Math in the Simpsons: Apu’s paradox

by birtanpublished on September 14, 2020

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 we do another a couple of videos on on mats and the simpsons and today it's actually going to be one on higher mats in the simpsons uh so let's just watch this clip and

Talk about it a bit but i was recruited by a card counting ring i want enough money to buy fake s.a.t scores i used to get into the real mit where i failed every class and was

Kicked out and had to move to springfield okay 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 going to talk about is

This blackboard that shows up in here okay that's a really really nice one okay so there's a lot of really nice mathematics in there and i'd like to make sense of it okay together with you

All right so let's have a really really close up look so let's just start out with that uh symbol up there so there's a squiggly sign in there okay that stands for sum

In mathematics in this case an infinite sum all right let's just take it apart so here we go so x equals 1 then 1 divided by 1 is 1 1 then put x equals to 2

1 it's one half and then next one put three in there you get a third and a fourth and the infinity up there tells you that you should go on you know so it just goes on forever

Right so that's an infinite sum right it's infinite sum and it starts out like one but then it gets smaller and smaller and smaller and kind of dies out as you get further and further this way

Okay so let's infinite sum somewhat to something who knows okay now what is comes next well then we do these sorts of manipulation there's a there's 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 okay now there's a bit of algebra happening here

And it's all okay right so as far as math is concerned what they're doing here is all okay 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 okay so there it is again okay and so uh let's forget about the rest uh there's something else here in this

Line move it together so that's basically what this all says so far right 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 right that makes this term here zero so that's the conclusion we draw from all this so far okay this guy is equal to zero now there's a

Bit more algebra which is all fine as far as math is concerned which then tells you that this guy here you know algebra is equal to one minus one half plus one third

And so on so basically the sum that we started with except that every second plus has been changed into a minus okay okay 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 that one minus one half is definitely something plus it's it's a half a percent positive and that guy here is also going to be something

Positive and that one's going to be something positive there's no negatives in there so it's going to 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 log2 this thing actually adds up to 0.69314 and actually if you have a

Really really close look at what's cancelled out here there's kind of a four on the end so i think that's that's what actually was supposed to be here right so there's a there's a

Contradiction here there's a contradiction here um you know basically but what it says so far and everything seems fine is that this log 2 is equal to 0 which is definitely

Not so does it mean that mit had a wrong board well let's let's figure it out okay let's figure it out but before we move on right um

I've just been commenting to no end on on the point in a 9.999 recurring equals 10 video and basically that is about infinite sums and there was like lots of people coming in saying

Well what's with all these infinite sums they you know they're not really of any use um you know so why why do you talk about them you know it's yeah well i 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 okay and just just to illustrate what's going on

You know one of the instances where this comes up is uh actually where this this insight here comes from um so actually turns out that you can represent the function log of something

As an infinite sum with a variable in it okay so um so you can that for for t you can actually put any number between um -1 and 1 in there and it will tell you what the

Corresponding log of that number here so for example if i put 1 in here that will tell me what log 2 is if i put you know one half in here that will tell me what log one one and half

Is and so on and well why is this good well these functions are very complicated right very hard to kind of calculate these sorts of things but if you have like 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 are very hard to calculate by just kind of chopping off this

Infinite sum at some point and then uh you know just calculating whatever is left over so maybe you just chop it off at 50 terms here and then it's just kind of uh you know products and and sums it's

Very easy to calculate and that will give you this approximation here um you know when you sometimes hear on the news pi has been calculated for another trillion digits

Well the way they do it is by infinite sums and just in general infinite sums are really good okay that doesn't mean that my has a trillion digits like that

No of course not just something calculation that's right that's right you just need a couple more of those terms and you get more anyway infinite sums are really good now how do we get out of this

Contradiction here well let's just have a look another look here so we've got here one number is equal to the same number plus log two okay now what's going on here everything

Seems to be all right what we're doing here there's another simpsons clip which actually gives you a hint okay let's just watch this one hey flanders it's no use praying i

Already did the same thing and we can't both win actually simpson we were praying that no one gets hurt oh well flanders it doesn't matter this time tomorrow you'll be wearing high

Heels nope you will pray not pray tell me great nudge rate's health grade none infinity rate so infinity plus one ah what's infinity plus one right it's infinity oh really okay

Hey right anyway so so this this this inside at infinity is infinity plus 1 gives us a really good hint of what's going on here right because if you put any number in here that's not going to work out

But if you put if you put infinity there that's going to work out so i mean actually the only resolution to this this whole thing is because all the other maths is fine is that this

Guy actually adds up to infinity which is really cool so this actually would make it right okay um okay well i mean it's a very complicated way of showing that uh this this sum is infinity so i want to

Show you another one which is actually a real classic it dates back 100 couple hundred years and it's been brilliant then it's still brilliant now i teach it interesting calculus

So proof that another proof the standard proof the nicest proof in many ways the nicest proof that this sum here is is infinity which at first sight is actually quite a surprising right because

These things really die out there so you might think well maybe it adds up to something finer but this guy no it's adds up to infinity okay so how do we do this okay so first thing is we create a

Second infinite sum okay 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 there so there's a 2 here a

4 here 18 and next one would be 16 next one will be 32 and so on okay all those stay the same now all the other terms that i haven't written here are going to actually be smaller than

The corresponding terms up there so this one third for example are replaced by one fourth one fourth this one-fifths i replaced by one-eighths makes it smaller that one here also by

One-eighths that's one by one-eighths okay now what about this one here what do i replace it by 1 16 1 16 and the next one 1 16 yeah and you just keep on going until you hit 1 16 and then it goes

Keeps on going with 1 32 yeah okay all right 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's not getting bigger right this one's definitely not getting bigger i mean

That's absolutely clear right and now uh the ingenious bts we also bracket things in a nice way we bracket them like this okay so uh what does this bracket add up to well there's only one term in it so it's one

Half what about this one here one quarter plus one quarter is it's a half uh what about this one here it's a half two and this one here it's going to be a half so every every single one of those rectangles has a half in it

So this second sum really adds up to one plus one half plus one half plus one half plus one half infinitely often and that's obviously infinity right infinity so that guy is definitely

Infinity and that one is bigger well it's got to be infinity there right so i mean that's just absolute beauty right and that's another maths in the simpsons video for you this

Time about higher maths in this instance and the next one we're going to do is going to be even higher man

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