Well, I love Simpsons, Giuseppe loves the Simpsons,
Everybody loves the Simpsons! And if you look at
the views: yeah, it's really everybody who loves
the Simpsons, so, I think, we'll do another
couple of videos on the math in the Simpsons.
And today it's actually going to be one on
higher math in the Simpsons.
So let's just watch this clip and talk about it a bit.
CLIP: "When I was recruited by a card-counting ring,
I won enough money to buy fake SAT scores I used to
get into the real MIT, where I failed every class,
and was kicked out and had to move to Springfield."
OK.
Now, we're not going to talk about card counting today.
We're going to do a video on card counting :)
Now, what we're gonna talk about is this
blackboard that shows up in here.
OK, that's a really-really nice one. OK?
So there's a lot of really nice mathematics in there.
And I'd like to make sense of it. OK? Together with you.
Alright. So let's have a really-really close overlook.
So, let's just start out with that symbol up there.
So there's this squirrelly sign in there, OK?
That stands for "sum" in mathematics. In this case,
an infinite sum.
Alright, let's take it apart. So here we go.
So... "x=1", then 1 divided by 1 is... 1.
Then put x=2: 1 divided by 2 is one half.
Then, next one, put 3 in there: you get 1/3. And a 1/4.
And that infinity up there tells you that
you should go on, you know... So, just going on forever.
So that's an infinite sum. Right? It's infinite sum.
And it starts out like "1", but then it gets
smaller and smaller and smaller, and kind of dies out
as you get further and further this way. OK?
So that's an infinite sum, sums up to something. Who knows?
OK. Now, what comes next? Well, then we do these
sorts of manipulation: there is a bracket with two terms
in it. What does that correspond to in this...
this way of writing it?
Well, it just corresponds to bracketing things
like that. OK?
Now, there's a bit of algebra happening here, and
it's all OK. As far as math is concerned, what
they're doing here is all OK. So I'm not going to go into
the details there. Let's just skip to this line here.
Something interesting is happening there: you actually
come across the same sum that you started out with.
So there it is again. OK?
And, so...
Let's just forget about the rest.
There's something else here in this line...
Move it together.
So, that's basically what this all says so far.
So we've got something here, we've got something
there, and then we've also got something here.
And obviously we can cancel those two guys out.
That makes this term here zero.
So that's the conclusion we draw from all this so far:
this guy is equal to 0.
Now, there's a bit of more algebra, which is all fine,
as far as math is concerned, which then tells you that
this guy here... algebra... is equal to
(1-1/2+1/3-...), and so on. So, basically,
the sum that we started with, except that every second
plus has been changed into a minus. OK?
OK. So, that's supposed to be zero.
Except it's pretty obvious that it's not, right?
Because if you bracket things like that, you see then
(1-1/2) is definitely something...?
GIUSEPPE: ... a half. BURKARD: ... positive.
It's definitely... it's a half, so something positive.
And that guy here is also going to be something positive
And that one is going to be something positive.
There's no negatives in there, so it's gotta add up
to something positive. Can't possibly be zero.
And actually if you know a little bit of calculus,
you actually recognize this infinite sum as...
something that adds up to something else very famous
ln(2). This thing actually adds up to 0.69314
And actually if you have a really-really close look at
what's cancelled out here, that's kind of a "4" in the end.
So I think that's what actually was supposed to be here.
So, there's a contradiction here, there's a contradiction
here... you know, basically, what it says so far
(and everything seems fine)
is that this ln(2) is equal to 0, which it's definitely not.
GIUSEPPE: So, does it mean that MIT had it wrong in their board?
BURKARD: Weeeell... Let's figure that out :)
Before we move on, right...
I've just been commenting to no end on the
"9.9999...=10" video.
And basically that is about infinite sums. There was, like
lots of people coming in saying "Well, what's these all
these infinite sums... They, you know, they are not really of any use.
So, why do you talk about them? It's nonsense."
Well, I said it there and I really want to say it today:
these infinite sums are... what makes calculus work.
Without infinite sums – forget about calculus.
And just to illustrate what's going on,
one of the instances where this comes up is
actually, where this insight here comes from...
Actually it turns out that you can represent the function,
log of something, as an infinite sum with a variable in it.
So you can... For t, you can actually put any number
between (-1) and 1 in there.
And it will tell you what the corresponding log of that
number is. For example, if I put 1 in here,
that will tell me what ln(2) is.
If I put, you know, 1/2 in here, it will tell me
what ln(1.5) is, and so on.
Well, why is this good? Well, these functions are very
complicated, very hard to calculate these sorts of things
But if you have a representation like this here,
if you have a representation like this, what you can do is
you can find very good approximations to these,
these numbers that we are here to calculate
by this kind of chopping of this infinite sum
at some point, and then, you know, just
calculating whatever is left over. So, maybe you just
chop it off at fifty terms here, and then it's just kind of...
you know, products and sums. It's very easy to calculate
And that will give you this approximation here.
When you sometimes hear on the news
"π has been calculated for another trillion digits"
well, the way they do it is via infinite sums.
And just in general, infinite sums are really good! :)
GIUSEPPE: Does that mean that when it has a trillion digits, like, that stops at a trillion digits...
BURKARD: No, of course, it keeps on going. That's right.
They just need a couple more of those terms, and
they get more :)
Anyway, infinite sums are really good. Now,
how do we get out of this... contradiction here?
Well, let's have another look here. So, we've got here
one number is equal to the same number plus ln(2). OK?
Now, what's going on here? Everything seems to be
alright, what we're doing here. There is another
Simpsons' clip which actually gives you a hint.
OK, let's just watch this one.
HOMER: Hey, Flanders, it's no use praying. I already
did the same thing and we can't both win.
NED: Actually, SimpSON, we were playing that
no-one gets hurt.
HOMER: Oh, well, FlanDERS, it doesn't matter. This time
tomorrow you'll be wearing high heels.
NED: Nope, you will.
HOMER: 'Fraid not.
NED: 'Fraid so.
HOMER: 'Fraid not.
NED: 'Fraid so!
HOMER: 'Fraid not infinity! NED: 'Fraid not infinity plus one!
BURKARD: Oh... What's infinity plus one?
GIUSEPPE: Ah, it's infinity!
BURKARD: Oh, really? OK :) GIUSEPPE: So they tie.
BURKARD: They tie, right.
Anyway, so this... this insight that infinity is
infinity plus one
gives us a really good hint of what's going on here.
Right? Because if you put any number in here,
that's not gonna work out. But if you put... infinity there,
this is actually going to work out. So, I mean,
that actually the only resolution to this whole thing is
(because all the other math is fine)
is that this guy actually adds up to infinity.
Hm, which is really cool.
So this actually would make it right.
OK, well, it's a very complicated way of showing that
this sum is infinity. So I want to show you another one,
which is actually a real classic. It dates back
a couple of hundred years. And it's been brilliant then,
it's still brilliant now. I teach it in the first-year calculus.
So, proof that... another proof.
The standard proof. In many ways, the nicest proof that
this sum here is infinity. Which, at first sight, is actually
quite surprising, right, because these things really
die out there, so you might think "well, maybe it
adds up to something finite". But this guy - no.
It adds up to infinity. OK.
So how do we do this? OK.
So, first thing is: we create a second infinite sum.
And I just put down a couple of the terms there.
So, this one here coincides with the guy on top with
all the terms that have a power of 2.
So there is a 2 here, a 4 here, an 8 here, and the next
one will be 16, next one will be 32, and so on.
All those stay the same.
Now, all the other terms that I haven't written here
are going to be actually smaller than the corresponding
terms up there. So this 1/3, for example, I replace by
one fourth. 1/4.
This 1/5 I replace by 1/8. Makes it smaller.
That one here also by 1/8. This one by 1/8.
Now, what about this one here?
What do I replace it by?
GIUSEPPE: 1/16. BURKARD: 1/16, and the next one?
GIUSEPPE: 1/16.
BURKARD: Yeah, and you just keep on going until
you hit 1/16, and then it keeps on going with 1/32.
Alright. So, we just keep on going like this.
And now if we just compare this sum to that sum,
you know, if this was a... if this added up to a number,
you know, we replace stuff here by things that
are smaller, so this should actually add up to something
that's smaller. Right?
Well, if it's infinity, then it might stay infinity or might
turn into a number. Doesn't matter. But definitely
this one is not getting bigger. Right?
This one is definitely not getting bigger, that's
absolutely clear.
And now the ingenious bit here is: we also bracket
things in a nice way. We bracket them like this.
OK, so... What does this bracket add up to?
Well, there's only one term in it, so it's 1/2.
What about this one here? 1/4 + 1/4 is... a half.
What about this one here? It's a half, too.
And this one here? A half. So every single one of those
rectangles has 1/2 in it. So this second sum really
adds up to "1+1/2+1/2+1/2+1/2+... infinitely often".
And that's, obviously, infinity. Right? Infinity.
So that guy is definitely infinity, and that one
GIUSEPPE: ... is bigger than infinity. BURKARD: Yeah, well, it's got to be infinity there.
So, I mean, that's just absolute beauty, right?
And that's another Math-in-the-Simpsons video for you.
This time about higher math in the Simpsons.
And the next one we're gonna do is gonna be
even higher math. :)