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.
I just want to say that I really appreciate this type of content - I'm just a maths undergrad, so I have by no means a high level of mathematical maturity, and while I was of course familiar with this identity the way he explained it was new to me. I feel like videos like this fill in the gaps in my mathematical understanding.
Thank you for sharing.
That was a pretty good explanation, the only part I didn't like is how he introduced some triangle magic to do complex multiplication without explaining why it works. Couldn't he just explain that when you multiply two complex numbers, you're just multiplying their magnitudes and adding their phase angles? Proving that would require some basic trigonometry, but that's something most of the people watching the video should already be familiar with. Is someone who doesn't already understand that supposed to just look at those triangles and take your word that it works like that? Because it doesn't really seem intuitive.
For all the love this is getting from math nerds I think this is actually a pretty good example of how poorly math is explained.
First, he substitutes pi for 1, but only in the fraction portion. If I'm not very familiar with (real) math I'm still figuring out (and falling behind) how he did this. OK, you can say it doesn't matter it will behave the same. But I've just spent the past 10(?) years losing points for not rounding the 100th decimal point correctly, what do you mean the actual numbers don't matter?!!
OK, I don't get it 100% but I'm smart enough to be able to accept it and still follow along, even if I'm playing catch up in my head. So, the terms are the same, substitute
mforn pi, follows well enough. But then you change the numerator and the exponent one? So you're sayingf(x) = (1 + 1/n)nx ~~ (1 + x/m)m ????
At which point your average student is just going to have to stare at this after class until it clicks.
However, pointing out that, no, m isn't a function of x, you're just going to pick an
mso that it's some number larger than x that will make the right side approach 0. Or maybe just reminding everyone, that at you're only focusing on whenx<=pi, and you're going to start changingm. Maybe that seems trivial, but this lets the students anticipate what's coming and get ahead of the topic than having constant stress as they feel like they're playing catch up all the timetl;dr: Even when approaching simple topics explaining where you're trying to end up, and not just what steps you're taking but why you're taking them and how you justify them can open up math to those who don't intuitively get it right away
twitches
I personally love this one, which IMO makes a much better job of redefining the math we know to explain new concepts, rather than bending existing methods to make sense of a non-intuitive concept (in the frame of classical operations)
Question for anyone who knows better than I: as the power of imaginary number tends to inf., does the result tend towards the golden ratio? Is there a link somewhere? The spiral that he was talking about at 13mins or so looked like that.
Edit: Not sure what the downvotes are for. It was a legit question
I don't get why pi makes it converge at -1. I get that it's the length of the semi circle but my mind isn't putting the two together.
This is the first time I've seen an explanation for this that I've felt fully satisfied with. Honestly, it makes my day. Thanks for positing it, and thanks to the creators for such a well done video.