 I'm Zor. Welcome to Unizor education. Today's topic will be a continuation of complex numbers. We were talking about a certain formal way to introduce complex numbers and right now I think that basically has covered more or less the more rigorous definition of the complex numbers. I would like to concentrate on their graphical representation now. We all know that real numbers can be graphically represented on a straight line with one particular point on that line marked as equivalent to number zero. Then there is a positive direction on the line and with a certain segment of a unit length we can just put all these marks one, two, three, minus one, minus two, etc. to represent integer numbers. Then we can divide each into certain fractions to represent fractions and rational numbers and fill it up in between with irrational numbers to complete the whole line of representation of real numbers. Now let's talk about complex numbers and how they can be represented. Well, complex number, generally speaking, is basically a combination of two real numbers and the imaginary number I squared, which is minus one. That's the general representation and right now I will not use the formal representation with curly brackets around these operations because these are actually true operations within the set of complex numbers where A as a complex number by itself is actually formally represented as, actually this is also curly, as a combination of A plus zero as far as the imaginary part is. So B is equivalently represented in the complex numbers. So I will be talking only about real operations of addition and multiplication and when I'm saying that the imaginary number is multiplied by real, that's not exactly true. Imaginary number in this case is multiplied by a complex number which is mapped into real number B. But these are parts of this more formal definition. So right now let's just abstract from it real addition between real numbers and imaginary number I so how can we represent this particular complex number graphically? Well, in theory I could choose another representation of this complex number. I can choose something like A comma B thinking that A represents the real number which is on the left and B is the real number which is supposed to be multiplied by imaginary part. This is another way to formally introduce complex numbers and I can add these two numbers together. I can multiply them, etc. So I can basically repeat everything except I'm not using these operations which I think I might actually find is convenient. But in this case this is another representation which can be used as well. But at the same time this reminds us that it's just a pair of real numbers and the pair of real numbers can be represented as a point on the plane with the Cartesian coordinates. So I will draw another line here and I can always represent this pair of two real numbers or if you wish this complex number as a point on the plane where its x-coordinate of this is A and coordinate is B, the y-coordinate. Okay, so since A and B can be any real numbers then all complex numbers basically fill out an entire plane because the x-coordinate and y-coordinate can be any real number. So basically all the points on the plane are filled out with representations of complex numbers. Alright, fine. And now certain very elementary properties of this type of representation. Now obviously we can always think about this particular segment which connects our real complex number with zero. Now obviously it has a length, it's a segment and it's a rectangular triangle with a straight angle. So basically the Pythagorean theorem can help us. Now this is A and this is B. So basically this length, let's call it C, this C is obviously the square root of A squared plus B squared. That's one thing which we can say about our complex number. So its representation is on a distance of this distance from zero. But at the same time let me make another couple of observations in this particular case. What if I would like to know the distance between two different points on the plane that represent complex numbers? Well obviously we can use something like this, Pythagorean theorem. And what's interesting about this is that for any complex number like this we can always talk about something which is called its modular. Now the modular in this case by definition is exactly this square root of A squared plus B squared. Now what's interesting is that this modular which can be defined purely algebraically is actually the same as in the geometrical, in the graphical representation of complex numbers is really a distance from zero to our number. Now a very similar thing can be observed if I would take a distance between two different points. For instance, what's the difference between this and this? In geometrical this is B, right? Approach this particular geometrical representation from the algebra of complex numbers viewpoint. If this representation of A plus B I, what is this particular point on the plane a representation of? Obviously it's x-coordinate is A and it's y-coordinate is zero, right? So we have this point which is represented here and we have this point which is represented here. We know that the distance between these two points, geometrical distance is actually B. But let's see what happens if I will use algebra of complex numbers. First I will just subtract from this number, I will subtract this number. What happens? From A plus B I, I subtract A plus zero I. What happens? Well, as you remember real parts are subtracted and A minus A gives zero and imaginary parts are also subtracted separately B minus this will be B I. Or if you wish to be more precise, zero plus B I. What's the module of this? Well, it's obviously the same thing, square root of zero square plus B square which is B. If B is a positive number, let's say B is a positive and A is a positive. I call it on this particular picture. So as you see, modular represents the same number in this case, modular the difference between these two numbers represents the same as geometrical distance between these two points. Generally speaking, this is a situation which is exactly the same for any two points. So modular in algebraic representation of complex numbers plays exactly the same role as the distance in their geometrical and their graphical representation. So if you will take another point, let's say here and call it C plus D I. Now what happens? Well, if you would like to know this distance, what you would probably do is you will build this triangle with the right angle, this. Now this is C and this is D, right? Since this is C plus D I, the projection onto the x-axis is C and on the y-axis is V. So what is the length of this? Well, okay, we will use the Pythagorean theorem and this segment is the difference between B and G. This segment is the difference between C and A. So we have C minus A squared plus G minus B squared square root. But again, this distance which is equal to this number can be obtained by just calculating the difference between these two complex numbers. So if you will do A plus B I minus C plus D I, now what's the difference will be? Well, obviously it's A minus C as a real part, B minus D as an imaginary part. So that's the difference and what's the module of this difference? Well, again, it's square of this plus square of this which is this. So the point is that geometry of the complex representation of complex numbers is very much resembles just algebraic operations on the complex numbers with a very important quality that the distance between two graphical representations of complex numbers is basically a module of their algebraic difference. Okay, that's one kind of side issue which I wanted to make with complex numbers. They can be graphically represented. What's interesting is that you can always represent a point on the plane using some other system of coordinates, not necessarily Cartesian coordinates, but you can also use polar coordinates and every point can be represented by the distance from the zero, from the origin of coordinates and this angle. So two parameters are which is a distance and alpha which is an angle always define our point on the plane. So I can always say that my complex numbers can be represented on the plane using this Cartesian coordinate system where A and B represent their correspondingly real and dimensionary part. At the same time I can say it can be represented by a pair of distance to the zero and the angle from x axis which I have to lift that particular segment to get to my point. Now if represented in this way for those who know some basic trigonometry it's obviously that A coordinate of this thing is equal to r times cosine of alpha. If alpha is this angle, right? So this is A, this is r and this is B and B is correspondingly r times sine of alpha. Now if you compare this with our algebraic representation of the number A plus B i but we can always say that this representation can be obtained following quickly. So what is the difference between these two, what is the length between these two points? So r, r is actually a module, right? So we can always have it as module A square plus B square multiplied by A divided by this square root plus B divided by this square root i. That's the same thing, right? I just multiply it by the module and divide it by module. Now this is r. So if A is equal to r times cosine of alpha then this, A divided by this square root of A square plus B square is the cosine of alpha and correspondingly D divided by the same thing, this is the sine of alpha. So that's how many different aspects, geometry, trigonometry and algebra are brought together in the same kind of very harmonious way because one complements another and no matter from which side you're approaching this problem you actually get more or less the same thing. So it proves that you are on the right way basically. If you can prove the theorem for instance in three different ways you are absolutely sure that you have proved it correctly. So this is three different ways to approach the representation of complex numbers. Trigonometrical way, Cartesian coordinate way and algebraic way and all lead to the same thing. Well that was just a small extra for graphical representation of complex numbers. There will be a couple of problems as well which I would definitely recommend to solve. Thank you, that's it for today.