 In this video, we delve into the very important question about what does it need to take a complex exponent? I mean, it makes sense to do things like i squared, because that's just i times i, which would be negative one in the situation. And we can even make sense out of doing something like what's i to the square root of two power, because that would be involving taking some limits and things, because we can do things like i to the one half, we can do radicals. And the square root of two as an irrational number is the limit of rational numbers. We can take, you can use calculus to make sense out of irrational exponents. So that takes care of any real exponent. But what if you wanted to do some type of imaginary exponent? Like, what if you wanted to do e to the i? What would that even mean, right? Or another option I have listed here is two to the one minus i. Well, the good thing about the second one is if, if we're gonna make any sense out of it, two to the one minus i, if we're taking exponents, you know, because let's say we've never, ever considered an imaginary exponent before, we would want whatever that new exponent rule is. It has to be compatible with the old exponent rules. So therefore, this should be the same thing as two to the first times two to the negative i, for which as this is just a real exponent, that's just gonna give us a two. And if you have a negative exponent, means you should do division. So you get two divided by two to the i. So whatever is in question right here, it really just depends on if we can do two to the i, we can do the rest of it, or same thing right here. If we can do e to the i, I think we can do the rest of it. And so if we look at this more general, you take a to, so a is some number raised to a complex exponent, x plus yi. Well, whatever complex numbers do as exponents, it should be compatible with the rules we already have. If you add exponents, that means multiplication a to the x times a to the yi. And when you have multiplication exponents, you should be able to factor it and you get something like a to the y to the i. Oh, we see that already. But we should be able to expand that a to the i y by exponents rules, right? This should look something like a to the y, which is a real number to the i. So we really only have to deal with the i-th power of things. That's all that kind of matters. But then what we can do even better is if you have a to the i y, you can actually replace a with e to the natural log of a. Notice that e to the natural log of a is just a again because the exponential and the natural log are inverse functions of each other. So we're gonna replace a with e to the natural log of a. And then by exponent rules, this becomes e to the i y natural log of a. And then kind of factoring this thing one more time, we're gonna get e to the i y natural log of a. So what I'm trying to say here is it turns out that all that we have to do is if we can figure out what e to the i is, then we actually can make sense out of every single imaginary exponent. And instead of just doing e to the i, we're actually gonna work with e to the i y, where y could be any real number. If we can take e to a purely imaginary exponent, it turns out we can make sense out of any complex exponents whatsoever. And although this might be bizarre, right? To even think about e to the i, what does that even mean quantitatively? Whether we ought to do it or not, turns out there is actually a very natural candidate on how to handle this creature right here. And the idea is we're actually gonna use power series to make sense of this imaginary exponent, because with power series, we can make perfect sense of what this is. And I'll show you what I mean in just a moment. And it turns out that this is really just gonna connect us to the polar form of complex numbers. And so it's not gonna seem so bizarre once we have the right perspective. So to help us do exponents, imaginary exponents for e, we're gonna introduce Euler's formula. Which notice if we have a complex number, z equals x plus yi, we'll call its modulus r and its argument theta. Then we can write a complex number in polar form, trigonometric form, z equals r cosine theta plus i sine theta. Now, this is what Euler's theorem says. Euler's theorem says that this is the same thing as r e to the i theta. That's the same thing as this thing right here. So more specifically, e to the i theta is just cosine plus i sine theta. That's the definition of an imaginary exponent. So we have to establish this principle right here. If Euler's identity holds, then this general statement about polar forms of complex numbers will follow very immediately. The proof of Euler's formula is something we are gonna omit for the time being. And the reason is, is that the most common proof, a student with C of Euler's formula comes from a technique about Taylor series, which Taylor series is a topic introduced in calculus two, which is beyond the scope of this trigonometric series. You can see a little bit of what's going on right here, but it turns out, if you start doing some infinite arithmetic, infinite sums, it makes a whole lot more sense why cosine and sine can combine in such a way to form an exponent like e to the i theta. And so I will delay that proof until many of you are ready to take calculus two. In fact, if you're dying to see the proof right now, click on the link you see on the screen, otherwise we're gonna move on. But let's just do some examples of this right here if you wanted to compute e to the i pi. Well, in this situation, your angle of play would be pi itself. So e to the pi i would look like cosine of pi plus i sine of pi. Cosine of pi is negative one and sine of pi is zero, in which case this would simplify just to be negative one, and that's it. It's quite a beautiful formula right there. e to the pi i equals negative one. I remember the first time I saw this fact, it was told to me in an AP calculus class when I was in high school. And I couldn't believe it. It's like, how in the world could the numbers e pi i interact in such a way that it gives you something so simple as negative one? In my disbelief, I turned to my graphing calculator, it was a TI-89, which could do complex arithmetic. I type it in e to the pi i, and then the calculator said negative one. And then I just accept it as gospel at that moment. It's like, oh, well, the calculator says it must be true. Of course, I've learned since then to not put so much stock in calculations, calculators themselves. But nonetheless, by Euler's identity, we can see very quickly that e to the pi i is in fact negative one, a very beautiful formula right there. Let's use a slightly more complicated exponent here. If you take e to the negative one plus i pi over two, well, since there's two parts, a real part and a magic part, we're gonna break it into two pieces. You have e to the negative one, and then e to the i pi halves. Well, e to the negative one, that's a purely real number, that'll just become one over e. And then we have e to the i pi halves, that becomes cosine of pi halves plus sine of pi halves. Cosine of pi halves, that should be zero. Sine of pi halves is one, so you can get one over e plus times one plus i. So that just gives you i over e, or one over e times i, whichever you prefer. So we can simplify this exponent to give us i over e right there. And then as a last example, this time let's say z is equal to two times e to the pi i over three. Same basic idea, we're gonna have a two sitting in front of it. We're gonna take e to the pi i over three. I should mention that in terms of complex numbers, this right here really is the polar coordinate of this thing. I should mention that if you take e to the i theta, this modulus is always equal to one. You can argue that if you take cosine theta plus r sine theta, its modulus will always equal one, because in that situation, if we kind of unwrap this a little bit, you're gonna get the square root of cosine square theta plus sine square theta, which sine square plus cosine square we know is one. So you see that e to the i theta is always a complex number whose modulus is one. And in fact, this is the general way of describing a complex number whose modulus is one that can always be written in this form e to the i theta. So then in general, you'll see complex numbers written in the form r times e to the i theta, for which you see right here the modulus and the argument of the complex number. And so this is commonly referred to as the polar form of that complex number. So continue on with the calculation. You'll get two times cosine of pi thirds plus i sine of pi thirds, which cosine of pi thirds is one half, sine of pi thirds is root three over two. You can distribute the two through and you recapture the complex number two plus i square root of three. And so Euler's identity is pretty awesome. It really is. And it shows you how you can actually use power series to calculate things that you might not realize power series are necessary for. Power series is a theoretical tool for extending functions from real variables into complex variables, which helps us get this complex, I should say polar form for complex numbers here.