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# Euler's formula & Euler's identity

## Video transcript

in the last video we took the Maclaurin expansion of e to the X and we saw that it looked like it was some type of a combination of the polynomial approximations of cosine of X and of sine of X but it's not quite because there were 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 to X of e to the 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 if we take this polynomial expansion of e to the X this approximation what happens and if we say if we say e to the X is equal to this especially as this becomes an infinite number of terms then it becomes less of an approximation and more of an equality what happens if I take e to the i-x and before that might have been kind of a weird thing to do we write it down e to the i-x because before it's like how do you define e to the I that's a very bizarre thing to do to take something to the X I power how do you even comprehend some type of a function like that but now that we can have a polynomial expansion of e to the X we can maybe make some sense of it because we could take I to different amounts to different powers and we know what that gives you know I squared is negative 1 I to the third is negative I so on and so forth so what happens if we take e to the i-x so once again it's just like taking the X up here and replacing it with an IX so everywhere we see the X and it's polynomial approximation we would write an IX so let's do that so e to the IX should be approximately equal to and we'll become more and more equal and this is more to give you 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 plus instead of an X will have an IX plus I X plus so what's IX squared so it's going to be so let me write this down what is I x squared over 2 factorial well I squared is going to be negative 1 then you can have x squared over two factorial so it's going to be minus x squared over two factorial I think you might see where this is going to go and then what is IX remember everywhere we saw an X we're going to place it with an IX so what is IX to the third power actually let me write this out let me not let me not skip some steps over here so this is going to be I x squared over 2 factorial actually let me I want to do it just the way so plus plus I x squared over 2 factorial plus I X to the third over 3 factorial plus I X to the fourth over 4 factorial and we can keep going plus I X to the fifth over 5 factorial and we could just keep going so on and so forth but let's evaluate these I X is raised to these different power so this will be equal to one plus I X I x squared that's the same thing as I squared times x squared I squared is negative one so this is negative x squared over two factorial and then this is going to be the same thing as I to the third times X to the third I to the third is the same thing as I squared times I so it's going to be negative I so this is going to be minus I times X to the third over 3 factorial and then so then plus then plus you're going to have what's I to the fourth power so that's I to this I squared squared so that's negative 1 squared that's just going to be one so I to the fourth is 1 and then you have X to the fourth so plus X to the fourth over 4 factorial and then you're going to have plus we ought to only write the plus it I to the fifth so I to the fifth is going to be 1 times I so it's going to be I times X to the fifth over 5 factorial so plus I times X to the fifth or or five factorial and I think you might see a pattern here coefficient is one then I then negative one then negative I then one then I then negative 1x the sixth over 6 factorial over six factorial and then negative I and then negative I X to the seventh over 7 factorial so we have some terms some of them have some of them are imaginary they have an eye there might be x I some of them are real why don't we separate them out why don't we separate them out so once again e to the i-x is going to be equal to this thing especially as we add an infinite number of terms well let's separate out the real and the non real terms all right 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 I obviously we could keep going on with that so the real terms here are 1 minus x squared over 2 factorial plus X to the 4th over 4 factorial and you might be getting excited now minus X to the sixth over 6 factorial and that's all I've done here but they would keep going so plus 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 eye over here actually let me just factor out so it's going to be plus plus 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're factoring out the eye so minus X to the third over 3 factorial over 3 factorial then the next imagine terms right over there we factor out the I plus X to the fifth over 5 factorial and then the next imaginary term is right there we factored out the I so it's minus X to the seventh over 7 factorial and then we obviously would keep going so plus minus keep going so on and so forth preferably to infinity so that we get as good of approximation as possible so we have a situation where e to the i-x is equal to 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 Maclaurin this was a mole or an approximation of cosine of X around zero or I should say the Taylor approximation around zero or we could also call it the Maclaurin approximation so this and this are the same thing so this is cosine of X especially when you add an infinite number of terms cosine of X this over here is sine of X the exact same thing so it looks like we're able to reconcile how you can add up cosine of X and sine of X to get something that's like e to the X this right here is sine of X and so if we take it for granted I'm not rigorously proving it to you that if you were taking an infinite number of terms here that this will essentially become cosine of X and if you took an infinite number of terms here this will become sine of X it leads to a fascinating fascinating formula we could say that e to the i-x e to the i-x is the same thing as cosine of X is cosine of X and you should be getting goose pimples right around now is equal to cosine of X plus I times sine of X and this is Euler's formula I always pronounce I'm right over here and this right here is Euler's or 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 from continuous compounding interest we have cosine and sine of X which are ratios of right triangles it comes out of the unit circle and somehow we've thrown in the square root of negative 1 there seems to be this cool relationship here but it becomes extra cool and we're going to assume we're operating in radians here's if we take if we assume Oilers formula what happens when X is equal to PI just to throw in another wacky number in there the ratio between the circumference of the diameter of a circle what happens when we throw in PI we get E to the I pi e to the I PI is equal to cosine of pi cosine of pi is what cosine of pi is cosine of pi pi is halfway around the unit circle so cosine of PI is negative 1 and then sine of pi is 0 so this term goes away so if you evaluated 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 could write like this or we could add 1 to both sides and we could write it like this and I'll write it in different colors for emphasis e to the I times pi plus 1 plus 1 is equal to I'll do that in a neutral color is equal to I'm just adding 1 to both sides of this thing right over here is equal to 0 and this this is thought-provoking I mean here we have just so you see I mean this tells you that there's some connectedness to the universe that we don't fully understand or at least I don't fully understand I is defined you know by engineers for simplicity so that they can find the roots of all sorts of polynomials as as as you could say the square root of negative 1 the square root of negative 1 pi is the ratio between the circumference of a circle and its diameter once again another interesting number but it seems like it come from a different place as I ecomes 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 to the X is also e to the 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 right over here you have 1 1 I don't have to explain why one is a cool number and I shouldn't have to explain why 0 is a cool number and so this right here connects all of these all of these fundamental numbers in some mystical way that shows that there's some connectedness to the universe so frankly frankly if if this does not blow your mind you really you have no emotion
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