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
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