e to the pi i for dummies

by birtanpublished on September 9, 2020

so 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 actually have been agonizing over this ever since and today I want to do just that I want to explain e to the PI I is equal to minus 1 –

Someone like Homer ok someone like oh ma who can only do addition subtraction multiplication division so we have to remind them or tell him two things the first one is that I is the strange complex number square of minus 1 I

Squared is 1 the second thing is just kind of a reminder if you've got a semi circle 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 so the first thing I have to

Explain to Homa is what is e e so to do that I give him a dollar and I tell him go to the bank now if arranged with the manager here to give him one her percent interest over a year ok so what happens to this $1 and Homer puts it in well

After one year he has one plus one is two dollars so he's got two dollars now it is actually not the best you can do with 100% interest you can do better if you find a better bank and we found a better event the second bank of

Springfield the second bank of Springfield they calculate and credit interest twice a year so after six months what happens you get 50% on what you've got there so it's $0.50 it's got one point five dollars now another six

Months path half of that is point 75 so I have to add that to one point five and it 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 spring

They do it three times so we 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 okay and it's actually quite easy to figure out a general formula for this

Well maybe Homer can't do it but I can do it so it's this one here and you can prove it to a tool if you watching this video so it's 1 plus 1 divided by n to the power of 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 check it for the simplest case is 2 and 2.25 so for n is equal to 1 got one plus one is two mmm it's – okay 1 plus 1/2 is 1 point 5 squared is 2

Point 2 4 ok works ok works in general now this is really good news but maybe what you think now and Homer definitely thinks this is well I divide more and more and I get more and more money so if I just divide enough maybe I get 2

Trillion of dollars at the end of the year said it doesn't work ok so for example if you divide into 125 parts you get that much money at the end of the year or have that much money here to the end of the year now if you crank up the

End what happens is well that number goes up but it was a very slowly and if she settles down to a number so if we push the whole thing to infinity I 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 it can't do any better than this that's e and that's also where e kind of comes up for first time historically exactly this sort of consideration ok

Cool so now we've got e we're ready to move on e to the PI I well not so fast let's just go to e to the 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 gala phones

Constant and eventually I definitely make a video about this one but just for today just ponder it a little bit what what does this actually say well it says weird number ^ another weird number and you're supposed to calculate this how do

You calculate something like this well 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 it well it would be doable if pie was equal

To three because then we know we just have to kind of multiply you know maybe chopped off bits here three times and together and we get rough approximation to what we're 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 we're compounding interest continuously that's what we want to know we can't go to sleep tonight if I don't know okay now

The trick is we have got this bit here which gets us closer close to e the more we crank up this n okay 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 I was

Close to the right thing as we want all right now that looks still pretty awful okay and well let's muck around a little bit with it so first thing we do is we multiply by PI here and there and since

We doors are on the top on the bottom obviously nothing changes and actually looks a bit uglier than before but what's nice here is that these two bits are the same and you know what you have to do now to get this is just crank up

To put in the box so I so N equals one we have this and equals two we have that and then that and it'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 you're going up towards infinity we can go up by nice numbers one two three they will also get us there and that's actually what we do and this here this is exactly

What we're looking for all right so it's a like really awful ^ awful but now we've just got addition division multiplication that's all we have to do right just a lot but that's basically all we have to do I told you

No getting there and of course the pie here stands for really any number whatsoever so what we've done now is actually we've figured out how to calculate the exponential function which basically nothing it was just this

That's that's where we just figure out that's pretty pretty good effort all right so well let's just graph this and a couple of those guys and see what happens it's a graph the exponential function

And I graphed the first one of these guy over the second one really where we take the M is equal to two so we've got this guy here not a terribly good fit but if you crank up the M now you can really see how good this gets and actually when

You press you know the button on the calculator that's what you calculator does at some level it just adds and multiplies and divides and does 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 doesn't work okay here we go almost there now just check in your pie 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 and here's we've got a nice number plus a nice number times I this weird was

Actually not very hard to multiply a couple of those things together I could teach you in a second actually I'm going to take teaching a second and just do it let's just do it on Mathematica and see what Mathematica spits out so for M

Equals one we get this number here it's also complex number it doesn't matter what you put up there doesn't matter what M is the results off is going to be a complex number huh so let's just crank it up now crank it

Up crank it up crank it up all the way to what did I do 100 okay and you can see that this first bit here gets closer and closer to minus 1 and the second bit here at a nice number in front of the I goes to zero so basically the the ugly

Part goes away and we're left with the minus 1 if you kind of go in we could stop here but actually I've got this really really really nice way of multiplying complex numbers which can apply to this multiplying complex

Numbers was triangle so I wanted to show you okay so here we go now complex numbers you can draw like 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 use it and he can really relate to it at this point in time so here we've got the real number line so there's zero there's one there's two and so on and well we've extended this real line by the complex plane it's

Just a small thing so every complex number corresponds to the point in here so for example 1.5 plus I is just a point where you go one point five over here along the x-axis and then one up Direction the y-axis okay and I just got

Here for example 1 plus 2i well 1 over here and then 2 2 units up ok now multiply those three things together right so what do we do well we do 1 times 1.5 is 1.5 then 1 times i is i to i times 1.5 is 3i and then the last one

That's where we have to kind of remember that I squared is equal to minus 1 so this is minus 1 times 2 is minus 2 now we just combine things together Robby's way so there and there and that's the product and of course that corresponds

To a point that guy out there hmm well how do you get from here to there not obvious right we can do this but you can actually see it at a glance you can see at a glance that these two guys get you up there how with triangles ok so to

Every point you to every complex number we associate a triangle and the corners are 0 1 and that point here ok so that's the first triangle let's save it second triangle 0 1 point there this guy now we align them like that stretch this one

The red one so that these two sides are the same and there's your product brilliant isn't it so you just kind of align and stretch these triangles and you know what what happens

And actually if you know the triangles it's pretty easy to predict where the products going to be let's do another example let's do this one here squared so what do you need 4 squared is the same triangle twice okay stretch it

That's a square now cubing and we're going to have higher power so we need to see what happens here so we just make it a triangle stretch it that's the cube of this number here all right now higher powers

That's the complex plane for higher powers this circle here the unit circle the circle of radius one around zero plays a very very special role why is that well here's a complex number on that unit circle the triangle that

Corresponds to it has two equal sides they aren't there so when you align two such triangles what happens we don't have to stretch all right doing some stretch let's just see what happens when I kind of raised this to the power of

Eight second power a third power fourth fifth sixth seventh eighth that's the eighth power of that guy here so this power well spiral or whatever it's just kind of wrapping around the unit circle so it

Doesn't matter how higher power you choose it's just going to end up somewhere here on that circle what happens when you could have moved it that guy here off the unit circle just move inside so we move it inside when we

Get well we get this nice spiral here kind of spiraling inside and actually if you go higher and higher it goes closer and closer to zero 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 okay so now let's go for the real thing the one that we're really

Interested in okay this guy here so that's the complex number here in the middle what is that well it's one over here and then you have to go up pi divided by M so that's kind of going up there let's just go for M is equal to

Three okay let's just draw this so there we go one over here three up there and then we have to do cubes right so three times same triangle scaling and so on what was just done and it gets us over there okay right now

What's going to happen when I make this M bigger okay well the one stays the same I make the M bigger that means that this number here gets smaller right get smaller it means that it's going to wander down here it's

Going to get closer and closer to this point and actually I can make it as close to this point as as I want as close to one as I want by making em bigger and bigger it's just going to wander down here down here down here

That he means the spirals going to wrap close to the unit circle hmm well let's just do it okay so crank up to four four triangles now crank up to five five triangles now it's wrapping close all right five six now let's just

Let it go and see how that guy here gets close and close to – well it's real magic happening about to happens we're ready to go for a magic okay there we go cranking it up all the way well not all the way up to a hundred you can see it's

Really getting closer and closer to minus one so it's pretty obvious why right I mean the bit that's obvious so fast that because that guy he 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 by why do we just go half ways around and for that you have to remember what I said at the very

Beginning this reminder about the lengths of this semicircle what's the length of the semicircle again it's pi okay it's pi and well what is this this is the M part of pi okay so basically we're starting out two M's part of part

Here and then we're doing this M times so pi divided by M times M is pi so we're going to eventually wrap around half ways smackle all right we're going to get into the pi is equal to minus one and I think this is the way to explain

It and hopefully well I don't know about Homer but you know hopefully you who are all masters of plus minus multiply and so on got something out of it

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