 Hi, I'm Zor. Welcome to UNISOR education. I would like to use trigonometry to represent complex numbers. Basically, it's quite an interesting topic because it actually combines together many different topics in mathematics. For me personally, it was really one of the manifestations of the beauty of mathematics. It's not just a memorization of formulas, etc. On the website UNISOR.com I decided to put together this combined picture of three different areas, three different fields in mathematics. Trigonometry, complex numbers, and geometry. Let me write the formula, which would be the quintessential formula which represents this type of synergy between these three different fields in mathematics. This is called the Euler's formula. Euler is the mathematician. He is a Swiss mathematician who worked almost all his life in Russia in, I think, the 18th century. So that's his formula. You see, it combines together so many different things which it's kind of hard to believe that you can combine them in one formula. First of all, there is an exponential function. But there is an I in the exponent, which is a complex number. It's an imaginary number. So it goes to a number theory, a complex number theory. And then you have trigonometric functions here. And the canonical representation of the real part and the imaginary part of the complex number. So it looks like we can actually use the exponential function with complex exponent, which we never addressed before. And as far as geometry is concerned, all I can say is that if x is changing, then the geometrical representation of this is a circle. So this is a circle. And this is a circle. If x is changing. So all these pieces together will be combined in the proper order. And my lecture right now is just the first lecture, which is supposed to basically introduce you to representation of this type. And eventually we will derive the Euler formula as well. All right. So let me just repeat my usual introduction. It's better to watch this lecture on Unisor.com website. It's part of the trigonometry, the representation of the complex numbers in trigonometry. Just because this website has not only the lecture itself, the video reference to the video, but also notes, which basically can be considered as a text book. And I usually put lots of problems, maybe not for every topic, but in any case, it's basically an educational website, which presents you an advanced mathematics for high school students. All right. Now, so let me just start with this particular trigonometric representation of complex numbers. So complex numbers is something which I have already addressed before in the course of algebra. And I do suggest you to refresh it. I mean, if you don't remember what complex numbers are or what I actually use or something like this. So you better refer to these lectures and refresh your memory. So I will assume that whatever is necessary from the complex numbers, you know. Now, the complex numbers are in general can be represented as this, where A and B are real numbers. I is an imaginary unit which has the property of this. Its square is equal to minus one. That's basically how we introduced the complex numbers. We could not extract the square root of minus one or any other negative number. And this actually helps us because now we can say that square root of minus i is equal to minus one is equal to i. Actually minus i or plus i. All right. So this is a canonical traditional representation of the complex numbers. Now, next is the graphical representation. Now the graphical representation is very simple. If we have two axes, we can put one real number, the real part of the complex number as the abscissa and another as an ordinate. And this point is a graphical representation of this number. So we assume that the unit of measurement along the horizontal x axis is just one. And the unit of measurement along the y axis is i. So I have A units along the real axis and B units along the imaginary axis. And AB is the point which represents my complex number in Cartesian coordinates. Now, let's now think about the coordinates and the plane and geometry of this thing. Now you know that not only the Cartesian coordinates can be used on the plane, but also polar coordinates. Now what is polar in this particular case? Well, there's also two parameters. One being the magnitude and another being an angle. So this is called the magnitude or modulus or absolute value of the complex number z. And the angle phi is called argument or a phase. Or just a polar angle. So basically in this particular case, now using the trigonometry in this right triangle, we can say that A is equal to r multiplied by cosine of phi, right? r times cosine phi plus i r. And the B is obviously r, which is a hypotenuse multiplied by a sine of phi. So this is another representation of this complex number z. Or you can actually factor out r. I'm using two different r's. Cosine phi plus i sine phi. Alright, that's another representation. So we have right now algebraic traditional canonical representation. We have geometric representation in Cartesian coordinates. We have a geometrical representation in polar coordinates, which leads us to trigonometric representation of the complex numbers. So again, two parameters. Now what's good and what's bad with these two different representations? The canonical representation and the trigonometric representation. Alright, here is a very simple thing. You can very easily add two numbers in both systems. Namely, if you have z1 equal to a1 plus ib1 and z2 equals to a2 plus ib2, two numbers, sum is very easy to calculate in this representation. What about the product? Multiplication is a little bit more involved. Okay, let's multiply them. Well, it's a1 times a2 plus a1 times b2 and i, a2 b1 i and i2 b1 b2 equals. Now, i squared, as we know, is minus 1. So this is minus 1 times b1 b2. So the real part is a1 a2 minus b1 b2, the real part of this product. And the imaginary part is a1 b2 plus a2 b1. Well, I'm not saying it's a complex formula, but it's a little bit involved. Let's do it in polar coordinates. So this is z1 and z2 is equal to r2 cos phi2 phi2 plus i sin phi2. Now, let's multiply them. It seems to be a little bit more complex, but it's not the case. It's actually simpler. Alright, let's follow. Well, obviously, you have to multiply the magnitudes. And whatever will be in the parentheses is cos times cos phi cos phi2 plus i cos phi1, sorry, sin phi2 plus sin phi1 cos phi2 and plus i2 sin phi1 sin phi2 equals r1 r2. Again, i2 is minus 1. Now, cos phi1 cos phi2 minus sin phi1 sin phi2. What is it? That's what it is. i times cos phi1 sin phi2 sin phi1 cos phi2. What is it? We know this is... So, what's interesting about this? Whenever I'm multiplying two complex numbers in their polar form, well, magnitudes are multiplied. That's obvious. What's interesting about the angles, angles are added together. It's really quite a remarkable thing. What does it mean geometrically? Here's what I suggest you to do. Let's have a unit circle and let's have only complex numbers on this unit circle. Now, what are these complex numbers? Well, obviously, these are complex numbers with magnitude equals to 1. Now, angle can be different. This is phi1 and this is z1. And this is, let's say, phi2 and this is z2. So, r1 and r2 both are equal to 1 because they're all on the unit circle. Now, what are we saying right now? That multiplying one by another is actually a rotation. So, to multiply z1 with phi1 as a face, as an argument. To multiply it by z2, which has phi2 as an argument, it's actually a rotation because we have to have some of these two angles and that's what will be my z1 times z2. And the angle, the angle would be phi1 plus phi2. That's what's interesting. So, in the polar form, multiplication of the complex numbers actually is a rotation. Now, as far as the magnitudes, magnitudes are multiplied, right? So, it's a stretching, so to speak. So, first you have to rotate one complex number by the angle which is the face of another. It's an argument of another complex number. But as far as the magnitude of the new of the product, it's just a product of two magnitudes, which means it's just stretching or squeezing, whatever. But if both of them have the same magnitude equal to 1, then it's just rotation, right? Okay, now, just as an example, let me just have a couple of cases. First of all, if you have real numbers, real numbers are those numbers where b is equal to 0, right? Or, if you wish, in the polar form, that's where sin is equal to 0. So, what's the argument, what's the face? When sin is equal to 0, it means that the angle is either equal to 0. So, it's all the points here or angle equals to pi, which is all the points here. So, obviously, on the plane where my complex numbers are represented as points, the numbers on the x-axis represent the real numbers because the imaginary part is equal to 0. Now, what if I would like to multiply any complex number z by some real number? Well, let's just think about it. Real number has the magnitude equals to its absolute value, and the face angle is equal to either 0 or pi. So, what does it mean? Well, let's consider the positive direction. If you're multiplying a complex number by a positive number r, so what we're saying is that the magnitudes are multiplied, so it's basically stretched r times. And as far as the angles are concerned, I have to rotate by the angle of this guy, but this angle is equal to 0, so there is no rotation. So, basically, this point, let's say r is equal to 3. So, this point is 3 times further from the origin of coordinates, but it's on the same line. Now, what if r is negative? Let's say it's minus 3. Well, again, we are stretching 3 times because the magnitude is equal to 3, but then we are turning by 180 degree by pi, so it goes this way, actually, the point. So, it's symmetrically transferred around the point 0 and stretched, in this case, 3 times. So, that's what multiplication by real number is. You are still remaining on this line plus or minus, I mean, into one direction where it was before or the opposite direction, and you're stretching by the number which is your real number you're multiplying by. Okay. Well, in particular, if you are multiplying by 1, obviously, it stays the same because the magnitude does not change and the angle is not changed. All right. So, what if you are multiplying by i? That's another example. So, let's say you have z1, whatever angles are, and z2 is equal to i. Now, what i is? Well, i is cosine of pi over 2 plus i sin pi over 2, right? Because the cosine of pi over 2 is 0, sin of pi over 2 is 1, so i is represented in the polar form as this one, and it has a magnitude of 1, right? So, when I'm multiplying, magnitudes are multiplied, which means magnitude remains the same, and angles are added. So, basically, if I want to, this is my z, I want to basically rotate it by pi over 2. So, it will be this one. This is z2 z1 times z2, when z2 is equal to i. So, multiplication by i, geometrically, means just turning to a perpendicular direction by 90 degrees. So, basically, what have we combined together? We combined complex numbers, trigonometry and geometry, and represented our complex numbers in this polar form. This is just an introduction to whatever this Euler's formula, which I have written in the beginning, to wet your appetite, basically. So, this is the first step. So, now we know how to represent complex numbers. Okay. So, please refer to Unisor.com, go to the same lecture again, and read the notes. They are like your textbook, basically, so you can learn again or refresh your memory. And one more thing, if you did not refresh your knowledge about complex numbers before this lecture, try to do it again right now, and then read the notes for this lecture at Unisor.com. And obviously, I do suggest you to enroll into the website, because if you enroll, then you can actually take exams and have some self-study programs, etc. Very beneficial for you. Thanks very much, and good luck.