 Hi, I'm Zor. Welcome to Unisor Education. This lecture is part of the advanced mathematics course for teenagers presented on Unisor.com and that's where I suggest you to listen to this lecture from. It has notes together with the lecture, so it's very useful for you. It will be like a textbook, basically. Now, today's lecture is dedicated to something which is quite a remarkable thing in mathematics. It's called Euler's Formula. Euler is a 17th, 18th century mathematician. He is Swiss mathematician, but he spent most of his life in Russia. Basically, he is one of the founders of Russian School of Mathematics. Actually, even right now, many textbooks which students in school are using, it still originated back in the Euler's time. That was a very, very long time ago. All right, so the formula I'm talking about is quite remarkable. It's actually beautiful, I would say, from my perspective. That's how it looks like. e to the power of ix is cosine x plus i sine x. Now, it's really remarkable in the way that it connects together something which is algebra and algebra of complex numbers and trigonometry, and also to explain how this whole thing actually is related. I will use geometry. So it's a synergy of many different mathematical subjects together into one particular formula. Frankly, I think that this formula is probably as beautiful, if you wish to use the statics term, as the famous Einstein's formula from theory of relativity, where energy is related to mass and the speed of light. I mean, it's really recognizable by many people, although it is really very, very not very obvious formula at all. So this particular relationship is something which was like some kind of very, very bright result, result of the very, very bright mind. So this formula in mathematics is also result of a very, very bright minds who basically came up with this formula. Now, Euler wasn't really the first one to mention something like this, but he basically put some very solid foundation. So let me talk about this formula. What's really unknown in this? Well, first of all, this. The exponent is complex. We have never actually seen the complex exponent. And we didn't define it yet. So how can we say that this is equal to that? If we know what cosine and sine is, but we have absolutely no idea what the left side of this equality is. We don't know how to raise the number into a complex exponent. So that needs to be defined. But to define it properly, we have to define it in a way so this definition actually complies with all the properties which we already know about exponential functions. So let me just start with something which is basically the properties of exponential function. And these are properties which we would like to preserve in our new definition. So what's the basic properties of exponential function? If you have something like this, where a is the base and x is exponent, well, you know the few properties. h to the power of 0 is always equal to 1, as we know. Also, if you have sum of two numbers x plus y, I'm talking about real numbers of course, it's a real function. So that's the product of two different exponents. And finally, if you have something like this, the product of exponent, that's actually a sequential usage of the power. Or if you wish since the product is commutative, you can put it a to the y and then to the x. So these are properties which no matter how I define my complex exponential function should actually be held. And this is basically a subject of something which is, you know, completely separate thing. So right now I will be talking about some reasonable approach to definition of this. Basically, I would like to add one more thing before I start. The real proof, if you wish, of this formula or a little bit more rigorous explanation of why the formula is such and such is basically related to calculus. However, I will try to bring some reason, some real foundation, why this formula actually takes place without using the methodology of calculus. Just using whatever we already know from trigonometry and the functions. So what do we know about this? First of all, what is E? E is a number which we have already defined before as being a limit of 1 plus 1 over n to the power of n when n goes to infinity. This is basically a definition of E and we have proven that this is a reasonable definition. If you remember, we have proved that this particular expression is between 2 and 3. It's bounded on both sides and it's increasing as n increases to infinity. So it must have a limit and that's why we have basically assigned this limit to the letter E. Also, we were talking about the function A to the power of x which has at point 0 certain steepness and the steepness in case A is equal to E, steepness is exactly equals to 1. So it's at 45 degrees, this particular tangential line. So these are properties of the E and that's the definition actually, n properties of E. Now, here is a very important factor. If we start E with this definition, I can actually continue this if you remember that E to the power of x is equal to limit of 1 plus x over n to the power of n. We have proved it in exactly the same lecture. It's a very simple consequence and the consequence actually being that you can put x here and x there and since our exponents are multiplied, that will be the same as n. Now, and this part goes to E as n goes to infinity. That's why the whole expression goes to E to the power of x. So we know that. Now, now I'm going to define my E to the power of ix. Actually, let me start into the power of i using this particular expression. Now, this particular expression was actually proved only for real number x. Now, if I would like to expand my definition of the exponent to complex numbers, what I'm going to do is I'll just use this particular property as a definition. So I'm basically defining this as, so now instead of real x, I'm using a complex x, the i, but what's the difference? Now, this, we don't know what it is. It's basically a undefined thing, right? Now, this, on the other hand, is something much more familiar because this is division of the complex number. This is addition of the complex number and this is raising of the complex number into an integer power. So we know how to calculate this and that's why we can actually use this as a definition of this. Now, obviously, I'm not going to really prove that the limit exists, for instance, although it can be done in some way. This is beyond the scope of this lecture. My point is that it's reasonable to assume that the definition of e to the power of i lies in exactly the same domain as basically e to the power of x for real x because now this can be calculated. That's what it is. All right, fine. Now, once this is defined, I can continue to interpret this thing graphically. Now, what's the graphical representation of complex numbers? Now, you remember that any complex number can be represented as a point on a coordinate plane with coordinates a as abscissa and b as an ordinate, a being the real part and b being an imaginary part. And this is our point a, which basically represents this number. So this is a graphical representation, right? Now, if I'm talking about this number, 1 plus i over n. Now, where is exactly this number located? Well, the a is equal to 1, b is equal to 1 over n, right? So if this is 1, this is a very small above the 1. This is 1, 1 over n. This is the point. And as n goes to infinity, this point goes closer and closer and closer to the x-axis, right? So what actually happens is that as we are moving to this particular limit, my 1 remains as 1 and my 1 over n gets closer and closer to 0. It's a very, very small number. Now, let's recall a very important property of trigonometric functions, which we have already learned. Remember this sign of x divided by x goes to 1 as x goes to 0. Now, in one of the previous lectures where I talked about trigonometry and geometry, basically we have proven this particular limit. So what I want to say that 1n goes to 0, right? As n goes to infinity. That means that sign of 1 over n and 1 over n is approximately equal to 1. So they are almost equal to each other. As n goes to infinity, sign of 1 over n is very, very close to 1 over n. Now, another thing. As n goes to infinity, 1 over n goes to 0, right? And cosine of 1 over n goes to cosine of 0, which is equal to 1. So, what I would like to say is that sign of 1 over n is approximately equal to 1 over n with large n and cosine of 1 over n is approximately equal to 1. And this approximation is increasing, is becoming better and better as n goes to infinity. And therefore, what am I doing now? I'm doing this. I'm substituting this particular number. One, I will substitute with a cosine of 1 over n. And I over n, I will substitute it with I sine over n, right? Because 1 over n is almost the same as sine of 1 over n. And 1 is almost the same. Now, why did it do it? Okay? Very simply. Again, let's go back to this. Limit of 1 plus x, well, actually I in this case, in Y over n to the power of n, right? We have substituted this piece with this. Why did we do it? Because I know how to raise this into the power of n. We have to raise it to the power of n, right? I know how to do it, because if you remember, the trigonometric representation of the complex numbers allow us to do this in the following fashion. I'm multiplying the argument, the angle, the phase, whatever you call it, by this particular exponent. And what do I have? I have cosine of n over n plus sine, I sine of n over n equals to cosine of 1 plus I sine of 1. And that is actually a definition of e to the power, e to the power of i. That's what we wanted to define, right? As a limit of 1 plus i over n to the power of n. So we have basically concluded, and again, this is not a rigorous proof. This is just an explanation. It's some reasonable consideration of the fact that this is equal to, let me write it again, that this is equal to this. So now what I can say is, let me define this particular expression, e to the power of i. Let me define it as a cosine of 1 plus i sine of 1. Now, it's a different story about how to prove the reasonability of this definition, that it actually takes all the properties of the exponential functions and remains, all these properties remain true for this particular definition. But we will do this in the proper time. Right now I'm just concentrating on this. And now let me continue if I would like to do this, what is this? This is the property which I presume is retained, right? And that's why it's this, to the power of x. And again, we know how to raise the complex number in its trigonometric form, how to raise it into a power. You just multiply the argument. And this is the famous Euler's formula. Now, again and again, I said many times that this is not a proof of the formula. These are some reasons which lead us to this definition, basically, of e to the power of i. And we are using the properties of the exponential function to derive further formulas. So, we will leave this as a definition, basically. Because from this definition, we can actually define anything for complex exponential functions. For instance, if I would like, for instance, to know what is e to the power of a plus bi, which is just any complex numbers. Well, I know that if I'm adding two different numbers in the exponent, it's the multiplication. So, it's e to the power of a times e to the power of bi. And e to the power of bi, we already know what it is. It's cosine of b plus i sine b. Now, what if I instead of e, I have some other number? What happens in this case? Well, let's talk about d. d is any real number. And I would like to raise it into power a plus bi. I will start with the same thing. I will multiplication d to the power of a times d to the power of bi equals. Now, instead of d, I can always substitute as e to the power of natural logarithm of d. Because that's the definition of the natural logarithm. The one which has a base e to the power of bi equals to d to the power of a times e to the power of b logarithm di. And we know what this is. It's d to the power of a times cosine of b natural logarithm d plus i sine of b times natural logarithm of d. So, that's basically how we can raise any particular number, real number d, into any complex power. Well, that actually ends my lecture, which defines the concept of complex exponential functions and the Euler's formula, which is, again, very famous formula in analysis. And what I wanted to do is just to introduce you to this formula in its original formulation without resorting to the methodology of calculus, which is a little bit more involved, etc. And we will probably do it in due time. But right now, I just didn't want to resort to these mechanics. I just wanted to use some reasonable approach, some explanation of this particular concept using whatever we already know by now. Well, that's it. Thank you very much and good luck.