What is Euler's formula actually saying? | Ep. 4 Lockdown live math

welcome back to lockdown math today we are going to be talking about Euler's formula and just to give you a little sense of where we're going to be ending up with this lesson I'm gonna go ahead and show you what we're aiming for at the end which is a certain visualization so I don't expect you to necessarily understand this immediately but the point is that this is something we're going to walk towards what we're going to analyze is an extension of the idea of Exponential's in a way that works in the complex plane and the illustration that you're looking at is showing very literally what the claim of Euler's formula is because what I want you to appreciate is what the actual statement says rather than letting it be shrouded in a certain mystery or a certain question of what the conventions are now needless to say this is kind of a confusing thing we've got this spiral of vectors and if it's not entirely clear don't worry about it I just want to give you a little sense of where we're going to be going with this but before any of that let's take a step back and remember where were we okay back in the end of the last lesson when we were talking about complex members one of the key types of complex numbers that we were looking at were those that existed on the unit circle so here I have a little complex plane drawn we've got the real number line with the points 1 and negative 1 indicated we've got the imaginary number line I being the square root of negative 1 and if you remember one of the main points that we emphasized last time is that when you have a number who's sitting one unit away from the origin at some angle theta multiplying by this number has the effect of rotating things by that angle this is incredibly important throughout physics throughout electrical engineering all throughout math you see these numbers everywhere they describe wave mechanics they're very important for polynomials it's really hard to overstate how important numbers that sit on this unit circle are now one way that you could write them is with the real and imaginary parts and based on lecture two if we know our trigonometry the x coordinate is going to be the cosine of that angle and the y coordinate which is the imaginary part is going to be I times the sine of that angle okay so you might think all throughout physics all throughout electrical engineering you see the expression cosine of theta plus I sine of theta in fact what you often see is another form of this almost always you see this written down as e to the power I times theta and this relationship is what's known as Euler's formula okay now he is a special constant of nature and I always remember in high school it was never crystal clear to me exactly what it was it was something that was just kind of handed down okay it's 2.71828 on and on and we were just taking you know we were to take this as a an analogue of Pi it's an irrational number that evidently the universe side finds significant I had one calculus teacher who would tell us it was the Andrew Jackson number because the president Andrew Jackson served two terms he was the seventh president and he was elected in 1828 so there you go I don't know if Andrew Jackson appreciated this relationship that you had to Euler but it always helped me remember that number now just a gauge sentiment because I'm not entirely sure where the audience is I want to know what your current relationship with this particular formula is so we're going to do a poll to start and this poll is mostly going to be helpful to me so pulling up some of the warm-up questions that we had we're evidently most of you if you were to choose one of Euler's formulas would have gone with the one that we're talking about today which is great you know appropriate given where we want to head as a poll that's actually going to be helpful to me let me ask you which of the following best describes your relationship with the formula e to the power I theta equals cosine of theta plus I times the sine of theta okay and the options here are that you've never seen it before totally understandable that you've seen it but you're confused by it that you still don't understand it but you've grown used to it or that you understand it well so as you can see answers are rolling in please participate this is going to be way fun if more of you participate you can go to three b-1b deco slash live link is also in the description that four would do to the place where you can answer this poll and why all the answers are rolling in I just want to remind us of a more famous variant of this expression that you often see which is basically what happens when you plug in PI and so if we say e to the I times pi we're now we're thinking of pi as an angle it's a it's a number of radians a distance around the unit circle you would plug in cosine of pi plus I times the sine of pi okay and the way to think about that is to look at the unit circle and ask what if you walked around until you'd walked a total distance of pi you know if it's a lake with the radius 1 and you walk around the boundary until you've got a distance of pi kind of by the definition of what pi is that takes you halfway around the circle so the X component cosine is going to be negative 1 and then the Y component sine is actually 0 so there's no imaginary part it's just negative 1 and this gets us you know what might be the most celebrity equation in all of math e to the I times pi is equal to negative 1 ok but it's baffling genuinely baffling because you've got this idea of raising a constant which is a little wishy-washy to start with what is e but you're raising it to an imaginary power and if you don't understand what that means you're in good company I think very few people in the world actually do now turning back to our poll where we've got a good number of responses and a pretty good spread too I'm genuinely curious to know who am I talking to right now okay cuz it looks like we have a pretty even distribution among all camps all right so the number one most common answer is C which is those who still don't understand it but they've grown used to it this I'm gonna guess includes people like math majors or engineering majors or people who have gone into a technical field where it comes up a lot so you have to grow used to it this is also revealing the fact that even though these lectures are targeted at high school students that this channel has a certain base demographic to start with so sometimes those sitting in the audience it's more like adults who have walk in it walked in and are sitting in the back of the class very happy to see that D I understand it well is a second most common answer I'm curious among those who answered e if by the end of the lecture they would say that what they understood about this is the same as what I'm going to teach because for example I would make the claim that this formula actually has nothing to do with the number II okay that the number II doesn't actually play a role in what this formula is computationally saying and I'd be curious if those who claim that they understand it well would agree with that statement after that we have those who've seen it but are confused by it and at the very end is the actual target demographic of those who've never seen it before I don't know if that's it's fair to call that the target demographic because I think sometimes you see it in a non circumstance here there but definitely those who answered a or B you're my people you're the ones that I want to talk to about this okay but let's say you see this can you see this weird formula e to the PI I or the generalization e to the I times some angle I think the healthy reaction to have to this okay if you're just seeing this for the first time the healthy question to ask is w.t.f okay what is the function at play and how is it defined okay what's the function because in this case the function is e to the X and we're plugging in certain imaginary inputs and I think a lot of people think that that refers to taking a number E and multiplying by itself some number of times and that X describes how often you're multiplying by itself and that yeah there's some notion of extending that to things like 1/2 or negative 1 or any kind of real number but that it's based in this idea of repeated multiplication now the thing that makes this equation misleading is that that's not the function that is not what each of the X is referring to let's emphasize that very heavily this is not what e to the X means this is not its convention instead what has emerged in math is that we use e to the X to be a shorthand for another function a function which I'm going to give the name X this is how you often see it in literature and it's defined to be a certain polynomial it's 1 plus X plus x squared divided by 2 plus X cubed divided by 6 plus X to the power 4 divided by 24 and in fact it's not a polynomial it's an infinite polynomial we add infinitely many terms each of which look like X to the N divided by n factorial the proper term for this is a series in practice if you're actually computing this because that denominator grows so quickly you can chop off the series pretty early to get an approximate value for what this actually is now right away we can see that this is going to work for lots of kinds of X that we could plug in anything where we know how to raise X to a power just a whole number power multiply it by itself and where we know how to divide by a factorial and add those together we can come up with a nice meaning for what this X function means but Before we jump into things like complex numbers and throwing that in I think it would be very unsatisfying to do that if we didn't first draw the connection to the number E and the idea of repeated multiplication because on the surface this infinite polynomial seems very different from the idea of some special constant of nature E and raising it to a power so let's build up a little familiarity we're going to take you know a couple minutes to do this and it's very it's a very healthy exercise I think to become friends with this function by starting to plug in a couple different values one of the first you might plug in is the number one so that way we get one and then X is 1 so that's one x squared is still 1 so that's one half and then one-sixth and then one twenty-fourth and in general we're adding 1 divided by n factorial in math we say that this is a series that converges in the sense that as we add more and more terms it approaches a certain value and it gets closer and closer to some specific value and that value ends up being around 2.71828 the Andrew Jackson number but at this point just think of it as it was going to be something X but one was going to be something this is what X but one happens to be at the moment there isn't necessarily anything special about that but let's say you wanted to plug in other values ok just to get a little practice with what that would look like if I wanted Expo - it would look like one plus two plus two squared over two plus two cubed over six and in principle if you were on a desert island and you just needed to compute Expo - you could work this all out by hand if you were comfortable with long division but of course we live in the modern world we have computers we have programming so I think to make this especially concrete let's go ahead and actually implement this function so that we can calculate a couple values and so that it's not a black box it's not just a calculator that's been handed to us we implement it so we know exactly what it's doing so for that let's go ahead and pull up a little Python this is desmos we might look at that a little bit later right now let's be a little bit more programmatic let's pull up some Python let's import some math because that's a sign you're always going to have some fun then I'm going to define the X function that's going to take in some number X and what I want is to return something that looks like 1 plus X plus x squared over 2 where in Python this double what do you call it an asterisk this double asterisk sign is how we do exponentiation and then X cubed over 6 and we kind of want to add that up a whole bunch of course instead of typing all that out we can use a little special syntax where I'm going to say I want you to return a sum of a bunch of terms each term is going to look like raising X to some power and it knows what X is because that's what was handed to it X to some power divided by the factorial factorial of N and that's built into this math package that we imported and I'm just going to do this for values of n that start at 0 0 factorial is defined to be 1 if you're curious we could talk all about factorials of weird values at a later date but that's all you need to know here and I'm just going to have it range up to 100 because a hundred factorial is going to be huge so that's going to be plenty big enough for a denominator to be small enough that those later terms don't contribute a lot okay so even if you don't know Python I hope that this is a reasonably clear way to turn math into something that our computer can chew on and crunch through the numbers so that we don't have to so for example if we type X above 1 we get the Andrew Jackson number 2.71828 on and on and I could type in expo 2 and it looks like that's around seven point three eight nine so maybe we even write that in our notes we go over here and say interesting X book two was about seven point three eight nine okay and if you spend a long time just kind of playing around with this okay I think it's not obvious that you might find this but if you were just plugging in a couple values X above three X before one important fact you might stumble across is that if I add two numbers in the input okay so in this case they get 1096 when I plug in seven which is three times three plus four that actually ends up being the same as if I plug in X but three times x before okay so adding the input corresponds to multiplying in the output and three and four weren't special here I could have done you know five point five and three point two and that would have gotten me some value and if instead I had added those together okay it gets the same value that is not at all obvious okay so I think that would be a genuine discovery to have with respect to this function with this polynomial if that was something that you found and it's it's important enough that I want to write it down X of a plus B is actually the same thing as X of a times X of B now if you just look at the polynomial this is not clear okay this is not something that like you look at it and say oh yes of course it couldn't have been any other way at the end of today's lesson though I'm gonna give you guys some homework and yes homework will make you learn better and what we're going to do is go through a couple problems that are gonna have you show that this fact is true that simply from the polynomial and the fact that it includes these factorial terms excuse me that's going to be enough to show this very special property now why am i calling it a very special property well I want to ask you a certain question that will hopefully hopefully make this clear and it really is important that you think this through yourself because I think if we if I just kind of tell you the implications here it's not going to sink in to the same extent as if you really noodle with it yourself so I'm going to pull up another question and I'm going to give you some time to think on this one what the question asks us is to suppose that we have some function f of X that has this special property we're adding two numbers in the input F of a plus B gives the same result as multiplying in the output F of a times F of B and that this is true for any real number a and B okay suppose you have some function that satisfies this property who cares how its defined whether it's through this polynomial through other means which of the following is true one of the options is that F of 5 is equal to F of 1 raised to the power 5 another is that F of 1/2 is equal to the square root of F of 1 and the other is F of negative 1 equals 1 divided by F of 1 now very important I'm not asking does there exist some function f where these 3 things will be true I'm asking which of these necessarily has to be true only from that property so you shouldn't be able to contrive some sort of adversarial function that doesn't satisfy it ok and then you have various options for which collection of these three things is true I'm going to give you some time to think about this because I really do think it's important so I'm going to turn up our pause and ponder music to get us in the mood and take a desperately-needed drink of water you while you're noodling on that I'm gonna go ahead and take a question from the audience where it looks like surfed asks what do you think is more interesting to someone who is a newcomer to higher level math special case theorems like e to the I PI or more general cases like e to the IX oh that is such a great question by the way anyone who wants to ask questions go to Twitter and just use the hashtag lockdown math those will be forwarded to me I think the best way to learn is to have specific examples and really let yourself understand the patterns represented by those specific examples and then generalize them for the specific one that you have here I think if you just if you just see e to the I pi that doesn't really count as a good specific example that explains the the generality it's more that you're plugging in one particular number into a formula so in this case e to the i-x as you'll see it has everything to do with circular motion and between this lecture and the next one we're going to talk about why that relation might be there so the form of having specific examples that aids your understanding wouldn't be you know a number like e to the I pi it would be building a relationship with circular motions in other circumstances like studying physics where you have a tetherball with some kind of centripetal force and it's orbiting or like orbital mechanics anything where you're you're really understanding the nature of circular motion that actually prepares you for understanding e to the i-x as a generality a little bit better so with all of that answers are still rolling in and I don't want you to feel rushed so I'm going to give a little bit more time here actually because we remember you need to really be sure that if you're saying that something like 2 or 3 or 1 is included in your answer that any function with this special property necessarily follows that ok oh let's let's see if we can grade it when the on the top answer is the year right it's a 2017 that's a little bit in the past 2013 we're losing the top answer interesting people are changing their mind it's like we're going back in time so it looks like a couple people are oh now we're in the future 2023 yeah so we're actually seeing people think about it and thinking hang on you know is it the case that option 3 here is necessarily true and they're really being critical about that so that's a good sign if you want to keep thinking about it please do but for the sake of continuing with the lesson I'm going to go ahead and lock things in here and see how people ended up answering so it looks like 20-30 of you the decade in the future believed that all three of these are necessarily true second most common answer was B they only believed that the first one is true and then after that people who either included number two or number three and the correct answer is that all of them are and it's very interesting to walk through why I do think it's very elucidating so let's let's go ahead and do this the first one which it seems like a majority of you believe is that if we plug in something like five to a function with this special property that it'll be the same as taking F of one raised to the fifth now the reason is that we could also write five as one plus one plus one plus one plus one and this property of addition in the input becoming multiplication in the output means that's the same thing as writing X both one multiplied by itself five times okay it lets you rewrite the whole thing in terms of X both one which I'm going to say five times this is the same as taking X both 1 raised to the power 5 and here when I'm writing Exponential's anytime you see a natural number a whole number it can literally mean multiplying by itself that number of times that's not the same as the fact that e to the X is a shorthand for this crazy polynomial so that can sometimes get a little confusing the exponent playing two different roles and we might give X both one a special name a shorthand let's just call it e for short that's not why we call this number e by the way it it's also not because this is what Euler's name starts with it's just because whenever Euler was using this the first time in a particular book he was partial to vowels and the vowel a had already been used so he was just his arbitrary letter okay so simply by virtue of this property we can see that expo five has to be written in terms of X both one well a little bit trickier was the question about plugging in one half okay and the key to solving this it's a little bit tricky is to think about what happens when we multiply that by itself X but 1/2 times X but 1/2 because of this property has to satisfy or I should say has to equal X both 1/2 plus X but 1/2 which is of course X both 1 which let's say we're using the shorthand and we call it e so what does that mean X but 1/2 has to be a number such that multiplying it by itself equals e well that's what we mean when we write square roots okay and the last one was talking about negative inputs so just as an example if we inputted something like negative 1 so the key here is to ask about multiplying it by the value when you plug in 1 by this rule that addition in the input turns into multiplication in the output that has to be the same as X of 0 now what is expo zero oh you know what I'm realizing hmm I think I might have actually made the enter the wrong answer there because in our case in our case Expo zero actually does come out to be the number one okay so if we plug in zero for X then all of these you add them together the only term that matters is one so X book negative 1 times X plus 1 is equal to X plus 0 which is 1 so this is true in the case of X I I can't actually think to myself right now if that's necessarily true you know if we look if we look at our question we're saying all we know about F is that F of a plus B is f of a times F of B if that necessarily implies that F of zero is going to be one and I I don't think it does I think we could construct a function yeah because we could just scale e to the X by some other amount okay yeah so actually what's graded here is is not entirely correct the correct answer would be only one and two I think someone correct me on Twitter if I if I'm wrong about that but very interesting so if we go back to our paper let's see in the case of X the this this value at 0 would be 1 what does that imply well if we call X both 1/e that means we're asking what is the number which when you multiply by e equals 1 and it would be 1 divided by e but yeah and for the general case but we need the added condition that when you plug in 0 you do get one now the point of all of this right the reason that I'm saying this is to emphasize why it's reasonable that we use the shorthand that when we say X both X for real numbers X Y it would be very reasonable to write this as e to the power x because basically this special property means that because you can use the number 1 to access pretty much all the real numbers by adding it to itself or dividing as needed or negating and then if you have certain continuity restrictions that will let you extend from rational numbers to all the reals but that's a technical point you don't need to you don't need to fuss over for what we're doing right now it lets you basically express everything that this polynomial can output in terms of what it outputs at the number 1 so you might read e to the X as just or whatever this polynomial is at the input 1 we can start exponentiating in terms of that so that's the connection right we could we could if we want to define the number e to simply be where is this polynomial at the number 1 and then all of this would follow but I really want to emphasize that only makes sense for real numbers because as soon as we start introducing things like complex numbers you can't just add to itself or subtract and divide and get the number I and at math you often do even crazier things where you plug in things like matrices into this polynomial which seems weird but you know you can take a matrix and square it you can divide it by two you can add all of those together and that's actually a very useful thing it's very useful for a field called differential equations which in turn is useful for physics in quantum mechanics you often plug in these things called operators which are kind of like the mature older brother of matrices and all of that looks totally nonsensical if we're talking about a constant raised to some kind of power but what you have to understand is that it's being plugged into this polynomial okay so I understand why we have the convention of writing this as e to the X it makes sense for real numbers but I think that's actually a bad convention as soon as we start extending it and I think that causes a lot of undo confusion so with all of that said let's finally have some fun and plug in some complex values okay and before we do that just as kind of a warm-up I want to make sure that we're comfortable with powers of I because that's what's going to be important here yeah so let's pull up our quiz and let's go ahead and ask one more question and then I'll take a question from the audience too while you guys think about this remember that I is defined to be a value that satisfies I squared equals negative one okay it's not it's defined for which values of n does I to the power n equal negative I okay so the spirit of this question is to have people thinking deeply about powers of I and I kind of I wanted it to be a question that's not totally obvious so one that you do have to put a little mental energy into so let's go ahead and take a question from the audience while I take a drink of water [Music] f of X equals zero doesn't work with F of negative 1 equals 1 over F of 1 yeah f of X equals zero oh interesting yeah okay so so they're saying that if f of X ever outputs zero actually I'm not entirely sure what say I'm saying f of X equals zero doesn't work with F of negative 1 equals 1 over F of 1 so I guess that fact that F of negative 1 is 1 over F of 1 doesn't imply that the output will ever be 0 so I could ask for some clarification but I think that might be giving at the general point of the mistake of the question which was to assume to extend the idea of X a little bit too much where the fact that expose 0 equals 1 gives us this other property and then Crispin Simmons says 3 is included because f of X plus 0 oh ok yeah ok great great yeah someone thought this through a little bit more deeply than I can while live and on camera and under some pressure f of X plus 0 is the same as f of X times F of 0 which implies f of X equals 1 awesome crispin you saved me he saved me on this one that's a beautiful way of thinking about it yeah all of these these questions where you have like a property of a function and you're supposed to deduce facts they can be very subtle and so in this case okay the question was graded correctly the property simply that addition in the input becomes multiplication in the output is enough to imply F of 0 equals 1 which in turn locks in the value for F of negative 1 oh it's so pleasing one little property can just lock in basic not the whole real number line because it didn't assume continuity but it locks in pretty much any value you want awesome excellent questions I love live stuff even if I make errors it does feel a little more interactive than the usual videos so let's see how we're doing on the quiz looks like we've still got some answers rolling in we're trying to understand which powers of I are equal to negative I and the options include multiples of 3 or multiples of 3 as long as they're positive or integers one below a multiple of four or those where you're restricted to them just being positive and I'm going to go ahead and grade this question and as always if you feel like you weren't done by the time I'm locking it in know that that's just for the sake of letting the lesson progress forward if you wanted to pause and think about it further yourself especially if you're watching this later that would be that would be totally in the spirit of these lessons okay so the majority or the plurality I guess of you answered that it's all integers one below a multiple of four which is correct and the second most common answer was basically the same but restricted to positives great so let's think through what powers of I actually look like because this is going to be the only thing we really need to plug in complex values into this X function and the way this can work if I pull out my infinite supply of unit circles is to remember that multiplication in the complex plane includes a rotation component and because I has a magnitude of one there's only rotation so multiplying by I has the effect of rotating 90 degrees if you multiply by I twice okay I squared you end up 180 degrees around reassuring it's supposed to have a square of negative one and this geometry lines up with that fact if we multiply by itself three times to get I cubed it ends up being a vector pointing straight down multiplying by itself a fourth time 90 degrees takes us back to one so I to the fourth is 1 and then after that they all start repeating so basically all you need to know is where's that exponent with respect to powers of 4 if it's a power of 4 you're at 1 if it's 1 above a power 4 you're at I now the the trickiness of the question was to also include the nuance of whether this is only for positive powers or negative powers but what a negative power means if we write something like I to the negative 1 that's defined to be 1 over I the number such that when you multiply by it you get 1 and you can see that that's actually the same as I cubed this is the same as I cubed because that's a number where when we multiply it by I we get one negative one that's also one below a multiple of four and in general anything that's one below all the negative multiples of four will also satisfy this property all right so it's a little tricky question but it's just because I wanted you to think deeply about this idea that powers of I of whole numbers just have us ticking forward by 90 degrees and finally we can have our fun okay we can take our crazy X X function and we could plug it into Python if we wanted to and see how things play there but first let me go ahead and visualize it for you so I'm going to let's see that circle is already suggestively drawn but I don't want that to be there I just want you to think about this polynomial where I'm gonna plug in I scaled by some kind of constant which very suggestively I'm going to give the name theta and then what that means is taking 1 plus I times theta plus I theta squared over 2 on and on and just as an example let's crank theta on up until it equals 1 and let's try to think pretty deeply about what this actually means computationally the first term is 1 which we might draw with this green vector on the bottom pointed one unit to the right the next term is going to be I times 1 so just I it's pointed straight up with magnitude 1 now think about what the next term is it's going to be I squared or I theta squared but theta is 1 so it's I squared which is negative 1 over 2 so that's why this next vector is pointed to the left and it has length 1/2 okay then after that I cubed is pointed straight down and we divide it by 6 so now we just have an itty-bitty vector that is of magnitude 1/6 and then they just get smaller from there the next one is pointed to the right but it has magnitude 1 over 24 and they just shrink a whole bunch and so we get something it's clear that this is a computation that makes sense to do it's not clear how we we know where it ends up that feels like it's we're going to require real math to have some kind of theory behind it we could be computational and pull up Python and remember that it actually has built-in support for complex numbers if write something like complex of to three it gives me a complex number with real part two and imaginary part three and what's cool is it it already knows the rules for multiplication so I could multiply this by other things it knows how to take their product very cool now given that our definition of this exponential function if you'll remember was only ever in terms of raising something to a power and scaling it by a factorial and then adding all those terms up well it should make sense to plug in something like complex zero one which is how we write I in Python and see what it pops out and indeed it crunches through the numbers and it gets for us something whose real part is evidently around point five four who knew and whose imaginary part is around point eight four and in terms of our visual you know that checks out it looks like the real part is a little above 0.5 so point five five four I believe that and the imaginary part is it claimed what point eight four yeah that seems about right so at this point we can actually think what is the claim of Euler's formula okay what is it telling us about the behavior here and my hope is that Whitley's formula might have started off mysterious but mysterious in a bad way okay because when you initially see this expression e to the I theta is equal to cosine theta plus I sine theta telling you that oh I guess imaginary exponents get us on the unit circle it doesn't leave you wondering about a real pattern it leaves you wondering what the convention is right it leaves you wondering WTF however right now what we can see is what this is saying is if I plug in I times theta evidently that's the same as walking around a unit circle now that feels very substantive there's a lot of content to be had in that expression because if we go to our visual what it's telling us is that if I if I crank up this value of theta the place that this sum of vectors converges always sits on a circle that's one of radius one that's not obvious if you just look at this expression and you think through what it's saying very literally that is definitely not something that just pops right out of it but it becomes a lot more beautiful in my opinion because you're no longer asking questions of convention you're asking questions of the universe of actual patterns so just to take the most famous input that there is plugging in PI around 3.14 let's think about what this is actually saying it's saying okay the first term is 1 so we've got our vector pointed 1 unit to the right the next term is I times pi okay so that's gonna be a vector that's pointed up because of the I with the length of pie around 3.14 which it looks like it is the next one is going to be I squared times pi squared over 2 so that I squared means that it's negative okay it's pointing to the left and I guess the magnitude is pi squared over 2 we could do a quick gut check if we wanted and just ask ourselves you know what is what is pi squared it's around 9.8 same as the gravity on earth in terms of meters per second squared which is always fun and dividing that by 2 it looks like it's 4.9 3 ok does that check out yeah I buy it that the the length of this vector is just less than 5 so 4 point 9 3 and then the next one because of the I cubed is pointed down and the magnitude is PI cubed over 6 evidently that's around the magnitude of PI cubed over 6 and so far they've the vectors have been getting bigger but it's at this point they start turning around because basically each time on the for each new term the numerator gets an extra pie thrown in but the denominator now gets an extra 4 thrown in like going from 6 to 24 you're multiplying by a 4 and then it's going to get an extra 5 and then a 6 so the denominator is going to start growing much more than the numerator which is why the vectors shrink and shrink and start to converge to a point so the I in there is what's letting these vectors all turn 90 degrees each time and give us this very pretty spiral sum and the claim of Euler's formula is this very fascinating fact that evidently when the vectors are rotating according to powers of I and when their lengths are changing according to powers of pi divided by n factorial they all conspire in just the right way that they land at the point negative 1 that's interesting still mysterious but mysterious in a good way and it's making an even stronger claim which is that any other angle that we might put in it puts us on a unit circle at that angle now this is especially mysterious when we consider the fact that as we let this value just continue increasing it's not even clear that the function would be periodic right because as theta increases more and more of the vectors become relevant because it takes longer and longer for the denominator of our terms to win out and yet they stayed very nicely constrained on to the unit circle I think that's interesting so just to just to do a last little concept check here I'm gonna ask you one final question to basically see if you've been paying attention see if the the claims-made stand to reason so getting rid of our powers of I the last question is a a wind down question of sorts for the evening which of the following values is closest to e to the 3 I okay so as usual this is the link you can go to which of the following values is closest to e to the 3 I [Music] [Music] you now as you're answering let me just remind you that the main takeaway I want you to have from this lesson is that the way that you read this is not to think about a constant e raised to an imaginary power because that doesn't really make sense when we're thinking of powers as repeated exponential repeated multiplication however what does let's see it seems like we have some controversy over the last question we'll address that in a moment but what you should think about here is the fact that e to the X is shorthand for this general polynomial okay and this general polynomial it turns out to be very relevant in things like calculus in a future lecture I'd like to start talking about why this idea of X to the N over N factorial and adding them up why is that useful why would anyone care why would it give rotation but if you know that that's how to read it hopefully this isn't a nonsense question and it's not one that just asks you to blindly believe convention it kind of asks you to blindly believe a claim about the fact that this follows a circle but we'll address that in the next lecture and I think that's enough time to go ahead and grade this and the correct answer is B which around 3800 W nicely got well done well it done indeed and it's around negative 0.99 plus 0.1 4i and so the expectation isn't that you're a machine who can calculate that in your head it's instead the idea that if we go over to our visual where we're plugging in values of theta and you ask what happens when you plug in 3 that the claim of Euler's formula is that we walk three units around a unit circle knowing that pi units takes us all the way around three units should take us so that the real part is really quite close to negative one and the imaginary part is something positive and of the options you were given those were the only ones in there now the very last thing that I would like to emphasize here is that nothing about this actual expression has to do with the number E has to do with the number 2.71828 on and on if we were to go to the computation taking place if I plug in something like X of a complex number whose real part is zero and whose imaginary part is pi okay which at first it might look like it's outputting something other than negative one but it's saying the real part is negative one and then some numerical error because partly because of how we defined it you know only doing so many terms and partly because computers can't can do infinite precision the imaginary part it might look like it's saying three point four five but really it's saying times 10 to the negative 16 that's what that means so it's basically zero if you wanted to see this we could maybe import numpy and do something like round it so I can round this thing to I don't know eight decimal places and we see that okay it's basically negative one so this is the function that we wrote showing us that e to the PI is negative one but what I'd like to emphasize is that nowhere in that function did we include the value 2.718 in fact nowhere in the computer's memory during the execution of this computation would it encounter the value 2.718 I would submit to you that that value is not relevant to the equation e to the PI I equals negative one which is kind of funny all right let's at least find out what the relationship is because you know I'm not going to say there's no connection between e and pi there is one but I think when you write the very famous celebrity equation e to the PI I equals negative one it's it heavily overstates the connection that there is between this e and that pi what's really going on is that we have this function X that we've defined and that when you plug in real values okay when you plug in real numbers for the reasons we talked about earlier it makes sense to write x of x in terms of whatever its value at one happens to be x but one raised to the X and that we often call that value X of 1e okay this is a property of the function another fact about this is that when we plug in imaginary numbers the claim of Euler's formula which I haven't shown why it's true but the the claim that it's making is that it's periodic basically that it walks around a circle and it's periodic with a period of 2pi I basically when you increase that input by 2pi you get back to where you started and even more specifically that you do so by walking along a circle so it's saying that each of these constants is a child of the function X of X right but the way that they're related kind of happens along different dimensions that I think is the healthy way to understand this formula now in the next lecture we are going to be talking about why this thing walks around in circles and that misses that necessarily involves a little bit of calculus so we're going to get a little bit of calculus into the mix here but as minimal as I can and before then I want to leave you with some homework because if you remember the core property that related this strange polynomial with factorials to the number E was that addition in the input is the same as multiplication in the output so before next time I won't grade this this is just you know to do to do on your own time in your own way question number one I want you to show that when you fully expand X but X times X of Y and you know you might have written down for yourself what this function is it's all those factorial things so that when you expand that out each term has the form X to the K times y to the M divided by K factorial M factorial for certain whole numbers K and M and remember zero factorial is defined to be 1 and question number to show that when you expand Expo of X plus y so addition in the input each term is going to look like 1 over N factorial times n choose K and times X to the K times y to the N minus K if you're not familiar with this term n choose K take a look at the binomial formula I'm sure there's plenty of excellent videos on YouTube about it I'll leave links in the description as I find them but show that that's true question 3 compare the two results you just found to explain this key property right the thing that relates this function to some special constant and the idea of raising it to powers and then if you're up for the the bonus question this is kind of the this is for all of the extra credit in the world justify see if you can say will this result still be true if x and y are complex numbers and will it still be true if they're matrices yeah that actually I think is a very important exercise to go through to understand when this is true for what kinds of inputs because like I said as you get deeper and deeper into math this X function remains very relevant and you start plugging in things that seem wilder and wilder it's useful it is genuinely useful and it's important to know what properties carry over and which ones don't no I was talking a little about this lecture with a friend before hand and I mentioned the spirit of the lockdown math series being like high school classes and you know he looks at me quizzically and is like is this is that a high school topic and the honest answer is I don't know what I can say is that it is the case that in high school we came across the formula e to the I theta I think the first time that I ever saw it was in you know learning about complex numbers it was just hot as the polar representation of complex numbers to emphasize this idea that when you multiply two things you you multiply their magnitudes and you add their angles but it was just this strange thing handed from on high the usual way that we teach a relationship between e to the X in this polynomial comes from a piece of calculus called Taylor series and you start with the function e to the X always defined in a little bit of a wishy-washy way depending on the course that you're taking and then you show that it's connected to this polynomial but I think there's no reason we can't go the other way around and I actually wish that I had seen this I wish that I had known from the beginning that through math especially through deeper math that this is what the expression e to the X actually means so with that I'm going to call an end to the proper lecture and just take a couple more questions from the audience to finish things off maintain the interactivity that is the spirit of our of our elections all right great great we have more discussion I love this Crispin's proof is wrong the constant function f of X equals 0 ah wonderful ah excellent counter example I think this is maybe niggling in the back of my mind it's like I don't think this is unique great so you have the case of a function that is constantly equal to zero and I think ok whoever was asking a question earlier that I didn't quite understand I think this must be what you were getting at and I didn't I just wasn't reading it properly but yeah that would be a function where f of 0 is not necessarily 1 beautiful alright so that question that I said was wrong and then I said it wasn't wrong it actually was wrong so thank you very much Erik moss and to everyone who can think more clearly about things than I can why alive on camera and then another person points out exactly the same thing I love this this is just like a real class I'm up here I get to make mistakes you're down there you get to correct them and I think it makes it better for everyone watching there's a legitimate discussion to be had what's relevant in the case of the X function is just that it's not constantly 0 that it does in fact equal 1 at the input x equals 0 but I love this discussion and I think I can imagine no better place to end things than right there so as always thank you for joining I hope this helped provide a little bit of a different perspective than what you usually see in other classes on what's a very popular topic on the Internet at least as math goes and next time we're going to talk about why it's true where the circle would come from I hope to see you then [Music] you [Music]

Channel: 3Blue1Brown

Views: 640,539

Keywords:

Id: ZxYOEwM6Wbk

Length: 51min 16sec (3076 seconds)

Published: Tue Apr 28 2020

Reddit Comments

[u/merlinsbeers]

It's saying there's a fundamental geometric relationship between logarithms and trigonometry.

But what is that saying?

[u/ziggurism]

What is this? 3b1b is going to livestream a math lesson tomorrow?

[u/M4mb0]

What Euler's formula says depends on what one assumes is known beforehand. Nowadays, at the university level, sine and cosine are often introduced as being precisely the real and imaginary part of exp(it). Then Euler's formula, by itself, is really not much more that splitting a complex valued function into real and imaginary part.

The real deal is showing that the function t → exp(it) is periodic with fundamental period 𝜏.

[u/babuchat]

I do not remember the name of that phenomenon, but today I had a math test on complex numbers and now this.

Getting those Matrix feels.

[u/ezpickins]

Is this out? Why is it posted if it isn't?

[u/Le_Martian]

Oiler’s formula says if your door hinge squeaks, you should oil it

[u/KishorRathva]

I am watching his videos like netflix series in lockdown ,this guy is genius 🙌🙌

[u/calcul8]

I love that he's sharing this with high school kids. I'm not sure about other locales, but students in my high school would never be introduced to this relation and get utterly lost in trying to memorize things like double angle formulas, half angle formulas, and all sorts of other relations.

Using this I was lucky enough to skip all that crap back then.

[u/theplqa]

The unit circle can be identified with rotations in the plane. Pick a point to be no rotation, (1,0), then every other point determines a rotation as how far you must go from (1,0). This is a group, we compose two points by composing their associated rotation amount. This group has a smooth parameter, the angle of the rotation, and thus forms a smooth manifold as well. We call these smooth groups Lie groups.

This group constructed above is isomorphic with SO(2), 2x2 real orthogonal matrices. Send the angle x to the element R(x) = ((cos x,sin x), (-sin x, cos x)). Then adding angles x+y is compatible with matrix multiplication, R(x)R(y) = R(x+y).

Using this explicit representation, we can look at the Lie algebra, the tangent space at the identity of the Lie group. For a connected Lie group, every element of the group can be obtained by exponentiating an element of its Lie algebra. One way to think of this is that if the Lie group is the position of the particle on the circle, then the Lie algebra is the velocity at (1,0), given this velocity we can obtain the motion of the particle if the group is connected.

The reason exponentiation is important is this. Say we have some velocity v, then we want to figure out how far we shift from 1 over some time t. If we do it once then we get 1 + ivt. But this isn't quite right, because v tells us infinitesimal information, not average information, thus we have to compose it over and over again. Doing it twice is (1 + ivt/2)^2. Keep going to get the limit as n goes to infinity of (1 + ivt/n)^n, which is just the definition of e^ivt.

Let's compute the Lie algebra now. This will just be R'(0), the general element R(x), take the derivative, and evaluate at 0 (because R(0) is the identity matrix = point (1,0)). J = R'(0) = ((0,1),(-1,0)).

Note that J² = -I. Thus using this matrix, we have an isomorphism of the complex numbers a + ib with 2x2 matrices of the form aI + bJ. Euler's formula in this form is e^xJ = cosx I + sinx J which of course identifies with e^ix = cosx + i sinx under the isomorphism.

Now the intuition is very simple. The Lie group is the circle and the special identity point is (1,0). The Lie algebra is the tangent space at the identity, which is the vertical line at (1,0). If we move to complex coordinates (x,y) -> x + iy then this vertical line is the purely imaginary line ix with x real. Now the exponential map on ix takes the line segment from (1,0) to (1,x) and wraps it around the circle. From this it's clear that it's periodic, exp(i 2pi) takes you all the way around, because the length of the line segment and the circumference of the circle are both 2pi.