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In the last video we took the MacLauren expansion of e^x, and we saw that it looked like some type of

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a combination of the polynomial approximations of cos(x) and of sin(x), but it's not quite, because there was

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a couple of negatives in there, if we were to really add these two together, that we did not have, when we

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took the representation of e^x. But to reconcile these, I'll do a little bit of a, I don't know if you can even call

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it a trick. Let's see, if we take this polynomial expansion of e^x, this approximation, what happens,

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and if we say e^x is equal to this, specially as this becomes an infinite number of terms, it becomes less of an approximation

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and more of an equality. What happens if I take e^(ix). And before that might have been kind of a weird thing to

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do. Let me write it down: e^(ix). Because before it's like, how do you define e to the ith power, that's a

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very bizzare thing to do, to take something to the xi power, how do even comprehend some type of a

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function like that. But now that we can have a polynomial expansion of e^x, we can maybe make

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some sense of it, because we can take i to different powers, and we know what that gives, you know,

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i^2=-1, i^3=-i, so on and so forth. So what happens when you take e^(ix). So once again, it's just like

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taking the x up here, and replacing it with an ix. So everywhere we see the x in it's polynomial

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approximation we would write an ix. So let's do that. So e^(ix) should be approximately equal to, and it'll become

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more and more equal. And this is more of an intuition, I'm not doing a rigorous proof here. But it's still

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profound... Not to oversell it, but I don't think I can oversell what is about to be discovered or seen in this

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video. It would be equal to 1+, instead of an x, we'll have an ix, +ix+, so what's

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(ix)^2? So it´s gonna be, so let me write this down, what is (ix)^2/2!? Well i^2 is gonna be -1 and

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you'd have (x^2)/2!. So it's going to be -(x^2)/2!, and I think you might see where this is gonna go. And then,

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what is, ix, remember, everywhere we saw an x we're gonna replace with an ix. So what is (ix)^3. Actually,

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let me write this out, let me not skip some steps over here. So this is going to be ((ix)^2)/2!. Actually let

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me... I wanna do it just the way... So +((ix^2))/2!+((ix)^3)/3!+((ix)^4)/4!+((ix)^5)/5! and we can just keep

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going so on and so forth. But let's evaluate these 'ix's raised to different powers. So this will be equal to 1+ix...

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(ix)^2, that's the same thing as (i^2)(x^2), i^2 is -1. So this is -(x^2)/2!. And then this is gonna be the same

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thing as (i^3)(x^3), i^3 is the same thing as (i^2)i, so it's gonna be -i, so it's gonna be -i(x^3)/3!. And then,

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so then +, you're gonna have, what's i^4? So that's (i^2)^2, so that's (-1)^2, that's just going to be 1, so i^4

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is 1 and then you have x^4 so +(x^4)/4!. And then you're going to have +, I'm not even gonna write the +

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yet, i^5, so i^5 is going to be 1i, so it's gonna be i(x^5)/5! so +i(x^5)/5!, and I think you might see a

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pattern here. Coefficient is 1, i, -1, -i, 1, i, then -1(x^6)/6!, and then -i(x^7)/7!. So we have some terms, some of them

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are imaginary, they have an i, they're being multiplied by i, some of them are real, why don't we separate

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them out? Why don't we separate them out? So once again, e^(ix) is gonna be equal to this thing, specially

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as we add an infinite number of terms. So let's separate out, the real and the non-real terms, or the real and the

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imaginary terms, i should say. So this is real. This is real, this is real, and this right over here is real. And we

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can keep going on with that. So the real terms here are 1-(x^2)/2!+(x^4)/4!, and you might be getting excited

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now, -(x^6)/6!, and that's all I have done here, but they would keep going, so +, and so on and so forth. So that's all of the

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real terms. And what are the imaginary terms here? And let me just, I'll just factor out the i over here. So it's

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gonna be +i times, well, this is ix, so this will be x, and then the next... so that's an imaginary term, this is an

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imaginary term, we are factoring out the i, so -(x^3)/3!, then the next imaginary term is right over there, we

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factored out the i, +(x^5)/5!, and then the next imaginary term is right there, we factored out the i so it's

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-(x^7)/7!, and then we obviously would keep going, so +, -, keep going, so on and so forth. Preferably to infinity, so

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that we can get as good of an approximation as possible. So we have a situation where e^(ix) is equal to

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all of this business here. But, you probably remember from the last few videos, the real part, this was the

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polynomial, this was the MacLauren approximation of cos(x) around 0, or i should say the Taylor

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approximation around 0, or we could also call it the MacLauren approximation. So this and this are the

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same thing. So this is cos(x), specially when you add an infinite number of terms, cos(x). This over here, is

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sin(x), the exact same thing. So looks like we are able to reconcile how you can add up cos(x) and sin(x) to get

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something that's like e^x. This right here is sin(x) and so, if we take it for granted, I'm not rigorously proving it

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to you, that if you'd take an infinite number of terms here, that this will essentially become cos(x), and if you

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take an infinite number of terms here, this will become sin(x), it leads to a fascinating formula. We could say

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that e^(ix) is the same thing as cos(x), and you should be getting goose pimples right around now, is equal to

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cos(x)+i(sin(x)), and this is Euler's Formula. This right over here is Euler's Formula, and if that by itself isn't

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exciting and crazy enough for you, because it really should be, because we've already done some pretty

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cool things. We're involving e, which we get from continuous compounding interest, we have cos(x) and

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sin(x), which are ratios of right triangles, it comes out of the unit circle, and somehow we've thrown in (-1)^(1/2),

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there seems to be this cool relationship here. But it becomes extra cool, and we are gonna assume we are

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operating in radians here, if we assume Euler's Formula, what happens when x is equal to pi? Just to

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throw in another wacky number in there, the ratio between the circumference and the diameter of a circle,

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what happens when we throw in pi? We get e^((i)(pi)) is equal to cos(pi), cos(pi) is what? cos(pi) is, pi is

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halfway around the unit circle, so cos(pi) is -1, and then sin(pi) is 0. So this term goes away. So if you

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evaluate it at pi, you get something amazing, this is called Euler's Identity!! Euler's Identity! I always have

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trouble pronouncing Euler. Euler's Identity!! Which we can write like this, or we add 1 to both sides, and we

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can write it this. And I'll write it in different color for emphasis. e^((i)(pi))+1=0. And THIS, this is thought

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provoking. I mean, here we have, just so you see, I mean, this tells you that there is some connectedness

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to the Universe that we don't fully understand, or at least I don't fully understand. i is defined by engineers

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for simplicity so they can find the roots of all sorts of polynomials, as, you could say, the square root of -1.

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pi is the ratio between the circumference of a circle and it's diameter, once again, another interesting

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number, but it seems like it comes from a different place as i. e comes from a bunch of different places.

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e you can either think of it, it comes out of continuous compounding interest, super valuable for Finance, it

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also comes from the notion that the derivative of e^x is also e^x, so another fascinating number, but once

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again, seemingly unrelated to how we came up with i, and seemingly unrelated to how we came up with pi.

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And then of course, you have some of the most profound basic numbers over here, you have 1, I don't

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have to explain why 1 is a cool number, and I shouldn't have to explain why 0 is a cool number. So this right

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here, connects all of these fundamental numbers, in some mystical way, that shows us that there's some

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connectedness to the Universe, so frankly, frankly, if this does not blow your mind, you really...

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you have no emotion.