So we are going to talk about what? The opposite of infinity? What is the opposite of infinity if it's not minus infinity then it is something that is the smallest thing ever What is the smallest thing that you could have and not quite zero? The closest thing to zero it's called an infinitesimal. is it even a thing? Because there is something about that, that makes you feel a bit uncomfortable It's been an idea that's been controversial for a long time, I mean thousands of years. Some people like it. Some people hate it. So what an infinitesimal would be? A smallest number possible close to zero as possible so it would be something that's bigger than zero but is smaller than all other numbers let's say 'r' for real number. It is smaller than everything else. That is slightly uncomfortable because of this: if you have two numbers, if we have 'a' less that 'b' there is always a number that I can fit in between let's call it 'c', there is always something I can fit in between and I can prove that because 'c' could be a+b divided by two and that's proven. There is always something that you can slip in between two numbers, no matter how close you think they are. - Can't we slip in something else? So the idea of having the next number to zero makes intuitive sense but mathematically it's alarm bells ring. So this has always been a problem. But people use them to solve problems like this: so let's say I have a circle and I want to know the area so this is one way you could do it. I can take my circle and I could split it up into triangles These are meant to be very small, thin triangles. These are wedges at the moment but that's the problem. We're going to pretend they are triangles. Now they are wedges, if I pretend it's a triangles that means that it has a straight base. But the thinner it is, then the closer it would be to a triangle. So, you kind of say, "Okay, well let's pretend they're really really slim." like the base is an infinitesimal. really slim, as slim as possible. And then what you do, you take this circle of triangles and you kind of unwrap it like a hand fan and you know, when you take a hand fan, shooo, like that. So that's what I'll do with this circle. That's my first triangle down here. So that's my centre. I take my first triangle there it is, down here at the bottom. Take the second triangle still connected to the centre I've kind of stretched it though so it looks like that. And I take the next triangle and it's a bit stretched again and it looks like that And I take the next triangle All right, so I take all my triangles I stretch them out They're all connected to the centre like
-they were before So they get really stretched out, really thin. So they turn into this, right. It's actually turned it into a triangle. Can you see that? Can you see that triangle I've done? My drawing is bad. You see that's a triangle. Now each of these triangles here, these slim triangles, have the same area as they did to begin with. They've been stretched. The perimeter is different but they have the same area because what is the area of a triangle? It's a half the base times the height Where they've been stretched, they have the same height up to here. They got the same base. So the area's the same. But the big triangle I've made it has height altogether the radius of the circle 'r' And the length of its base is the circumference of the circle. So what is the area of this triangle? It is a half at the base is the circumference, the height is the radius. That's the area of the triangle. The circumference? Two pi r. Right? So it's a half times "two pi r" - the circumference, times r. And you get pi r squared. And that is the area of the circle. You get the right formula for the area of the circle. And so you did it by splitting it up into thin bits and kind of adding them together with this clever idea. The guy who came up with that idea was Kepler. Do you know Kepler? The guy who worked out the elliptical orbits of planets. So he was interested in working out the areas of an ellipse. There is something that should make you feel uncomfortable though because there's always that small error, isn't there? I mean I was saying that these triangles were approximating the wedge. There was always the -
- and this is what the problem people have had with infinitesimals. There's always, - shouldn't - not quite - yeah what happened to all those curved bits? So what they did - what the mathematicians were doing when they were using it, is they were ignoring them. So these mathematicians who were using infinitesimals were saying, "Well it's really close to zero. We'll ignore it." And then there were other mathematicians saying, "Well hang on!" "You can't just ignore it. It might be close but it's not the same." "And they can't be ignored." And this is the problem people had. But yet, it seems to have worked. I'll show you a different idea that we used to do. Right, I'm just throwing my money around. Because I'm so rich I can throw my pennies around. Yes, the idea is that I've got my stacks of pennies here, and they're meant to be two cylinders, right. but if I took this one and started to mess it up a bit so it wasn't a cylinder and to do something like that. Okay, so that's not a cylinder any more. And the idea was, what mathematicians said was, "If you took it, if you compared them slice by slice, then each slice has the same cross sectional area." like the area of a penny here. So you can say that this has the same volume as a cylinder. It's not a cylinder but because you can compare it slice by slice, instant by instant, they have the same volume. But those slices don't have depth, they don't have volume themselves. So you are comparing cross sectional area to cross sectional area. So it's kind of like adding up all these cross sectional areas but they don't have depth. You're adding up all these zero - - these things with zero depth. So how can you add them all up and get a volume? But that's something they would do. You can compare two shapes like that. And if each instant has the same area we can say they have the same volume. So it was this idea going around as well. And infinitesimals were really important when Newton and Leibniz started to use them when they discovered or invented calculus, which is a massive subject in maths; really important. So in the seventeenth century they discovered calculus. This was trying to work out the area under a curve. Right. So let's try to work out the area under a curve. So what you could do, is you could take this curve, if you want to find the area underneath it here. So you could split this up into thin strips. And you were saying that each of these strips were like a rectangle. If you turned that into a rectangle you do get a little bit of error but the thinner the rectangles the smaller the error. So you can add up all these rectangles and find areas under curves, which is something mathematicians do. So what Newton and Leibniz were doing - I might use a different colour So chop it up, and let's say you have a little more here, let's make it a different colour. So the base of this rectangle is something small. Infinitesimal perhaps. And I'll call that delta. That's actually a traditional notation. So let's do that. And what you're saying is if this curve - we call this curves things like f of x. So what I'm saying is if this here is x, then this is going to be x + delta, x plus a little bit. And the width of the base is delta. If you work out this area, all the way up to x + delta. Let's take the area of all the way up to x + delta and I subtract all this green bit here that's less than x. So subtract the area, that is here, below x. That is equal to the area of this strip. So I've subtracted all the other strips. And what is the area of that strip? It is f(x), that's the height of the rectangle multiplied by this little base. Or to put it another way you could say the function is [Area at] x+delta minus the area at x divided by delta. So what we've noticed is that the function was rate of change of area. It might sound a bit abstract. But this is the big discovery that Newton and Leibniz made. That was actually hugely important because that means you could work out area by working out rate of change. And rate of change was something that they could do. And Newton and Leibniz were both working on this at the same time. And they both knew of each other's works, kind of, but Leibniz published first. And then Newton published three years later his Principia Mathematica. And there was this big debate about who came up with it first. And it got really nasty. Newton really wanted to protect his discovery. And Leibniz was almost - he couldn't be bothered to argue any more. And in the end they set up this committee to work out who discovered calculus first. And it was all Newton's friends, and Newton himself wrote the report. And do you know what? They discovered that Newton came up with it first. And then Leibniz then died with his discovery, I guess, taken away from him the honour of the discovery taken away from him. And historians have looked at their notes and decided that they came up with it completely independent of each other. So now they both get credit. But this uses infinitesimals. It's the same problem. And Newton knew this as well. There's always this little bit of error. It doesn't quite work. And in the end, about two hundred years later they started to tidy this up, and they got rid of the infinitesimals They realized you don't need to use the infinitesimals. They had this new idea called limits where these could get thinner and thinner and thinner and they would approach zero. And this was called a limit, and this was mathematically rigorous and consistent. And it meant you didn't have to use infinitesimals to work out these areas and they could be thrown out of the theory. And not needed any more. And there are no infinitesimals in the real numbers. They were gone and then they made a comeback. It's the "can't keep a good idea down." It made another comeback in the 20th century. And it took til the 20th century, for it to made another big comeback In the 1960s, a guy worked out a system where you could include the infinitesimals. You could have a mathematically rigorous and consistent system. You can solve problems using this system. The guy was called Abraham Robinson. It was in the 1960s. He could only do this in the 20th century because it required mathematics that had been done in the twentieth century. They took the real numbers and then they added in infinitesimals into what was already there. So infinitesimals don't exist in the real numbers but they added them in. So they said, "an infinitesimal is something that satisfies this. Right." It is something that is less than all the other real numbers and bigger than zero. So they added this idea, so it's an addition. It's called the hyperreals. So you got these infinitesimals - let's call them 'e' this is an infinitesimal and you also have infinites in this system. So you can have one over an infinitesimal is an infinite and that's allowed, that's included. If you have two numbers that are infinitely close together, if you said x and y are close together, that means that x minus y is an infinitesimal. So there is an idea of being infinitesimally close in this system. In the real numbers zero is the only infinitesimal. In the real numbers we know we would never do this, would we? One divided by zero is not infinity. We would never do that. And that is still true in the hyperreals. Everything that is true in the reals is still true in the hyperreals. So that is not true in the real numbers and it's not true in the hyperreals either. So something like this: x plus y equals y plus x that's true in the real numbers and it's true in the hyperreals. If you have x squared is greater than x, x has to be bigger than one, that's true in the real numbers and it's true in the hyperreal numbers. We know that 2x is not equal to x that's true in the real numbers and it's true in the hyperreal numbers which means if you had an infinitesimal an infinitesimal is not the same as two times the infinitesimal, and it's not the same as three times, or four times, or n times. These are all different. They'll all infinitesimals but they're all different. So these things are not the same. So there's lots of infinitesimals. There's not just one infinitesimal there are lots. There's this nice little quote, it's this guy Abraham Robinson wanted to get into the mind of Leibniz that's the quote that I've read. He wants to get into his thinking. And the thinking of infinitesimals and can we make this work and can we use it. Now this system, finally, is consistent, it is rigorous. It can be used properly. It can be used to solve problems. It's called non-standard analysis. People fans of this systems say it's more intuitive and they say it can solve problems with shorter proofs. It has finally made another comeback. Because what about now, before we divided by 2 if k was even But in the real numbers, all the non-zero elements, and k is certainly not zero, all the non-zero elments you can take their reciprocals, you can divide by them.
