Recently we did a video on the most mysterious and beautiful identity in
mathematics
e to the pi i is equal to minus one. Comes up three times in the Simpsons
which, of course, makes it even more important. Now,
afterwards a few people challenged me to come up with an explanation that even
Homer can understand and I've actually been agonizing over this ever since.
And today I want to do just that, I want to explain
e to the pi is equal to minus 1 to someone like Homer. Ok, someone like Homer
who can only do addition, subtraction, multiplication, division.
So, we have to remind him or tell him two things. The first one is that i is this
strange complex number square root of -1,
i squared is -1. Second thing is just kind of a reminder. If you've got a
semicircle of radius 1 then the length of the semicircle is pi.
Ok, so keep those two things in mind, we have to use them later on.
The first thing I have to explain to Homer is what is e. So to do that I
give him a dollar and tell him "Go to the bank". Now I've arranged with the manager here
to give him 100% interest over a year, ok.
So what happens to this one dollar when Homer puts it in?
Well after one year he has 1+1 = 2 dollars.
Now this is actually not the best you can do with a 100%
interest, you can do better if you find a better bank. And we found a
better bank, the Second Bank of Springfield. At the Second Bank of
Springfield they calculate and credit interest twice a year. So after six months
what happens?
You get fifty percent on what you've got there. So that's 50 cents which gives 1.5 dollars.
Now another six months pass.
Half of 1.5 is 0.75, so you have to add that to 1.5 and that gives you 2.25. That's what
you've got
at the end of the year if you calculate and credit twice. Now, at the Third Bank
of Springfield they do it three times.
So what do you get? After four months you get that, after
eight months you get that and at the end of the year
you've got that, even more.
And it's actually quite easy to figure out the general formula for this.
Well, maybe
Homer cannot do it, ... , but I can do it. So it's this one here and you can probably
do it, too, if you're watching this video.
So, it's (1+1/n^n. So, if you credit n times
throughout the year
that's how much money you have at the end of the year. Let's just check this for the
simplest cases 2 and 2.25. So for n=1, we've got 1+1=2
hmm is 2 - okay.
1+1/2 = 1.5 squared is 2.25, ok, works.
Ok, works in general. Now, this is really good news but maybe what you think now,
Homer definitely thinks this is: Well I divide more and I get more money. So,
if I just divide enough maybe I get a trillion dollars at the end of the year.
Sadly that doesn't work.
So, for example, if you divide in 125 parts you get that much money at the end
of the year, or have that much money at the end of the year.
Now, if you crank up the n what happens is, well, that number goes up but it goes up
very slowly and actually settles down to a number. So, if we push the whole thing to
infinity,
take the limit of this, we get this number here: 2.718...
dollars and that's the absolute maximum, that's "continuously compounding interest",
that's what it is, so we can't do any better than this,
that's e. And that's also where e comes up for the first time
historically, exactly this sort of consideration, ok. Cool, so
now we've got e. We're ready to move on. e^(pi i). Well, not so fast. Let's just
go to e^pi first which is actually almost as mysterious as e to
the pi i. Why is that? Well it's got a special name
it's called Gelfond's constant and eventually I'll definitely make a video about
this one, but just for today
just ponder it a little bit. What does this actually say. Well it says weird
number to the power of another weird number
and you supposed to calculate this. How do you calculate something like this? I give
it to you on a piece of paper and you don't have a calculator.
That's ... strange.
I think nobody will be able to do this. Well, it would be doable if pi was equal
to 3 because then we know we just have to multiply,
you know, maybe chopped off bits here (at the dots), three times together and we get
a rough approximation to what we are looking for. But, no, we have this one here.
So we really want to calculate this,
we want to really know for some strange reason
how much money Homer has after pi years if you're compounding interest
continuously.
That's what I want to know, I can't go to sleep tonight if I don't know.
Ok, now the trick here is, we have got this bit here, which gets us closer close to e
the more we crank up the n.
Ok, and so if I put that one up here and put a large number in here we get the
right thing, or approximately the right thing, or as close to the right as we
want.
Alright now that looks still pretty awful, okay, and, well, let's muck around a
little bit with it.
So, the first thing we do is we
multiply by pi here and there, and see when you do this on the top and the
bottom
obviously nothing changes and it actually looks a bit uglier than before. But what's nice
is that these two bits are the same and, you know, what you have to do now
to get this is to just crank up the bit in the box.
So n = 1 we have this, n = 2 we have that, and then that, and .... that's still
pretty awful !
Except what's really important here, and that's a really really nice trick, is,
what's really essential here is, that we're going up.
It doesn't matter how we're going up, as long as we're going up towards infinity,
we can go up via nice numbers: 1, 2, 3, ...
That will also get us there, and that's actually what we do and this here is
exactly what we're looking for. So here it's like really awful to the power of
awful, but now we've just got addition, division, multiplication
that's all we have to do ... just a lot. But that's basically all we have to do,
so we are getting there.
And, of course, the pi here stands for really any number whatsoever.
So what we've done is actually we've figured out how to calculate the
exponential function with basically nothing, with just this.
That's what we've just figure out, that's a pretty pretty good effort.
Alright, so let's graph this (e^x) and a couple of those guys (the functions on the right) and see what
happens.
So I've graphed the exponential function and I graphed the first one of these guys,
well the second one really where we take m = 2.
Not a terribly good fit but if you crank up the m
you can really see how good this gets. And, actually, when you press, you know,
the button on the calculator that's what your calculator does at some level.
It just adds and multiplies and divides and these sorts of things. That's all you
can do. Anything, anything complicated in mathematics, you know, when you do it numerically
has to be reduced to just basic arithmetic
otherwise it doesn't work. Okay here we go.
Almost there now. Just chuck in your pi i, that's what we're interested in, and go
for it. And actually we could go for it at this stage. It's actually not very hard
to multiply things like this.
Well, this is basically a complex number in here, so we've got a nice number plus a
nice number times i. It's actually not that hard to multiply a
couple of those things together I could teach you in a second actually i'm going
to teach you in a second.
Let's just do it on Mathematica and see what Mathematica spits out.
So for m = 1 we get this number, it's also a complex number. Doesn't
matter what you put up there,
doesn't matter what m is, the result is always going to be a complex number. So,
let's crank it up now.
Crank it up, crank it up, crank it up, all the way to ... what did I do, a hundred.
And you can see that this first bit here gets closer and closer to -1 and
the second bit here, that nice number in front of the i goes to 0. So
basically the ugly part goes away and were left with the -1 if you kind
of go to infinity. We could stop here, but actually I've got this really, really
nice way of multiplying complex numbers which we can apply to this, multiplying complex numbers with triangles.
Let me just show you.
Ok so here we go. Now complex numbers you can draw.
Real numbers you can draw on the number line, complex numbers you can draw in the xy-plane.
Actually Homer stands right on top of the xy-plane so we might as well
use it and he can really relate to it at this point in time.
So here we've got the real number line there is 0, there is 1, there is 2, and so on. And, well, we've
extended this real line by the complex plane. It's just this whole thing.
Every complex number corresponds to a point in here. For example, 1.5+i is just
the point where you go 1.5 over here along the x-axis and then one up in the direction of the
y-axis. And then this guy here, for example, 1+2i. Well, 1 over here
and then 2 units up.
Ok, now multiply those two things together.
So what do we do? Well, we do 1 x 1.5 is 1.5 then 1 x i is i. 2i x 1.5 is 3i
and then the last one
that's where we have to remember that i squared is equal to -1, so
this is -1 x 2 is -2.
Now we just combine things together in the obvious way, so there and there and that's
the product. And, of course, that corresponds to a point, that guy out there.
Well how do you get from here to there? Not obvious, right, we can do this, but you can
actually see at a glance, you can see at a glance that these two guys get you up
there. How? With triangles!
Okay, so to every point, to every complex number we associated a triangle and the
corners are 0, 1 and that point here.
So that's the first triangle. Let's just save it.
Second triangle 0, 1, point.
Now we align them
like that, stretch this one, the red one, so that these two sides are the same and
there is your product.
Brilliant isn't it. So you just kind of aline and stretch these triangles and you
know what happens. And actually if you know the triangles
it's pretty easy to predict where the product is going to be. Let's do another
example. Let's do this one here squared. So what you do for squared is
this triangle twice.
Ok stretch it, that's the square. Now,
cubing and we're going to have higher powers so we need to see what happens
here, so just make another triangle, stretch it,
that's the cube of this number here.
Alright now higher powers. That's the complex plane. For the higher powers this
circle here, the unit circle, the circle of radius 1, around 0 plays a very
very special role.
Why is that? Well here is a complex number on that unit circle. The triangle that
corresponds to it has two equal sides, there and there. So, when you align two such
triangles
what happens? Well you don't have to stretch, right.
Let's see what happens when I kind of raise this to the power of 8 ... 2nd power, 3rd
power, 4th, 5th, 6th, 7th, 8th. That's the 8th power of that guy here.
So this power spiral, or whatever you want to call it, is just kind of wrapping around the unit circle.
So it doesn't matter how high a power you choose, it's just going to end up
somewhere here on that circle.
And what happens when you move that guy here off the unit circle?
Let's just move it inside. So we move it inside, what do we get? Well, we get this nice
spiral here, kind of spiraling inside. And, actually, to go higher and higher it goes
closer and closer to 0.
If we move this guy outside, well it's always going to be a spiral, but the
spiral kind of spirals outside. Main lesson to take away from this is that
the closer you start at the unit circle the closer the spiral, this power spiral will
wrap around the unit circle. So now let's go for the real thing, the one
we're really interested in.
Ok, this guy here. So there's the complex number, here in the middle.
What is that? Well it's 1 over here and then you have to go up
pi /m so that's kind of going up there. Let's go for m=3.
Ok, let's just draw this.
There we go 1 over here, pi/3 up there, and then we have to do cubes, right.
So 3 times same triangle, scaling, and so on, what we've just done. And it gets us over
there.
Ok, right. Now what's going to happen when I make this m bigger?
Well, the 1 stays the same. I make the m bigger,
that means that this number here gets .... smaller, right?
That means that it's going to wander down here, that it's going to get closer and closer
to this point, and actually I can make it as close to this point as I want, as
close to 1 as I want
by making m bigger and bigger. It's just going to wander down here, down here, down here.
This means that the spiral is going to wrap close to the unit circle.
Well let's do it. So, crank up to 4, four triangles. Now crank up to 5,
five triangles now. It's wrapping closer, right.
5, 6 now let's just let it go and see how that guy here
gets closer and closer to -1. It's real magic about to
happen. Ready to go for the magic?
There we go, cranking it up all the way. Well not all the way, up to a hundred :)
We can see it's really getting closer and closer to -1 and it's pretty
obvious why, right?
I mean the bit that's obvious so far is that because that guy here wanders down and
down, it gets closer and closer to the unit circle
we should get a closer and closer wrap around the unit circle but what's not
clear at the moment, maybe, is why we don't wrap further or closer. Why do we only
go halfways around. And for that you have to remember what I said at the very
beginning this reminder about the length of this semicircle. What's the length of
this semicircle again?
It's pi, okay it's pi. And what is this? This is the mth part of pi.
So, basically we're starting out with the mth part pi here and then we're doing
this m times so pi/m times m is pi.
So we're going to eventually wrap around halfways, smack on, and we're going to get
e^(pi i) = -1 and I think this is the way to explain it.
Hopefully,
well I don't know about Homer but you know hopefully you who are masters
of plus, minus, multipl,y and so on
got something out of it.