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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
a combination of the polynomial approximations of cos(x) and of sin(x), but it's not quite, because there was
a couple of negatives in there, if we were to really add these two together, that we did not have, when we
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
it a trick. Let's see, if we take this polynomial expansion of e^x, this approximation, what happens,
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
and more of an equality. What happens if I take e^(ix). And before that might have been kind of a weird thing to
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
very bizzare thing to do, to take something to the xi power, how do even comprehend some type of a
function like that. But now that we can have a polynomial expansion of e^x, we can maybe make
some sense of it, because we can take i to different powers, and we know what that gives, you know,
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
taking the x up here, and replacing it with an ix. So everywhere we see the x in it's polynomial
approximation we would write an ix. So let's do that. So e^(ix) should be approximately equal to, and it'll become
more and more equal. And this is more of an intuition, I'm not doing a rigorous proof here. But it's still
profound... Not to oversell it, but I don't think I can oversell what is about to be discovered or seen in this
video. It would be equal to 1+, instead of an x, we'll have an ix, +ix+, so what's
(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
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,
what is, ix, remember, everywhere we saw an x we're gonna replace with an ix. So what is (ix)^3. Actually,
let me write this out, let me not skip some steps over here. So this is going to be ((ix)^2)/2!. Actually let
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
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...
(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
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,
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
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 +
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
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
are imaginary, they have an i, they're being multiplied by i, some of them are real, why don't we separate
them out? Why don't we separate them out? So once again, e^(ix) is gonna be equal to this thing, specially
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
imaginary terms, i should say. So this is real. This is real, this is real, and this right over here is real. And we
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
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
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
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
imaginary term, we are factoring out the i, so -(x^3)/3!, then the next imaginary term is right over there, we
factored out the i, +(x^5)/5!, and then the next imaginary term is right there, we factored out the i so it's
-(x^7)/7!, and then we obviously would keep going, so +, -, keep going, so on and so forth. Preferably to infinity, so
that we can get as good of an approximation as possible. So we have a situation where e^(ix) is equal to
all of this business here. But, you probably remember from the last few videos, the real part, this was the
polynomial, this was the MacLauren approximation of cos(x) around 0, or i should say the Taylor
approximation around 0, or we could also call it the MacLauren approximation. So this and this are the
same thing. So this is cos(x), specially when you add an infinite number of terms, cos(x). This over here, is
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
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
to you, that if you'd take an infinite number of terms here, that this will essentially become cos(x), and if you
take an infinite number of terms here, this will become sin(x), it leads to a fascinating formula. We could say
that e^(ix) is the same thing as cos(x), and you should be getting goose pimples right around now, is equal to
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
exciting and crazy enough for you, because it really should be, because we've already done some pretty
cool things. We're involving e, which we get from continuous compounding interest, we have cos(x) and
sin(x), which are ratios of right triangles, it comes out of the unit circle, and somehow we've thrown in (-1)^(1/2),
there seems to be this cool relationship here. But it becomes extra cool, and we are gonna assume we are
operating in radians here, if we assume Euler's Formula, what happens when x is equal to pi? Just to
throw in another wacky number in there, the ratio between the circumference and the diameter of a circle,
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
halfway around the unit circle, so cos(pi) is -1, and then sin(pi) is 0. So this term goes away. So if you
evaluate it at pi, you get something amazing, this is called Euler's Identity!! Euler's Identity! I always have
trouble pronouncing Euler. Euler's Identity!! Which we can write like this, or we add 1 to both sides, and we
can write it this. And I'll write it in different color for emphasis. e^((i)(pi))+1=0. And THIS, this is thought
provoking. I mean, here we have, just so you see, I mean, this tells you that there is some connectedness
to the Universe that we don't fully understand, or at least I don't fully understand. i is defined by engineers
for simplicity so they can find the roots of all sorts of polynomials, as, you could say, the square root of -1.
pi is the ratio between the circumference of a circle and it's diameter, once again, another interesting
number, but it seems like it comes from a different place as i. e comes from a bunch of different places.
e you can either think of it, it comes out of continuous compounding interest, super valuable for Finance, it
also comes from the notion that the derivative of e^x is also e^x, so another fascinating number, but once
again, seemingly unrelated to how we came up with i, and seemingly unrelated to how we came up with pi.
And then of course, you have some of the most profound basic numbers over here, you have 1, I don't
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
here, connects all of these fundamental numbers, in some mystical way, that shows us that there's some
connectedness to the Universe, so frankly, frankly, if this does not blow your mind, you really...
you have no emotion.