[00:00] Today we are going to be talking about Euler's formula. ending up with this lesson, I'm going to go ahead and show you So, I don't expect you to necessarily understand this immediately,
[00:16] but the point is that this is something we're going to walk towards. of exponentials in a way that works in the complex plane. very literally what the claim of Euler's formula is.
[00:33] rather than letting it be shrouded in a certain mystery or a certain question of what the Now, needless to say, this is kind of a confusing thing.
[00:45] We've got this spiral of vectors, and if it's not entirely clear, don't worry about it. But before any of that, let's take a step back and remember where were we.
[00:57] Back in the end of the last lesson, when we were talking about complex numbers, at were those that existed on the unit circle. We've got the real number line with the points 1 and negative 1 indicated.
[01:13] 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,
[01:27] multiplying by this number has the effect of rotating things by that angle. throughout electrical engineering, all throughout math. They describe wave mechanics, they're very important for polynomials.
[01:41] 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. the x coordinate is going to be the cosine of that angle,
[01:56] is going to be i times the sine of that angle. So you might think, all throughout physics, all throughout electrical engineering, you see the expression cosine of theta plus i sine of theta.
[02:10] 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.
[02:24] Now e is a special constant of nature, and I always remember in high school, It was something that was just kind of handed down, okay, it's 2.71828, on and on.
[02:37] And we were just taking, you know, we were to take this as an analog of pi. It's an irrational number that evidently the universe finds significant. because the president, Andrew Jackson, served two terms.
[02:52] He was the seventh president, and he was elected in 1828. I don't know if Andrew Jackson appreciated this relationship that he had to Euler, Now, just to gauge sentiment, because I'm not entirely sure where the audience is,
[03:08] 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. if you were to choose one of Euler's formulas,
[03:22] would have gone with the one that we're talking about today, which is great, you know, 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
[03:37] e to the power i theta equals cosine of theta plus i times the sine of theta? And the options here are that you've never seen it before, totally understandable, that you still don't understand it but you've grown used to it,
[03:53] So, as you can see, answers are rolling in. This is going to be way fun if more of you participate. that forwards you to the place where you can answer this poll.
[04:09] And while 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. So, if we say e to the i times pi, now we're thinking of pi as an angle,
[04:27] 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.
[04:39] 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? boundary until you've gone a distance of pi, kind of by the definition of what pi is,
[04:55] 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,
[05:12] Okay? idea of raising a constant, which is a little wishy-washy to start with, And if you don't understand what that means, you're in good company,
[05:29] 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, Because it looks like we have a pretty even distribution among all camps.
[05:45] which is those who still don't understand it but they've grown used to it. This, I'm going to guess, includes people like math majors or engineering majors or people who have gone into a technical field,
[05:59] where it comes up a lot, so you have to grow used to it. 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
[06:14] Very happy to see that d, I understand it well, is the second most common answer. I'm curious among those who answer d, if by the end of the lecture they would
[06:28] say that what they understood about this is the same as what I'm going to teach. formula actually has nothing to do with the number e, okay? in what this formula is computationally saying.
[06:43] understand it well would agree with that statement. and at the very end is the actual target demographic of those who've never seen it before.
[06:55] because I think sometimes you see it in an odd circumstance here or there, You're the ones that I want to talk to about this, okay? You see this weird formula e to the pi i, or the
[07:11] 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?
[07:26] What is the function at play and how is it defined, okay? Because in this case the function is e to the x, And I think a lot of people think that that refers to taking a number e and multiplying
[07:45] by itself some number of times, and that x describes how often you're multiplying by And that yeah, there's some notion of extending that to things but that it's based in this idea of repeated multiplication.
[08:02] Now the thing that makes this equation misleading is that that's not the function. Let's emphasize that very heavily. This is not its convention.
[08:15] Instead, what has emerged in math is that we use e to the x to be a shorthand for another function. This is how you often see it in literature.
[08:29] It's one plus x plus x squared divided by two plus x cubed divided by six plus x to the power four divided by twenty four.
[08:45] 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. In practice, if you're actually computing this,
[09:00] the series pretty early to get an approximate value for what this actually is. work for lots of kinds of x that we could plug in.
[09:13] 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 exp function means.
[09:26] 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 Because on the surface, this infinite polynomial seems very different from
[09:43] the idea of some special constant of nature e and raising it to a power. We're going to take, you know, a couple minutes to do this. this function by starting to plug in a couple different values.
[09:59] So that way we get 1 and then x is 1, so that's 1. x squared is still 1, so that's 1 half.
[10:11] And then 1 twenty fourth. And in general, we're adding 1 divided by n factorial. sense that as we add more and more terms, it approaches a
[10:28] And that value ends up being around 2.71828, the Andrew Jackson number.
[10:40] x of 1 was going to be something. At the moment, there isn't necessarily anything special about that. Just to get a little practice with what that would look like, if I wanted x of 2,
[11:00] it would look like 1 plus 2 plus 2 squared over 2 plus 2 cubed over 6. And in principle, if you were on a desert island and you just needed to compute x of 2, you could work this all out by hand if you were comfortable with long division.
[11:15] 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.
[11:28] We implement it so we know exactly what it's doing. This is Desmos. Right now, let's be a little bit more programmatic.
[11:42] 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 exp 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,
[11:58] where in Python this double, what do you call it, an asterisk, this double asterisk sign is how we do exponentiation. And we kind of want to add that up a whole bunch.
[12:10] 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.
[12:22] and it knows what x is because that's what was handed to it. Factorial of n. And that's built into this math package that we imported.
[12:36] And I'm just going to do this for values of n that start at 0. We could talk all about factorials of weird values at a later date, And I'm just going to have it range up to 100 because 100 factorial is going to be huge.
[12:54] be small enough that those later terms don't contribute a lot. So even if you don't know Python, I hope that this is a reasonably clear way to turn the math into something that our computer can chew on,
[13:08] and crunch through the numbers so that we don't have to. So for example, if we type x of 1, we get the Andrew Jackson number. And I could type in x of 2, and it looks like that's around 7.389.
[13:25] We go over here and say, interesting, x of 2 was about 7.389. And if you spend a long time just kind of playing around with this, okay,
[13:43] I think it's not obvious that you might find this, but if you were just plugging in a couple values, x of 3, x of 4, one important fact you might stumble across is that if I add two numbers in the input,
[13:58] okay, so in this case I get 1096 when I plug in 7, which is 3 plus 4, that actually ends up being the same as if I plug in x of 3 times x of 4.
[14:10] So adding the input corresponds to multiplying in the output. And 3 and 4 weren't special here, I could have done, you know, 5.5 and 3.2, and that would have gotten me some value.
[14:24] And if instead I had added those together, okay, it gets the same value. So I think that would be a genuine discovery to have with respect to this function,
[14:36] And it's important enough that I want to write it down. Exp of a plus b is actually the same thing as exp of a times exp of b.
[14:57] Now if you just look at the polynomial, this is not clear. oh yes, of course, it couldn't have been any other way. and yes, homework will make you learn better.
[15:11] that are going to 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.
[15:27] Well, I want to ask you a certain question that will hopefully make this clear. Because I think if we, uh, if I just kind of tell you the implications here,
[15:41] it's not going to sink in to the same extent as if you really noodle with it yourself. going to give you some time to think on this one.
[15:53] some function f of x that has this special property. gives the same result as multiplying in the output, f of a times f of b.
[16:07] Okay? Whether it's through this polynomial, through other means. One of the options is that f of five is equal to f of one raised to the power five.
[16:24] Another is that f of one half is equal to the square root of f of one. And the other is f of negative one equals one divided by f of one. some function f where these three things will be true.
[16:38] I'm asking which of these necessarily has to be true only from that property. of adversarial function that doesn't satisfy it. And then you have various options for which collection of these three things is true.
[16:55] 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.
[17:25] Where it looks like Serft 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 i x.
[17:41] to Twitter and just use the hashtag lockdown math. I think the best way to learn is to have specific examples and really let yourself
[17:53] 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 see e to the i pi, that doesn't really count as a good specific example that explains the generality.
[18:08] 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 we're going to talk about why that relation might be there.
[18:23] So the form of having specific examples that aids your understanding wouldn't be, It would be building a relationship with circular motions and other some kind of centripetal force and it's orbiting or like orbital mechanics.
[18:40] Anything where 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 So with all of that, answers are still rolling in and I don't want you to feel rushed.
[18:56] So I'm going to give a little bit more time here actually. Because 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
[19:09] any function with this special property necessarily follows that. Oh, let's see if we can grade it when the top answer is the year.
[19:23] 2013, we're losing the top answer. 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.
[19:37] hang on, you know, is it the case that option 3 here is necessarily true? So that's a good sign.
[19:52] 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. believe that all three of these are necessarily true.
[20:09] So, second most common answer was B, they only believe that the first one is true. And then after that, people who either included number 2 or number 3. And it's very interesting to walk through why.
[20:25] The first one, which it seems like a majority of you believe, is that if we plug in something like 5 to a function with this special property,
[20:37] that it'll be the same as taking f of 1 raised to the fifth. Now the reason is that we could also write 5 as 1 plus 1 plus 1 plus 1 plus 1.
[20:51] And this property of addition in the input becoming multiplication in the output means that's the same thing as writing x of 1 multiplied by itself 5 times. It lets you rewrite the whole thing in terms of x of 1, which I'm going to say 5 times.
[21:12] This is the same as taking x of 1 raised to the power 5. whole number, it can literally mean multiplying by itself that number of times.
[21:24] That's not the same as the fact that e to the x is a shorthand for this crazy polynomial. And we might give x of 1 a special name, a shorthand.
[21:36] That's not why we call this number e, by the way. It's just because whenever Euler was using this, the first time in a particular book,
[21:48] he was partial to vowels and the vowel a had already been used. Okay, so simply by virtue of this property, we can Now a little bit trickier was the question about plugging in 1 half.
[22:06] is to think about what happens when we multiply that by itself. Exp of one half times exp of one half, because of this property,
[22:18] has to satisfy, or I should say has to equal exp of one half plus exp of one half. Which is of course x of 1, which let's say we're using the shorthand and we call it e.
[22:31] x of 1 half has to be a number such that multiplying it by itself equals e. Well that's what we mean when we write square roots. And the last one was talking about negative inputs.
[22:47] 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,
[23:04] Now what is exp of 0? Ooh, you know what I'm realizing. I think I might have actually made the, entered a wrong answer there.
[23:18] Because in our case, in our case exp of 0 actually does come out to be the number 1. you add them together, the only term that matters is 1.
[23:31] So exp of negative 1 times exp of 1 is equal to exp of 0, which is 1. I, I can't actually think to myself right now if that's necessarily true.
[23:46] 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 0 is going to be 1.
[23:59] Yeah, because we could just scale e to the x by some other amount. The correct answer would be only 1 and 2.
[24:12] Someone correct me on Twitter if I'm wrong about that. So if we go back to our paper, let's see. In the case of exp, this, this value at 0 would be 1.
[24:27] Well, if we call exp of 1 e, that means we're asking And it would be 1 divided by e. condition that when you plug in 0 you do get 1.
[24:44] 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. why it would be very reasonable to write this as e to the power x.
[24:59] Because basically this special property means that because numbers by adding it to itself or dividing as needed or negating.
[25:11] will let you extend from rational numbers to all the reals. But that's a technical point, you don't need to fuss over for what we're doing right now.
[25:23] can output in terms of what it outputs at the number 1. So you might read e to the x as just, whatever this polynomial is at the input 1, we can start exponentiating in terms of that.
[25:38] the number e to simply be where is this polynomial at the number 1. But I really want to emphasize that only makes sense for real numbers.
[25:51] you can't just add 1 to itself or subtract and divide and get the number i. you plug in things like matrices into this polynomial.
[26:03] you can divide it by 2, you can add all of those together. It's very useful for a field called differential equations,
[26:15] In quantum mechanics you often plug in these things called operators, which are kind of like the mature older brother of matrices. talking about a constant raised to some kind of power.
[26:30] But what you have to understand is that it's being plugged into this polynomial. So I understand why we have the convention of writing this as e to the x. But I think that's actually a bad convention as soon as we start extending it.
[26:45] So with all of that said, let's finally have some fun and plug in some complex values. I want to make sure that we're comfortable with powers of i.
[26:58] 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.
[27:11] Remember that i is defined to be a value that satisfies i squared equals negative one. For which values of n does i to the power n equal negative i?
[27:24] So the spirit of this question is to have people thinking deeply about powers of i. And 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.
[27:39] So let's go ahead and take a question from the audience while I take a drink of water. F of x equals zero doesn't work with f of negative one equals one over f of one.
[27:57] Yeah, okay. So they're saying that if f of x ever outputs zero. F of x equals zero doesn't work with f of negative one equals one over f of one.
[28:17] f of one doesn't imply that the output will ever be zero. So I could ask for some clarification, but I think that might
[28:29] Which was to assume, to extend the idea of exp a little bit too much. Where the fact that exp of zero equals one gives us this other property.
[28:42] And then Crispin Simmons says three is included because f of x plus zero. Yeah, someone thought this through a little bit more deeply F of x plus zero is the same as f of x times f of zero, which implies f of x equals one.
[29:01] That's a beautiful way of thinking about it. property of a function and you're supposed to deduce facts. And so in this case, okay, the question was graded correctly.
[29:17] The property simply that addition in the input becomes multiplication in the output is enough to imply f of zero equals one, which in turn locks in the value for f of negative One little property can just lock in basic, not the whole real number line
[29:31] because it didn't assume continuity, but it locks in pretty much any value you want. Even if I make errors, it does feel a little more interactive than the usual videos.
[29:43] Looks like we've still got some answers rolling in. We're trying to understand which powers of i are equal to negative i.
[29:55] 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.
[30:08] And as always, if you feel like you weren't done by the time I'm locking it in, If you wanted to pause and think about it further yourself,
[30:21] especially if you're watching this later, that would be totally in the spirit 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.
[30:36] And the second most common answer was basically the same but restricted to positives. because this is going to be the only thing we really need to plug in complex values
[30:51] 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.
[31:03] And because i has a magnitude of one, there's only rotation. If you multiply by i twice, okay, i squared, you end up 180 degrees around.
[31:18] 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.
[31:31] Multiplying by itself a fourth time, 90 degrees, takes us back to one. So i to the fourth is one, and then after that they all start repeating. that exponent with respect to powers of four.
[31:46] If it's one above a power of four, you're at i. of whether this is only for positive powers or negative powers.
[31:58] But what a negative power means, if we write something like i to the negative one, that's defined to be one over i, the number such that when you multiply by it, And you can see that that's actually the same as i cubed.
[32:13] This is the same as i cubed because that's a number where when we multiply it by i, Negative one, that's also one below a multiple of four. negative multiples of four will also satisfy this property.
[32:29] So it's a little tricky question, but it's just because I wanted you to think deeply just have us ticking forward by 90 degrees. We can take our crazy exp function, and we could plug it
[32:45] 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, uh, let's see, that circle is already suggestively drawn,
[32:57] I just want you to think about this polynomial where I'm going to 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 one plus i times theta plus i theta squared over two,
[33:12] And just as an example, let's crank theta on up until it equals one. And let's try to think pretty deeply about what this actually means computationally. The first term is one, which we might draw with this green vector on the bottom,
[33:26] The next term is going to be i times one, so just i. Now think about what the next term is.
[33:39] It's going to be i squared, or i theta squared, but theta is one, So that's why this next vector is pointed to the left, and it has length one half.
[33:51] Then after that, i cubed is pointed straight down, and we divide it by six. So now we just have an itty bitty vector that is of magnitude one sixth, The next one is pointed to the right, but it has magnitude one over twenty four,
[34:06] And so we get something. It's not clear how we would know where it ends up. That feels like it's going to require real math to have some kind of theory behind it.
[34:20] and remember that it actually has built-in support for complex numbers. it gives me a complex number with real part two and imaginary part three,
[34:32] 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,
[34:44] 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.
[35:00] something whose real part is evidently around point five four, who knew? And whose imaginary part is around point eight four, It looks like the real part is a little above point five,
[35:16] And the imaginary part is, it claimed what, point eight four? So at this point, we can actually think, what is the claim of Euler's formula?
[35:29] And my hope is that Euler's formula might have started off mysterious, Because when you initially see this expression,
[35:41] e to the i theta is equal to cosine theta plus i sine theta, telling you that, it doesn't leave you wondering about a real pattern, it leaves you wondering what the convention is, right?
[35:56] 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,
[36:13] 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 the sum of vectors converges always sits on a circle that's one of radius one.
[36:26] If you just look at this expression and you think through what it's saying But it becomes a lot more beautiful, in my opinion, you're asking questions of the universe, of actual patterns.
[36:42] plugging in pi around 3.14, let's think about what this is actually saying. got our vector pointed one unit to the right.
[36:55] Okay, so that's going to be a vector that's pointed up because of the i with a length of pi around 3.14, which it looks like it is. The next one is going to be i squared times pi squared over two.
[37:11] So that i squared means that it's negative, okay, it's pointing to the left. We could do a quick gut check if we wanted and just ask ourselves, It's around 9.8, same as the gravity on Earth in terms of meters per second squared,
[37:28] And dividing that by two, it looks like it's 4.93. Yeah, I buy it that the length of this vector is just less than five, so 4.93.
[37:41] is pointed down and the magnitude is pi cubed over six. And so far the vectors have been getting bigger,
[37:53] but it's at this point they start turning around because basically each time on the, But the denominator now gets an extra four thrown in. And then it's going to get an extra five and then a six.
[38:10] 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. 90 degrees each time and give us this very pretty spiral sum.
[38:24] 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,
[38:37] they all conspire in just the right way, that they land at the point negative one. Still mysterious, but mysterious in a good way. And it's making an even stronger claim, which is that any other
[38:52] 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?
[39:08] 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 stay very nicely constrained onto the unit circle.
[39:21] So just to do a last little concept check here, I'm going to ask you one final question to basically see if you've been paying attention, So getting rid of our powers of i, the last question
[39:37] is a wind-down question of sorts for the evening. Which of the following values is closest to e to the 3i?
[39:50] Which of the following values is closest to e to the 3i?
[40:20] 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.
[40:32] thinking of powers as repeated multiplication. However, what does...
[40:45] We'll address that in a moment. to the x is shorthand for this general polynomial, okay?
[40:57] relevant in things like calculus in a future lecture. x to the n over n factorial and adding them up. Why would it give rotation?
[41:13] 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. but we'll address that in the next lecture.
[41:27] And I think that's enough time to go ahead and grade this. And the correct answer is b, which around 3800 of you nicely got. And it's around negative 0.99 plus 0.14i.
[41:43] 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,
[41:55] that the claim of Euler's formula is that we walk three units around a unit circle. three units should take us so that the real part is really quite close to negative 1,
[42:09] 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.
[42:23] If we were to go to the computation taking place, if I plug in something like exp of a complex number whose real part is 0 and whose imaginary part is pi, which at first it might look like it's
[42:37] outputting something other than negative 1, but it's saying the real part is negative 1, 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 do
[42:51] The imaginary part, it might look like it's saying 3.45, but really it's saying times 10 to the negative 16th, that's what that means. If you wanted to see this we could maybe import numpy and do something like round it.
[43:09] So I can round this thing to, I don't know, 8 decimal places, So this is the function that we wrote, showing us that e to the pi i is negative 1.
[43:21] 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.
[43:35] I would submit to you that that value is not relevant to the equation e to the pi i equals negative 1, which is kind of funny, right? Let's at least write out what the relationship is, because, you know,
[43:49] 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 1, it heavily overstates the connection that there is between
[44:02] What's really going on is that we have this function exp that we've defined, and that when you plug in real values, okay, when you plug in real numbers,
[44:15] for the reasons we talked about earlier, it makes sense to write exp of x in terms of whatever its value at 1 happens to be, exp of 1 raised to the x.
[44:27] And that we often call that value exp of 1 e, okay? Another fact about this is that when we plug in imaginary numbers,
[44:39] the claim of Euler's formula, which I haven't shown why it's true, but 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 2 pi i.
[44:59] Basically when you increase that input by 2 pi i, you get back to where you started, So it's saying that each of these constants is a child of the function exp of x, right?
[45:12] But the way that they're related kind of happens along different dimensions. Now in the next lecture, we are going to be talking about why this thing walks around in circles.
[45:26] so we're going to get a little bit of calculus into the mix here, but as minimal as I can. Because if you remember, the core property that related this
[45:42] addition in the input is the same as multiplication in the output. So before next time, I won't grade this, this is just,
[45:55] you know, to do on your own time in your own way. Question number 1, I want you to show that when you fully expand exp of x times exp of y, and you know, you might have written down for yourself what this function is,
[46:10] Show 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.
[46:24] And remember, 0 factorial is defined to be 1. Question number 2, show that when you expand exp of x plus y, so addition in the input, each term is going to look like 1 over
[46:38] n factorial times n choose k times x to the k times y to the n minus k. take a look at the binomial formula, I'm sure there's plenty of
[46:50] excellent videos on YouTube about it, I'll leave links in the description as I find them. Question 3, compare the two results that you just found to explain this key property.
[47:03] constant and the idea of raising it to powers. this is for all of the extra credit in the world, justify,
[47:15] 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? through to understand when this is true for what kinds of inputs.
[47:30] this exp function remains very relevant and you start plugging in things that seem to know what properties carry over and which ones don't.
[47:45] Now, I was talking a little about this lecture with a friend beforehand, and I mentioned the spirit of the lockdown math series being like high school classes. is this, is that a high school topic?
[47:58] What I can say is that it is the case that in high school, I think the first time that I ever saw it, it was in, It was just taught as the polar representation of complex numbers to emphasize this idea
[48:16] that when you multiply two things, you multiply their magnitudes and you add their angles. The usual way that we teach a relationship between e to the x in
[48:28] 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 And then you show that it's connected to this polynomial.
[48:42] And I actually wish that I had seen this. especially through deeper math, that this is what the expression e to the x
[48:54] So with that, I'm going to call an end to the proper lecture and just Maintain the interactivity that is the spirit of our lectures.
[49:07] I love this. The constant function f of x equals zero. I think this was maybe niggling in the back of my mind.
[49:20] So you have the case of a function that is constantly equal to zero. I think this must be what you were getting at.
[49:34] But yeah, that would be a function where f of zero is not necessarily one. So that question that I said was wrong and then I said it wasn't wrong, And to everyone who can think more clearly about things than I can while live on camera.
[49:53] I love this. I get to make mistakes. And I think it makes it better for everyone watching.
[50:05] What's relevant in the case of the exp function is just that it's not constantly zero. That it does, in fact, equal one at the input x equals zero.
[50:17] And I think I can imagine no better place to end things than right there. I hope this helped provide a little bit of a different perspective than what you usually
[50:29] 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. 844 00:50:54,660 --> 00:50:39,160 .