In a recent video we discussed polynomial equations, or functions; basically anything that looks like this, x raised to a power. We explained if you graph these things the solution or the root or the zero to that polynomial can be easily seen as the point or points where the plot crosses the x axis. Professor Eisenbud also showed us that an equation to an odd power must have extreme points down here and up here, meaning at some point it must cross the x axis. So all polynomials to an odd degree have at least one root among the real numbers, maybe more if we get a little kink like this. Great! But what about even degreed polynomials? For example, this one; well that's never going to cross the x-axis. I mean, are there polynomials for which there are no solutions? Well strap yourselves
in because thinking about that is leading us towards one of the most important theorems in all of mathematics.
Let's do it. So I'm going to start with that equation, that equation of even degree that didn't have any roots. f of x equals x squared plus 1.
- (And that's not lost cause?) That is not a lost cause. People probably have heard of the square root of -1,
there's a very famous mathematician, Sir Michael Atiyah, who says that the square root of -1, i, is the most important single invention in all of mathematics. It certainly solves this equation because if I claim that I have i equals square root of -1 then by definition i is a root of that polynomial.
- (But there) (is no i on here, we haven't got it.)
- We haven't got an i so we have to find out where i lives. And by the time of Gauss in 1800 it was pretty well understood that you should think of a plane of numbers called the complex plane. And if you put in axes there will be a real axis and a complex. And 1 will be in its comfortable place there, and i will be here. And you can do ordinary arithmetic on these things, you write some number - people usually use z for some reason for complex numbers - so z would be a certain real part a and a certain imaginary part, maybe b. And then I would write z equals a the real part plus bi the imaginary part. And complex numbers, you multiply them just the way you think. If I want to multiply 1 plus 2i times 2 minus i let's say, then I just multiply them in parts just as you would an algebraic expression if i were x. You can do arithmetic, so I can plug any complex number, the things in this plane are called complex numbers things like z, I can just plug them into any polynomial and I get another complex number because I can multiply them and I can add them and do all the things that I have to do. I can also divide them and they satisfy all the comfortable laws of arithmetic like commutativity, a times b is b times a and all those things. So I can- just comfortable arithmetic. And I can use them in polynomials. If I do this I might want to separate the source of the polynomial, the place where the x's live, with the values, the place where the f of x's live. So I'm going to use a red plane for that one. And now if I draw it like this, so here's the again the
real axis and the imaginary axis. It's call the imaginary axis, not because it isn't there, it's just as real as the real axis, but the numbers on it are the pure imaginary numbers, the ones without any real part. And then I can do things like squaring the number, I can for example, I could square i. And I would record the value of the square over here; so for instance if my polynomial f - let's use green for the polynomial - so if I have f of a blue x is a red x squared for instance, then f of i would be i squared, which by definition is minus 1. So I would I would say that f sort of takes this point on the real axis over here, here's here's f the green line, over to the minus 1 point there. So I think of f as taking each point in this plane to some point in that plane. So if I want to think about the roots of a polynomial now, a polynomial is one of these things that takes a point here to a point here, and the roots are the points in the blue plane that it takes to the zero point in the red plane.
- (Right in the middle there) (in the bullseye?)
- Right in the middle right, here's zero. That's our goal. Here's the Fundamental Theorem of Algebra: it says, this is due to Gauss, again I like to write the word theorem. About 1800 I forget the exact date.
- (He was pretty) (handy that Gauss guy wasn't he?)
- He was pretty good, yeah. People always like to say, well, he's no Gauss but that's not a not much of an insult. Anyway, the theorem says that any polynomial with complex coefficients - can I spell coefficients? Yes. - having degree d, that is the polynomial not the coefficients, has at least one complex root but actually it follows from that that it has d complex roots computed with multiplicity, but let's just say has at least one complex root, then we don't have to talk about multiplicities. And this is the Fundamental Theorem of Algebra because it's the basic connection between algebra and geometry. Roots are points somewhere, so they're geometric objects; and polynomials are algebraic objects, so this is this is the connection and makes all of algebraic geometry work. It makes me happy because I'm an algebraic geometer, it gives me something interesting to do.
- (So what) (you're saying is there's no polynomial) (that I could concoct that you wouldn't) (be able to find roots for, as long as) (you're allowed to have complex numbers) (in your tool belt?)
- That's right except for you're saying that I could find them. I'm going to show you they exist, I'm not going to find them for you. If you want to actually find the root then Newton's method of fastest descent, things that- numerical methods are very powerful. But it's hard to find roots of polynomials, much harder than proving they exist. So I'm just going to prove they exist. So how can we do that? We can't do this trick with lines but we can do a trick that's a lot like it. And for that I have to explain something about what happens when you multiply complex numbers. So I think of a complex number, here's my blue complex plane, and here's a complex number. So I think of- I like to think of it as a vector drawn from the origin. And if I think of the vector, the vector really has a length, and then once I've told you the length, which is called the absolute value of the complex number, so z. So if you know the Pythagorean theorem you know the length of this vector is gotten by taking the square of the real part plus the square of the imaginary part and taking the square root of that. So it's- so if z is a plus bi and the absolute value of z,
I'll write it out, it's the magnitude it's called, is the square root of a squared plus b squared. Just Pythagorean theorem. Now there's another thing that you need to determine where that vector is, and that's the angle, let's call it theta. Let's say we knew that the magnitude was 1, then we could compute the sine of theta and the cosine of theta and if the magnitude were 1, if this were of length 1, then the sine which is the opposite side over the hypotenuse would just be the opposite side of the triangle which is the imaginary part. And the real part would be the cosine of the angle. So when I multiply things, multiply complex numbers, I'm doing these tricks with the real and imaginary parts, some- I should think of that as something happening to sines and cosines. And if you remember your high school trigonometry, which I certainly don't, then there's some addition law for sines and cosines and it comes out to saying this: if I take two complex numbers z and w, here's w, and I take their- the angles, let's call
that one psi; then when I multiply them I multiply their lengths, I multiply the magnitudes, and I add the angles. So I would take psi and then I would add theta to it and I would get something this way and the length would be out here someplace, right? So this is z times w. So it's really easy to multiply complex numbers geometrically, you just multiply the lengths and add the angles.
And that's some formula which you can
figure out, but it's- I'm going to take it for granted. Now we're ready to go; and I want to take an arbitrary polynomial - do you have more of this nice paper, Brady?
- (I do...) And I want to think of this as going from a blue complex plane, here's 1 and here's i, to a red complex plane.
- (So David) (before, when we were doing the
video before, when you) (were doing this, we had two axes;) (here you're representing-)
- So each- this this is one axis and this as the other axis. If we wanted to make a real graph like that we'd need to do it in four dimensional space - and it's hard to imagine four dimensional space. I want to graph this function, I want to tell you what happens when I apply x, and I want to find the point here which- or at least to convince you that there is a point, a blue point, which will go to zero on the red side. So that's what I want, where is that point going to come from? How am I going to find a point which
does that? (To prove one exists?)
- To prove one exists. So I start by scratching my head and thinking, I don't know anything about this polynomial just like before. But I do know that the- I do know the leading term, that's x to the n, and I know if x is very big that's likely to be dominant. On the other hand if x is very small I can do something too;
if x is zero then the value will be C. So I know that the zero point over here in the blue plane goes to C over in the red plane. So here's a nice red C, let's say C is down here. C is a complex number so it's just someplace in that plane. And I know that 0 goes there. Now that's not terribly helpful because I wanted to get a point to go to 0 not to go to C. So what can- what else do I know? I I know that just as before, if I take x to have enormously large magnitude, then x to the n will have that magnitude to the nth power, that'll be much much much bigger than x to the n minus 1. So for really large x's, x's way out in a big big circle around this, those are the x's with big magnitude. So this is magnitude of x much bigger than 0. And now it's a circle because it could have any argument, it could have any angle. I know that x to the n will be even bigger and I'm going to shrink the scale on this red plane so that I don't have to draw it off the- off the table but it'll go to some huge circle. Now, what will happen when I take x and I take a walk with x? I think of it as a dog on a leash and I lead it around the circle? So we know what happens, the magnitude - remember I'm pretending that this polynomial is x to the n because it's like x to the n - the magnitude
will stay the magnitude of x to the n, the same thing it was before. So we will walk around this circle and the real polynomial will stay close to the circle, it'll sort of dance around like a dog would on a leash after all. Something else is happening though, as I move x around this one moves n times as fast because I'm multiplying the angles by n. So it actually goes around n times. For the purposes of this proof it's enough to know that it just goes around but we'll see why the n is important there too. So as I move around here once, I move around the circle actually many times. Okay, that was nice but did it help? Not obvious. Now comes the trick, here's- the rest of the proof once you see this you you just know it's right. I'm going to shrink the blue circle down to the point 0 little by little. And I'm going to watch what happens on the right hand side. On the right hand side this circle, and remember the real values of the function were dancing around this circle somehow but they're a nice continuous curve that meets up to itself again. This circle will begin to shrink too, because I know that when x is zero or very small this circle has to be very close to C. So somehow the circle shrinks down in all directions, in some very uneven way, but at the end of the day it forms a little circle around C if x is performing a little circle around zero. Now in that process somewhere this this circle has to pass through the origin, and that's where I catch my roots. So somewhere along the line as it shrinks I have to see a picture like that. And that's the point which is- which I was looking for. That point happened maybe from over here and it will be a point on a small- a smaller circle here where this function actually goes to zero, where it has zero value. We said that it really goes around n times; so actually this path is going around n times and as I shrink it will keep going around n times so there'll be n of these curves that go- that move through the point one at a time or maybe all at once and so there'll be actually n roots. And so that's the full Fundamental Theorem of Algebra. If you're still watching this video you've got the sort of attention span that makes me think you might enjoy the Hello Internet podcast which features fellow youtuber CGP Grey and myself. We talk about all sorts of things, very often we talk about what it's like making YouTube videos. If you'd like to find out more check out the links, I'll put things in the video description.
I didn't watch the whole video, but there is a proof with Louvile's theorem.
Then if f(x) is a polynomial without any roots then 1/f(x) is entire an entire and bounded function. So that 1/f(x) is constant. Thus f(x) is constant.
This shows that if a polynomial has no roots then it is constant. So any non-constant polynomial has at least one root.