 Hi, I'm Zor. Welcome to Unizor Education. This lecture is a continuation of the topic related to trigonometric representation of complex numbers. In this particular lecture, we'll talk about exponentiation of complex numbers in trigonometric form. And obviously, this is part of the whole course of advanced mathematics presented on Unizor.com, where I actually suggest you to watch this lecture from, because the site contains, for every lecture, there are notes which basically serve as a textbook. Well, besides, obviously, there are exercises and exams, etc. Alright, so we continue talking about trigonometric representation of complex numbers, and we'll talk about raising into power. Now, last lecture was very much introductory. I was just talking about how the complex numbers are represented in trigonometric form, or polar form, if you wish, which looks like this, where r is an absolute value, or modules, and phi is the argument or a phase of the complex number. Now, raising into power, obviously, it's based on multiplication. And multiplication, as we know from the previous lecture, is very easy to accomplish in polar coordinates in trigonometric form, because if you have two different numbers, r1 and r2, cosine phi2 plus i sine phi2, then their multiplication is very easily expressed in this format. You multiply the modules, and you add the arguments. Now, analogous formula of multiplication in canonical representation looks much more complex. It's AC minus BD as a real part, and complex part is BC plus AD. It's a little bit more cumbersome, and you can't really remember it quite well, but this is a very natural, and it has a geometric representation. Remember, if r1 and r2 are equal to 1, then basically we are just turning the point on the unit circle, which is represented by this particular representation, where this is the angle of phi1. We are turning by angle phi2, getting phi1 plus phi2. So, multiplication on the unit circle with r1 and r2 is equal to 1 is basically a turning rotation by this particular point. Now, this is multiplication, and as you see, it's easy, and that's why it's very easy to define the power. Now, we will start with, obviously, natural power, integer positive power n, and what does it actually mean? Well, I would like to make a very interesting comment here. What we are talking about, we are talking about certain things which meet the definition. Like, for instance, if we are talking about multiplication of the real numbers, we start with multiplication of integer numbers, actual natural numbers, positive integer, then we add the negative based on certain properties of the multiplication which we would like to preserve, and then we expand the irrational numbers, then the irrational numbers, and same thing with any other operations. First, it's defined for a simple case, and then we examine the properties in that simple case, and we are trying to define for more complex objects in the same manner to preserve these properties. Now, what are the properties of the exponentiation, of raising into power which we would like to preserve? Well, obviously, we know that something like this. We did prove this for rational numbers, actually for all real numbers, but let's talk about rational numbers, where m and n are any rational numbers, right? And a was a positive number. Now we are talking about complex numbers and raising the complex numbers into certain powers, and that's not easy, obviously. If you remember, within the real numbers, we did not really even allow the negative numbers as a base, primarily because we cannot raise minus one into power one-half, which is actually a square root. In complex numbers, we can allow even that, because this is equal to i in the complex numbers, right? Because i-square is equal to minus one, by definition. Again, we are defining, we are expanding our universe, but we are trying to preserve the good properties which we already observed in the smaller universe which we had before. So, now we are expanding to complex numbers, and we would like to preserve properties like this, and properties like this, and properties like this. So, these are very good properties of the exponents as they are defined in a simple case with real numbers, or even with integer numbers, if you wish, even with positive integer numbers, because that's where it comes from. It comes from the positive integer numbers where all these are obvious, right? Because positive integer number raised into positive integer number is basically a multiplication by itself, m times and m times, which is actually m plus n times, right? So, for a simple case, it's obvious, and then we expand the universe to preserve these properties. I'm going to preserve these properties for the complex numbers in trigonometric form, and I will define these operations correspondingly. And again, let me start from something simple. And simple is where we are talking about raising the complex number into positive integer power, which is basically multiplication by itself and times. And we know how the multiplication looks like in trigonometric form in polar coordinates, right? So, if I have r times cosine phi plus i sine phi, if I want to multiply by the same number, I will have to multiply my absolute values, right? And I have to add the angles, so phi and another phi, so it will be 2 phi, right? So, that's the formula of raising to the power of 2. Now, how about power of n? Well, obviously, it's a trivial exercise in induction. You can define, you can actually prove this by induction. So, this is, for n is equal to 1, that's basically an identity, right? Now, for n is equal to k, if we assume it, then n is equal to k plus 1 would be the same thing for n multiplied by this single expression, right? And that means that we will have to multiply the absolute value of one times another, which is r to the n times one more r, or r to the k plus 1. And this will be k and again plus 1, so it will be k plus 1. So, the formula is trivial proven by induction. Okay, so that's fine. This is trivial. Now, let's expand it to different values of exponent. First, let's talk about negative exponent, right? So, let's say we want to do this. We want to define it in some way so it's reasonable. And all the laws which I was talking about before are supposed to be preserved. Now, how can we do it? Okay, let's just think about exactly the same way as we were talking about the similar problem in real numbers. What I did was, I used this property. Now, what if my n is equal to minus m? I will have a to the power of m plus minus m is equal to a to the m times a to the minus m, right? According to this formula, if I would like to preserve this property, this must be true. If I want to expand to negative exponent. Now, what is this? This is a to the power of 0, which is, again, there is a property of being equal to 1, which means that this a to the minus m is 1 over a to the power of m. And this might be a definition. And if this is a definition of negative exponent, then everything is fine. All my rules, laws, whatever properties are preserved. Now, I will do exactly the same thing here. So, I define this as 1 over r times cosine phi plus i sine phi. Okay, right? So, that's easy. Now, let's think about how I do this. Well, 1 over, yes, to the power of m. Now, 1 to the power of, 1 to the r to the power of n is retained. But now, I have to express this somehow in a power of minus m. Sorry, the power of n. Now, let me go to the properties of multiplication of the complex numbers in trigonometric form. What do I know about this? Well, I already know that this is equal to cosine n phi plus i sine n times phi, right? So, here I can replace it with this, n phi n phi. Okay, fine. So, let's talk about this. For instance, I have this expression. What if I will multiply it by this expression? What happens? Well, we know that if I'm multiplying two different complex numbers in trigonometric form, my absolute value is multiplied, which is 1 and 1. But my angles, my phases, my arguments are added together. So, what will be, will be cosine of sum of this, which is 0, plus i sine 0, which is equal to, now this is 1, cosine of 0 is 1, sine 0 is 0, so this is equal to 1. So, what follows is that 1 over this is this. So, I can replace this with this. And to be even more explicit, I will use this. 1 over r to the n is r to the power minus n. That's the real number's power, right? So, what do we have right now? We have this same exact situation. I have this exponent, which is negative. And the formula basically looks exactly as if it was the positive. So, I raise my absolute value to this power, and I multiply my phase, my argument, by this power as a multiplier. So, the formula is exactly the same as just a second ago I was talking about for positive integers. So, that's quite interesting actually. The formula is exactly the same for positive and for negative. You raise the argument, sorry, the absolute value to the power, and you multiply the argument by the power. Okay, fine. How about rational? Well, let's start rational slowly. Let's do only rational as 1 over n. So, my purpose is r cosine phi plus sine phi to the power 1 over n. How can that be defined properly? Okay, well, let's just think about, let's take this and raise it to the power of n. What happens? We don't know how to raise to the power of n, right? We just used it, regardless, by the way, of the fact whether n is positive or negative. We have to raise this absolute value, the modulus, to this power. So, r to the power of 1 over n to the power of n would be what? We have to multiply, right? This is r. So, the result would be, the absolute value would be r, and the face, the argument, I should multiply my angle, my face, by the power, right? So, it would be cosine phi over n times n plus i sine phi over n times n, right? Now, what can we say about this? Well, if this to the power of n is equal to this, now, by definition, what is 1 over n? It's a number which is being raised to the power of n would give me this one, right? If I will raise this to the power of n, my powers are supposed to be multiplied, right? It will be 1 over n times n, which is 1, which is just exactly this piece. So, what I can say from here is that this is equal to this piece. Since, as I just said, if I multiply, if I will raise this to the power of n, I will get this, which means this raised to the power of 1 over n will give me this. Now, again, let me repeat, these are definitions. These are not theorems. They are definitions of new operations, how to raise the complex number into positive integer, to negative integer, to irrational power. Now, irrational would also be nice, obviously, to complete this particular thing. Well, that's a little bit more involved. And as I was explained in the irrational power of real numbers, real positive numbers, actually, you have to really use the limit theory. And every irrational number can be considered as a limit of certain sequence of rational numbers. And basically, the approach should be exactly the same thing. So, you can define the irrational power as a limit of rational powers where the rational powers tend to the irrational number if there is one. All right? So, these are formulas for raising this particular complex number in trigonometric form in any power. We had integer positive, negative, we had this one. Now, I would like to, again, point that the form, exactly, the form of this formula is exactly the same. You raise the absolute value to this power and you multiply the argument by this power as a multiplier. So, it's exactly the same formula. Now, the last one which I would like to add is to put m over n here, right? Now, what happens in this case? Well, these are just two different rules. You have to apply them sequentially because what does it mean if you do m, m over n? You can consider it as a to the power of m and then to the power of 1 over n, right? Or vice versa, first to the power of 1 over n and then the result would be the power of m. So, you're applying two different formulas. One formula gives you, let's say, we are considering this as 1 over n and to the power of m. I have too many parentheses and brackets, etc. So, I have this to the power of 1n and then to the power of m. Now, what is this? Okay, first we raise it to the power of 1 over n to get this one. Now, this I have to raise to the power of m and I get what? My absolute value is supposed to be raised to the power of m, which is this. And my argument is supposed to be multiplied by plus i sine. So, again, you see exactly the same formula. You raise your absolute value to the power, whatever the power of the whole complex number is, you use this power as a multiplier for the face, for the argument, right? So, in general, I can basically derive this formula for any kind of power, any kind of real power. Integer, negative, rational, irrational, r times cosine phi plus i sine phi to the power of x, let's say. Where x is any real number is r to the power of x times cosine x phi plus i sine x phi. That's better to use brackets here. So, this is a general formula for all real x. Now, and that's very important right now, you see, in trigonometric form, that looks quite easy. Try to do it in regular canonical representation like a plus bi. Even this, something like to the fifth degree, even this would look extremely complex because you have to multiply a plus bi five times by itself. It's a lot. It's a to the fifth degree and etc. I mean it's a big formula, which obviously nobody remembers. But in the trigonometric form, in the polar form, that's raising into any kind of a power, the operation of exponentiation, raising into power, looks really simple, right? Well, and that was my point actually for today's lecture. What I suggest you to do is go to this website and read the notes for this lecture. It's like reading the chapter in the textbook, just to make sure that you properly understand it. So, what do we know right now? We know that we can use any kind of an exponentiation to a real power for complex numbers. So, if something like minus one to the power of one half was impossible to do in the realm of real numbers, in the complex numbers, it's very easy to do. What is it? Let me just, you know, make an example. Minus one is here. So, it's one times cosine of the angle of pi plus i sine pi, right? So, forget about one. We don't need it. Now, I'm raising it to the power one half, which means what? Absolute value is one, so it doesn't matter. I will have cosine pi over two plus i sine pi over two. Now, this is zero, right? And this is one. So, I have i. You see, i to the power of two gives me minus one. Square root of minus one is i. So, in complex numbers, it's easy. Okay, that's it. Thank you very much and good luck.