what we mean by something squared, which we all know. So two squared is two plus two, two lots of two three squared is three plus three plus three. Four squared would be four, lots of four. So in general for any number XX squared is just x plus x plus x plus x like x times. Okay? So now here comes the calculus. Just differentiate on both sides. If you remember your calculus, don't worry if you don't, you can just be, use one of the other proofs, that one equals zero, you won't lose anything. Uh, differentiate on both sides. And we all know when we differentiate X with respect to X, when we differentiate x, we get one. So a bunch of ones on the left hand side. And when we differentiate x squared, you may remember the answer is two x. So now on the left we've got X lots of one, and on the right we've got two x. So for any value of x, we just have x equals two x. So I'm gonna substitute X equals one and that gives you one equals two. And then you just subtract one from both sides, one equals zero. So listen, I've given you three proofs. I think it would frankly be childish if you didn't now believe that one equals zero. So let's assume you do believe that one equals zero. That allows us to do something very interesting. We can now prove that everything is true. Everything it's a corollary, this is a a, a theorem that's a consequence of a previous theorem. So here's the corollary, let s be any statement then s is true. So this solves all of our worries and we can prove everything. So here, let's do that. So take any false statement, whatever you like, any false statement at all. F uh, now if I happen to know, if I've managed to prove that this thing that at least one of f and s is true, if I know that that statement is true, then the thing that's true must be S because F is false, right? So if I can prove that at least one of s and F is true, then we will be able to deduce that S is true. Okay? Alright. Now let's consider a particular false statement. One equals zero. So that's a false statement. I think we probably probably agree, but we also now have to agree 'cause I've proved it three different ways that one equals nor is also true. We just proved it three times. So this happens to be also true. So f is true in that case it's definitely true to say that at least one of f and F is true 'cause F is true marvelous. But we just said earlier and nobody told me I couldn't, that that if we can prove that statement that at least one of s and F is true, then s must be true. So you have to now agree that s is true, okay? QED every statement is true. So this is how high the stakes are, right? If we let even one false statement into, into mathematics, then it just renders everything. Uh, you know, all bets are off, everything is true. So we gotta be really, really careful and try not to prove anything that's false. I, I try to do that in my mathematical life. Okay? So with that in mind, we've already proved everything is true. So, you know, maybe we could stop there, but let's carry on, let's carry on. Um, 'cause I wanna, I wanna show you six impossible facts to, 'cause I admire the red queen so much. You know, we try and practice these things every day, okay, so impossible fact number five, um, it turns out that zero is a positive number. Didn't know that before. Now we do. Zero is greater than zero, okay? So let's, let's prove that. Okay, we're gonna think about these two theories. So X is gonna be one plus a third, plus a fifth plus a seven plus a. So kind of the odd one over the odd numbers added together. And y it's gonna be one over the even numbers added together a half plus a quarter plus a a six plus an eighth and so on. So there are those two things. We don't know what those ah, but let's just call the X and Y. Alright? What happens if we double Y? So now we've got just double each term. So one plus a half plus a third plus a quarter and so on. So this is actually the combination of X and Y because we've got one over all of the odd numbers and interleaved with one over all even numbers. So two y is actually equal to x plus Y. So that's, if we solve that, well not solve it. But if we rearrange, just take away two y from both sides, we get X minus y equals zero. Okay? So that's all good, but there's a slight different way we can think about it. Here's the different way. So now think about x minus Y and just, you know, x and YX is above Y here. So we can just do like compare the terms or take away the terms pairwise. So we can have one minus a half and a third minus a quarter and a fifth minus a sixth, dot, dot dot. That's also x minus y. Now the reason I put that bracketing there is because if you look at each bracket, then each bracket you've got something that's strictly greater than zero, one minus a half is bigger than zero. A third minus a quarter is bigger than zero, right? A fifth is bigger than a sixth, a seventh is bigger than an eighth and so on and so on. So we are adding together infinitely many things that are each bigger than zero. So if we add together infinitely any positive numbers, we're gonna end up with something bigger than zero for sure. So X minus Y then is bigger than zero, but hang on a minute, we just saw above that it equals zero. So again, we've got, we've got this fantastic result that zero is bigger than zero. So that's, that's that proof all very nice. So I think it's now time, uh, we already saw an infinite sum doing something a little bit suspicious, let's say right at the beginning here we've got another bit of something odd going on with infinite, infinite sums. So it's now time I think to talk about the challenge of infinite series. What, how can we understand what's going on? We have, we really proved that zero is greater than zero, I hope not.<laugh>, I hope we haven't proved that one equals an all. So what, what do we do with infinite series? And the basic, the problem, one of the problems is if you think about this, this one, so one plus two plus four plus eight plus what powers of two, we can see that that's getting bigger and bigger and bigger. It feels like intuitively, like the sum of that ought to be infinity. Okay? We are just getting, we're adding more and more and bigger and bigger numbers. Um, we could, we could plausibly say that equals infinity. Um, well okay, that's our x if we work out one plus two x, um, that's one plus twice everything in there. So one plus two plus four, plus eight plus so on. And if you multiply all those things by two, it becomes 2, 4, 8, 16, so on. So actually one plus two x is just X again. So now we've got one plus two x equals x. And if you solve that, you get x equals minus one. So am I saying that infinity equals minus one? I mean computer scientists say that, don't they? Maybe, maybe, maybe that's true. Uh, but it, it feels like that isn't quite right. So what could, what could we do? What which bit of this reasoning is, is something that we should disallow. So we could say, well let's not do, let's say we are not allowed to do calculations with infinity for example. That would, that would get us outta it. Just say we, this is banned because it leads us to weird paradoxical ideas. You could do that. Um, but that wouldn't help you with the first example we saw. It wouldn't help you with this ones and one minus one, one minus one issue because there nothing zooming off to infinity. We, we did it one way and got zero, we did it another way and got one, but there's no infinities in there in the calculation. So that feels like we could banish infinity, but that doesn't fully fix the problem. Uh, so maybe it's something else that we need to do. Um, you could say, well I'm just not going to allow adding up infinitely many numbers that's clearly problematic and I'm just going to ban it. Um, it doesn't sound very intellectually curious. So we don't wanna do that. But also we can't really afford to do that because we do need these things, not these ones precisely, but we need infinite series. We use them all the time. Um, if we want to work out, it's kind of useful to be able to know and think that three tenths plus three hundredths plus three thousandths plus three 10000th plus.dot, what is that? The decimal N 0.33, three n 0.3 recurring. That's the decimal expansion of one third. It's kind of useful to have that. Or you might say, okay, well we won't do that, we'll just have fractions, we'll just write a third, you know, we've got something right there. But of course that will not work for numbers like pie, which are irrational. There is no fraction, uh, that exactly is pie. So we need these, these.dot dots. We need to somehow incorporate them into mathematics in a way that leaves us, puts us on a firm footing where we don't accidentally prove that n equals one, which is takes the shine off your afternoon if you accidentally prove that. Um, or maybe it makes it more fun. So we need to be able to work with infinite series and therefore we have to work out how to avoid these, these apparent paradoxes. So how do we do this? Well, let's think about an example. So if I took a half plus a quarter plus an eight plus a 16th, so one over the powers of two add those together plus.dot, um, what we do with this or other series is we say, well we obviously we can't actually add infinitely many numbers together so we make better and better approximations. And what we do is to work out the sum to a given point call, we'll call that partial sum. We work out these partial sums, so the sum to one term, sum to two terms, three terms and so on. So if we do this, I've actually plotted the first few of these on a graph just to show you what happens. So the first term is just a half. If we had a quarter, the sum is three quarters, a half plus a quarter, plus an eighth is seven eighths, then we get 15 sixteens and so on and so on and so on. And you can see I've, I've done, so this is the graph of the partial sums that we are working out. And you can see it looks really rather like we are getting closer and closer and closer to one. And in fact if you can work out an expression for what is the nth partial sum sum to end terms and it's one minus one over two to the N. So as n gets bigger, that one over two to the end is getting smaller and smaller and smaller. We really are getting closer and closer and closer and staying close to one. So I mean I, you know, we could give a a formal very formal definition of, of what we want to say. But the basic idea is we would say that this series does have a limit. The limit is one and it converges to one. And the way we prove that is that we show that however tiny tinier number you pick as as a distance away from one at some point, eventually those postal sums will stay, will become and stay as close as you like to that the, the limit that we think it is. So here we think the limit is one and you can prove that however tiny a number you give me, I can guarantee that there will be a point beyond which all of those sums stay closer to one than your tiny teeny number. And so that's our, I mean informally I've defined the concept of convergence there that this, there is convergence, uh, to this limit, which in this case is one. So that's the basic idea and we can see that this does now help us with our one minus one problem. What about this? Well, if we work out those partial sums, first one is one and then it's zero and then it's one again and then it's zero again, it just keeps flip flopping between zero and one. It's never gonna converge because it never settles down to being very, very close to a single number. It just keeps flip flopping between, um, after an even number of terms it's zero. And after an odd number of terms it's one. So this thing does not converge with this, with this definition we're working with. Um, so if it doesn't converge, what's the opposite of that? It diverges. This is a divergent series even though it doesn't zoom off to infinity, but it never settles on one value. So it's meaningless to say it equals zero or equals one. Um, it never act doesn't equal anything 'cause it never, it never sits still. Alright? So we can think, well when, when would things, how do we know if something does converge? I mean that's clearly quite important. So certainly if we are adding up all these things and we've got to stay within a smaller and smaller distance from our, what we think is the limit, um, the things that we are adding on are gonna have to get smaller and smaller themselves. Uh, so if we think about something like this, this is called the harmonic series. So we're adding, um, one plus a half plus a third plus a quarter plus a fifth plus. If you look at the partial sums here, and I've plotted the first about a hundred of them, you can see that it kind of won and then one and a half, three over two, and then you add a third. So the bits you're adding on get less less each time. Kind of the grainy of this thing is settling down. It hasn't settled down to anything yet, but it sort of looks as if maybe it might give it enough time. You know, it's getting flatter and flatter the line. So it might, it might settle down to something. Um, but it doesn't. And I, let me explain, lemme explain why it doesn't. So if we think about this, this series, we can group the terms. Um, so we've got one and then a half and then, and then we're gonna group the next two terms, a third and a quarter, then the next four terms up to one eighth, then the next eight terms, the next 16 and so on and so on. The reason we're doing this is because inside each bracket everything before the last term is strictly greater than that. So one third is bigger than a quarter quarter, uh, one fifth, one sixth, one seventh. Those are all bigger than one eighth. So whatever this sum is, it's bigger than what I've just written here where we've replaced a third with a quarter, we've replaced everything in that bracket by one eighth and so on, and four lots of one eighth is half, two, lots of quarter is a half. All of the brackets add up to a half now. So we know whatever this sum is, it's bigger than one plus a half, plus a half plus a half forever. And that's clearly going to be bigger than any number you like to mention. If you want this to be bigger than a million, you just do 2 million of these brackets and you are, you're bigger than that. So even though it slows down, it grows slower and slower. This series gets to be if you wait long enough, bigger than any number. So this cannot converge. This is also a divergent series, so it's not enough to just have the terms getting smaller and smaller. We need more than that. So you do have to be quite careful. There is a tweak we can make to this series. Um, if we alternate pluses and minuses, you get what we call the alternating for obvious reasons harmonic series. So this is one minus a half plus a third minus a quarter plus a fifth minus a six and saw. Now again, I've plotted some partial sums here and I've gone up to, uh, the 200th partial sum. And you can see this isn't proving anything, but it is looking quite good for converging 'cause it's sort of oscillating around and then settling down to, to a number that seems to be somewhere just slightly less than not 0.7. And you can actually prove, uh, properly that if you work this out, this does converge, we'll call it s it converges to uh, the logarithm of two, log two, uh, which is about 0.69, something like that. So this does converge, it's a convergence series, but there's an issue and mathematicians discovered this are, they're looking at this and they said, yes, it converges to to log two fine. But let's think about calculating that in a slightly different way. Is there is a different way of calculating this thing. So, um, we just, we're gonna add all the same things up but just in a slightly different order. Um, like we, you know, we know we can do this if we're adding two or three numbers, it doesn't matter which one we add first. So, you know, seems, okay, so what have I done? I want to, I want to explain or I want you to be convinced that we have got all of the numbers in there, all, all of the terms in there. So each of these brackets has three terms. The first term that I'm highlighting, uh, in green is one over the odd numbers. So we've got, um, in the, in the original ordering it's one minus a half plus a third minus a quarter plus a fifth. So we've got, we are adding all the one over odds and you can see that they appear at the beginning of each of these brackets. So you've got one plus a third, plus a fifth, plus a seventh and so on. And then the other two terms are the one over even numbers just in the original order, but we're putting them in pairs. So minus the half minus a quarter in the first bracket, minus six, minus an eighth, minus a 10th, minus 12th and so on. So we have got everybody. Now let's think about these brackets then. Uh, the first two terms in each bracket, one minus a half is just a half. Um, what have we got? We've got a third minus sixth, that's a sixth, a fifth minus a 10th is a 10th and so on. So we can simplify a bit. This is still the same sum and now we've got the, this calculation to do. You'll notice that all the denominators now are even, so we can bring out a factor of a half and then we'll see we've got a half of one uh, minus a half plus a third minus a quarter plus a fifth minus six. Hmm, that sounds familiar. That's our original sequence or aboriginal uh, series. So we just seem to have shown that S is equal to half of S Um, but if s is a number that sort of implies that S is zero, which have we just proved that the logarithm of two is zero, we hope we haven't <laugh> that that would be bad. And this was a genuine issue that mathematicians realized, oh my goodness, we worked out this thing and then somehow now it also seems to be equal to zero what is going on? Because as I've said, you can, if you've got, like if you're doing one plus two plus three, you can definitely, you can rearrange that, you can do it as three plus two plus one or three plus one plus two. You get the same answer. So why can't we do this now? Well sometimes you seem to be able to get away with it, sometimes you don't. What's going on and what is going on was finally worked out by uh, the great mathematician Bernard Reman, uh, in what is now known as the Reman series theorem. So this is gonna solve all our problems. Well perhaps just the ones about infinite series, not like all our problems. Uh, but it's a great theorem. I need to give you one bit of terminology and it's this, if you've got a series like the ones we've seen, um, there's this concept of being absolutely convergent and that's not like really convergent <laugh>, you know, you can either be convergent or not. Um, but absolutely convergent is indicating taking the absolute values of the terms of the series, the absolute value of the number. It's just like it's magnitude ignoring the signs. So the absolute value of minus seven is seven, for instance. So we say that a series is absolutely convergent. If the series where you add the absolute values, then the ModuLite of the numbers is convergent in the way we've talked about with limits. So as an example, that alternating harmonic series we saw that's convergent to log two possibly. Um, but it's not absolutely convergent because if you turn those minor signs into plus signs you retrieve the harmonic series that we saw was divergent went off to infinity. So that's the definition we need. And now here comes the theorem. It says if a series is convergent but not absolutely convergent, like our alternating harmonic series, this is really amazing. It can be rearranged to converge to any limit at all or even to not converge to diverge. So we saw a rearrangement of the alternating harmonic series. We started off, they converged to log two, then we rearranged it and it seemed to end up being zero actually you couldn't rearrange it to sum to anything at all to converge to any limit at all, which is pretty amazing <laugh>. Now what, what's does this mean that this, this log two thing is wrong? No, what it means is that when you rearrange it, you get a different answer. So if, if you've got this kind of series that converges but not isn't absolutely convergent, you cannot rearrange the terms.'cause if you do it, you know, any arguments you make become null and void, you have changed the sequence of partial sums by rearranging. So you're just not allowed to do that. If you don't do that, then you're okay and you can trust the answer that you get at the end. By contrast, these absolutely convergent series are brilliant because for them you can rearrange it as much as you like and it will not affect the answer. So they're like the gold standard of series, that's what you want. You want something to be absolutely convergent, then you can do whatever you like to it. So this really resolved everything and it means we can avoid these scary apparent paradoxes and accidentally proving that that zero equals one. So this deals with the kind of infinite series sort of problem. I wanna give you one more impossible fact and then I know that I owe you a couple more explanations from the beginning. So this one, uh, we're gonna prove that half equals a third. Um, now seems unlikely. This proof is based on one that was devised by a French mathematician, uh, Joseph Bert Hong, who wanted to illustrate some difficulties with probability. So it think about this question. So you take a circle and you have to find the probability p that's a random chord of that circle. So just a line drawn between two points on the circumference that any random chord is longer than the side of the inscribed equilateral triangle. Okay? What's the probability that a random chord will be longer than the side of that trunk? So there's, I'm gonna show you how to, how we can work this out. So there's take your random chord, how do you do that? Well, you pick two random points on the circumference, right? So pick your first point and then you pick another point and that gives you a chord, okay? So to, to think about whether this is gonna be longer than the side of the triangle, I'm gonna just rotate the triangle. Doesn't matter if I rotate that picture on the right to make it fit in with whatever I need to do. So I'm gonna do that so that one vertex of the triangle coincides with the, the first point of my chord. And then we can see it's quite easy because the chords gonna be longer than the side of that triangle precisely for the other end of it. That the second point that we chose lies in this arc, which is between the two other triangle vertices, and that's a third of the the circumference. So there's a one in three chance of that happening. So clearly the probability is one third of this thing happening. Okay, great. Now I'm gonna give you a different argument. So the different argument goes like this. We can, we can define a chord in a slightly different way or choose our chord in a different way. Uh, because if you take a chord of a circle, if you bisect it taking perpendicular bisect, you might remember that uh, line will go through the center of the circle. So you can define a random chord in a slightly different way. You can pick a random radius and pick a random point along that radius and draw a perpendicular at that point that will give you a chord. And so every chord is uniquely determined, um, by picking a radius and a point along the radius. Okay, so how, what's the criteria then? How do we know if that chord is longer than the side of the inscribed triangle? Well, again, I can, I can put my inscribed triangle wherever I like. It doesn't change the size, so I'm gonna put it so that, uh, its base is at right angles to this radius I've got. And then we can see that the cord will be longer than that side precisely when the point that I choose, uh, is inside that triangle. So above wherever the triangle meets the radius. So where does the triangle meet the radius? Well, uh, let's quickly do a bit of trigonometry. Um, if we join a little radius there, we can see what we've created. There is a RightAngle triangle which has a 30 degree angle and sign of 30 is a half. And so the height of that little triangle is half the radius. So the triangle meets this radius halfway along. Any point when we are making the chord, if we draw the point, um, in the first half of the radius, it'll be longer than the side of the triangle. Otherwise not so clearly. The probability is a half. And I'm convinced by both those arguments and there are arguments that we can see as well. Neither of them is wrong. Neither of them is right. The problem here is that we haven't defined the problem correctly. And this is a real issue in probability. If you've got an infinite amount of things to choose from, when you're trying to select randomly, you have to be very, very careful because the way you are making your choices often will affect the answer. So we have two perfectly valid bits of reasoning. If we choose our chord by picking two random points of circumference, the prob is one third. If we choose our chord by this other way of, um, radii and point on a radius, the probability is a half. We've got to define how we're choosing those random things from this infinite collection. And this is one of the, you know, a major step in understanding probability theory, that actually sometimes we don't have enough information and it doesn't make sense to define a thing. And this deals as well with that one of the proofs that one equals zero. I gave where I said that, you know, zero to the zero is one and zero to the zero is zero. Now both of those kind of make sense that they're, they're reasonable things to say. It doesn't mean that one equals zero because actually if you don't say how we are going to define zero to the past zero, then you can't, you can't trust the answers you get. So normally, in fact, what we say is zero to the zero is undefined. We just don't define it because we'd like, you know, we want it to fit in with something to the X is one. Um, sorry, X to zero is one, but we'd also like to fit in with zero to the anything it zero. And you can't make both of those happen. So we just say we are not gonna define that there be dragons. And sometimes you have to do that. That's the only thing that makes sense. Okay, so we've got a few errors to clear up in the last few minutes. We, um, we talked about whether the smallest number really is one. Uh, I, I'm sorry to have to say it isn't <laugh>, it isn't. But what went wrong? Well, what went wrong is really what we've got here is a proof that there is no smallest positive number. It's a proof by contradiction, a reductive ad absurd because we started with an assumption that there was a smallest number and it led us to something ridiculous. It didn't actually prove the smallest positive number is one. Um, so we could rephrase this as an actual theorem. There is no smallest positive number. And the proof is we say, well let's imagine there were and do a thought experiment. Uh, so let's assume there isn't, sorry, let's assume that there is a smallest positive number and we'll call it X and then we did our little deduction and show that X would have to be one. But that's clearly ridiculous because you know, a half is a positive number that is smaller than one. So we, we ended up in the realm of something false, a contradiction. And so our assumption that there is a smallest positive number must have been wrong because it led us to this, this false statement, this contradiction. So proof by contradiction are really wonderful things in mathematics. Some of the best proofs there are involve this technique. You just take an idea for a walk and if it fails, you know, once you've eliminated the impossible, whatever remains, <laugh> has to be the truth. So I didn't want this next thing I wanna show you another proof by contradiction. I didn't want to say it's an impossible fact 'cause I fully believe it's conclusion. All positive intes are interesting. So let's call this the optimist theorem. So let's prove this. Let's imagine a ridiculous thing to imagine. Let's imagine that's not true then there's at least one boring positive integer. But so if there's at least one of these, we can look at the whole set of them and look at the smallest one. We'll call that n So n is the smallest of all the positive integers. N is the very smallest first boring positive integer. That's pretty interesting, pretty interesting fact about n So, so n is interesting which contradicts what we said. So there, there, so they clearly, as we all, no no, there's no, uh, boring numbers at all. They're all interesting. That's cast iron proof. Okay, so there's one proof left that we haven't dealt with and that was the amazing fact that all triangles are isosceles. And here, well let's begin at the beginning. Let that, that diagram I drew here. Here's an iso leaves triangle and we're gonna see what happens when we move the top. So, so A, B, C, we're gonna see what happens when we move that top vertex a so that it stops being ISOs leaves the red line. You see that is bisecting the, the side bbc. So that's gonna stay fixed. But the angle bisector, which is currently hind behind the red line because when it's an is OES triangle, those two lines are the same. The angle bi sector's gonna move and we're gonna look at where they intersect, where the angle BI sector intersects this, uh, perpendicular bisect. So, um, we'll just watch this happen and you'll see that that place where they meet isn't actually inside the triangle. Now you can see we, it gets close, closer, click on. No, it's not gonna make it, it is not inside the triangle. Now that on its own isn't necessarily an issue.'cause maybe all the, you know, the calculations or whatever we did didn't rely on the diagram necessarily, but well we'll see. So the diagram ought to look a bit more like the thing on the right, not the thing on the left. However, all of those calculations or or deductions about the congruent triangles, they are all still completely valid. Those were correct. All of those pairs of triangles really are congruent to each other. So we still genuinely, it's true that AP equals AQ and PB equals qc, but the problem is guess what? It's a rogue minus sign. I knew that there'd be one somewhere. So yes, ab if you look at the diagram on the right, AB does equal AP plus pb and AP does equal aq and PB does equal cq, but that does not equal ac.'cause Q is outside of the, of the triangle. Now we had to extend that line to drop a perpendicular. And so AC does not equal AQ plus qc, it equals AQ minus qc. And that of course <laugh> changes everything. So here the diagram, it wasn't that it was wrong, but that it led us to think something that was not guaranteed to be correct. So if you see some geometry, just make sure to draw a diagram that uh, that's perhaps quite accurate and see, see what you get. Um, well let's, let's finish up with another theorem. There exists a triangle with three right angles. Now we've proved all triangles, right? Softly. This should be a dole. Well, now this is kind of the final thing that I want us to think about. That when we prove something, we start with initial assumptions. When we prove that the angles in a triangle add up to 180 degrees, you may remember this proof, it relies on the parallel postulate 'cause you have your triangle and you the top point of that. Um, you draw a line through that top vertex parallel to the opposite side, and that then you do some stuff and you get very quickly the angles add up to hundred 80. If you do not have the parallel postulate, then you can't prove, at least not in that way, that the angles in a triangle add up to hundred 80 degrees. That particular proof doesn't work. And it turns out that there are geometries, for example, the one we have on the surface of a sphere in which the angles in a triangle do not add up to 180 degrees. And here is one that has three right angles. And that exploring the idea, the question, when is this true? When could this be true? It's actually one of those most powerful questions in mathematics. So it's really fun to explore when might this thing be true. Now, this is obviously a very non-nuclear juncture is a very deep and exciting and interesting area of mathematics. I thought I'd give you a slightly less highfalutin puzzle to finish with, um, along the line, what I call fake proofs of true facts. So fractions again, and sadly they exist. Um, we all know how to cancel fractions, right? If it's the same thing on the top and the bottom, we can cancel it. So 16 over 64 does equal one quarter because we just cancel those sixes. Yeah, that's how cancellation isn't that? No. Okay, we'll do another one. We'll do another one. I I, okay, 19 over 95, right? Cancel the nines. Yes. So it works, uh, does not always work, but when does it work, right? It's quite fun. In fact, you can extend this if you have a one and a bunch of nines and then the same number of nines on the bottom and then a five, you can cancel all of those pairs of nines and you do magically get the answer one fifth. There's a little proof of that underneath very quick thing. It's actually a true proof for once. So you take n nines and then a five, that's 10 to the power NN plus one, take away five, bring out the factor of five. Then you get two, uh, times 10 to the N. So two with n zeros minus A one, and then that equals exactly five, lots of one. And then N nines. Okay? So you can actually prove this. So the challenge is find other, uh, true facts with fake proofs like this. And in particular, I don't know what the answer to this is. Is it ever true that you can add fractions in the way that some eight year olds and nine year olds would like to add fractions, which is A over B plus C over D equals A plus C over B plus D. So that's a little challenge to keep you going until my next lecture, which is my very fine aggression lecture Next time, uh, on June the fourth, we're gonna talk about some real proofs and, and you know, the beautiful ideas that we can exploit and enjoy in mathematics. And I'm gonna show you some of my favorite proofs, but we'll stop there. Thank you very much. Thank you Sarah. Um, I'm Martin Elliott. I'm the provost of the college and it's always an immense treat for me to come here, Sarah, speak. I'm also very reassured that I now know how the chancellor works out the next budget.<laugh>. Tremendously powerful. I'm gonna take some questions online first, is there anything other than an infinite series that can cause false statements to be proven true? Well, uh, yes. Proven there's got to be a hole in your proof somewhere, but I mean, yes. And as we've, we've seen some of the ways that that can happen. I did not divide by zero at any point today, but that's another way that you can often make false things happen to true. You disguise it, you do some algebraic calculation. You say, you know, let x equal Y at the beginning and then you wait a few lines until everyone's forgotten and then you divide by X minus Y and hope nobody notices. So there's a lot of ones that involve division by zero, but I thought this is a very superior audience and they're not, they are gonna spot division by zero just straight away. So I I didn't do that. Let's take some questions from the floor before I go back to those online. So anybody who have one Y the, um, the, the one I, um, uh, hadn't seen before, and I'm not quite sure of the, the problem, the, the differentiation one mm-Hmm. Um, the x plus X plus X equals x squared. Can you you say? So? I think so this is where you have x, lots of X and then on the right hand side you have X squared. Now, one problem is that I had X, lots of X, and when I differentiated all those little Xs, I got ones, but I didn't do anything about the fact that I had X of them. And I think that's, that's the basic issue. So I ended up with X, lots of one, but it's not really clear how you make a function that says here are X, lots of X and actually you sort of force yourself to write X squared and that's <laugh>, that's the issue. So I mean, if you were to try and do this from first principles and properly represent the left hand side, I ignored the fact that I had x lots of this thing and I didn't do anything about that. That's really what's gone wrong there. Any more questions than the before? Otherwise I'll read you one out. Uh, why are there irrational numbers that can't be expressed as a ratio of two other whole numbers? And does it show us anything or prove anything? Uh, well why are there, you know, it might have to ask a higher power than me, but there definitely are <laugh>. There are, and, and so you can see this, the, um, if you take it just a square whose sides of one and the diagonal of that square cannot be expressed as, as a fraction, however hard you try it, square of two. Um, and these, so these are called irrational numbers and they are everywhere. Uh, but you know, nothing to be afraid of. It's just that we, our, our human minds who thought of using fractions to write things just didn't realize that there are these other numbers out there. So yeah, they, they, they are, they vastly outnumber. In fact, the numbers that come over written as fractions, almost every number that you pick up random off the street is irrational. How, how frequently do these things happen in the conversion of pure mass as it were into applied mass? Apart from the first example you showed of us landing on Mars, how frequently disasters happen when people get these sequence of assumptions wrong? So I think we are quite lucky in that most of the time I would say we, we've got away with things over the years, let's say <laugh>. Um, so in the kind of 200, 300 years ago, people would kind of play around with infinite series and assume everything would be fine. And it, because the series they were playing around with happened to be quite well behaved. Mostly you can get away with it, but that's, I mean, that's no way to do mathematics. But it took a while for to really come up against these issues for real. And I think one risk that can happen if you're transferring kind of from pure mathematics to applied mathematics is maybe the pure mathematicians have this cast iron thing that says, under these conditions and these conditions alone, you are allowed to do this. And then perhaps that may slightly get lost sometimes. And then you just say, oh, I know I can do that. And I'm sure it's fine. Just every so often it isn't fine. Um, but you know, in the, in the real world, of course, uh, you, you have mathematical models, you have approximations and you can test and everything and make sure that, that it will, it will work in real life. But we, it is important to, to make sure, yeah, that we are not gonna end up with <laugh> with, with proving impossible facts. So yeah, occasionally it crops up. Uh, thanks for that. Is there a governing body or anything to accept a proof? Oh, okay. Yeah, there should be. Um, well what, what in practice, what happens is when you're a mathematician and you, you think you've got a theorem, you will write that up as a, an academic paper. You will send it to a journal and the journals then will send that out to referees. So, so you know, people, other mathematicians who will look independently at what you've written and try and follow your argument and if, if there's something that they don't understand or why, why is that allowed? Then they send it back and say, can you explain how you got from here to here? And if you can't, maybe you realize there's a mistake. So the idea is, you know, peer review will capture, um, any errors that you might have made. And you know, as you go on in your mathematical career, hopefully you make fewer gross, logical, uh, <laugh> mistakes, you know, it still can happen and, and you know, not, not perhaps not these kind of things, but maybe other things. And of course you can always, everyone can make mistakes, um, accidentally. So yeah, it's peer review that happens and then things are published and your other fellow mathematicians will read them. So that's the closest we get. I guess Last thing you said there, a great example of that was Andrew Wildes. Yes, Yes. So Andrew Welles, so famously the proof of firm last theory in which, which came some, some years after firmer claimed he'd, uh, got a nice little thing that didn't fit in the margin. Um, but Andrew Wells approved firm's lost theorem, but his first proof as submitted had a fatal flaw and happily that was able to be fixed. But it, the first original proof was that like 93, um, had this, had this mistake in it that was, it wasn't like, oh, you've forgotten to do something, some trivial thing. It was genuinely a fatal flaw in the proof, but happily he did manage to correct that. It took a lot of work and then the correct version was, was submitted and everyone does now believe it. So I've got one last question, which borders on the philosophical, oh, <laugh>, the basis of mathematics is the concept of zero unity and infinity. Do those concepts apply universally? Alright, well, uh, if you accept, certainly those are very important concepts, zero, one and infinity. And you can almost, so I remember, uh, James Joyce, famous mathematician <laugh>. He, he had a little notebook and he, he wrote these sort of links between these three concepts. So zero divided by one is zero, one divided by zero is infinity. And you know, infinity divided by one is, so all of these things you can sort of relate to each other. So they are, they're wonderful concepts and in a way they're all linked to each other. But, um, I say I think that they are universal because, you know, and not zero IE you don't have anything that's clearly a thing that can happen anywhere. Um, one, the basic unit from which everything else is built, I feel, you know, that's got to, if you have any alien civilization and they count, they're gonna start at one like we do, you know, these things, these things are not really choices. And there's a gone who, what's the name of that mathematician who said that, uh, God invented the, the natural numbers and everything else <laugh> comes from man, right? But yeah, I think these things are universal concepts. Thank you. Well, I'm sure you'll agree with me that Sarah is more than one in 1,000,001 in infinity. Thank.