Last class, we started with complex numbers and I gave you a brief introduction about complex numbers. In fact, we also did representation of complex numbers. Isn't it point form, polar form, Euler's form. So today we are going to start with operations on complex numbers. So I think of whole class today will go in operations on complex numbers because there are many of them to take care of. So let's start with operations on complex numbers, operations on complex numbers. The first operation that I would like to discuss here on complex numbers is comparison of complex numbers. Comparison of complex numbers. Sir, you should never compare anything, sir. Okay, so comparison when you talk about comparison, we talk about two types of comparison one, are they equal or are they not equal. So equality and inequality are the two types of comparison, which normally we talk about. So if you talk about equality, so when you say a one complex number is equal to another complex number, let's say Z1 and Z2 are equal complex numbers, then it can only happen. Then it can only happen when their real parts are equal and their imaginary parts are equal. So this can only take place when A1 is equal to A2 and B1 is equal to B2. Okay, that is to say, the real part of Z1 should be equal to real part of Z2 and imaginary part of Z1 should be equal to imaginary part of Z2. So usually under these two conditions, or only when these two conditions are simultaneously met, you can say one complex number is equal to the other. However, a very, very good thing here is that there is nothing like inequality of complex numbers. So this operation is invalid. This operation is invalid. Please do not write such kind of operations. In fact, you cannot say greater than equal to also if you say greater than equal to equality makes sense inequality doesn't make sense. It is like you can't say one vector is more than the other vector or one vector is less than the other vector. Of course you can say their magnitude, one vector's magnitude is more than the other or lesser than the other. But as a vector itself, you can't compare two vectors. Similarly, you can't compare two coordinates, right? So vector coordinates, they basically are all the same thing because coordinate are treated as position vectors. So you can't say one coordinate is more than the other or one coordinate is lesser than the other. Yes, you can definitely say the distance of one coordinate from the origin is more than the distance of the other coordinate from the origin. That you can say, but as I can say a complex numbers or let's say as points or as vectors themselves, there is no inequality comparison between them. Okay, is it fine. So this is about the comparison of complex numbers. Those who joined in so we just started with the operations on complex numbers and this is the very first operation which I'm discussing it. This kind of operation is very useful in solving complex number equations which we are going to take up towards the last part of our chapter. Okay, the next class will be ending this chapter and we'll be starting with limits and derivatives. So in that session we'll be talking about how does this particular operation help us to solve complex number equations. Okay, so can I now move on to the next operation which is addition of complex numbers. Any questions any concerns here do let me know. So no down no inequality between complex numbers. Wish our country was also the same no inequality between any two people. Isn't it. So no inequality in complex numbers. See complex numbers teaches us to be equal equal or don't be equal but no inequality. Okay, next is, next is addition of complex numbers addition of complex numbers. Okay guys, and girls, please be attentive in this part looks to be very simple from school point of view, but not that simple I mean it has got a lot of insights which you are going to talk about. So let us say we have two complex numbers a one plus IB one and a two plus IB two. Okay, when you add to complex numbers. So addition always happens between similar things, isn't it, you can add mangoes to mangoes you can add gawas to gawas, but you can't add mango to go right. Anyway, when you're adding you add real to real, and you add imaginary to a binary. So basically, addition here will lead to a one plus a two plus IB one plus B two. Okay. Now what is important is not this mathematical operation, but what is important is what is the geometrical meaning of this. So, if I ask you, where is z one plus z two located on the argon plane, if you know the location of z one and that too, what will your answer be. Let us say I draw z one and that too. Let's say this is my argon plane, real z axis, imaginary z axis, and let's say z one is located over here. Okay, so let's say one is located over here, let me call it as a point a. Okay, and z two is located over here. Let's say I call it as a point B. Okay. Where do you think will be z one plus z two located. That is more important for us, because my many of the questions asked in the competitive exam will be revolving around your geometrical understanding of these concepts. Okay. So obviously, the answer is wherever a one plus a two comma B one plus B two is located, right, but geometrically how do you figure it out. I mean, of course, plotting you will work, but is there any relation by which you can connect over z one z two and that one was it to exactly make a list the same way as adding two vectors, right. Now see, most of you have rightly said that you have to locate a one plus a two comma B one plus B two but where will it be located. So for that, I'll do a small construction here and we will I will justify that construction also. So let me join the origin to these positions and B. Okay. And now I'll complete a parallelogram. I'll complete a parallelogram here. Okay. And let's say I get this point C. Okay, I claim that. And I can justify it also that this point C is actually representing your z one plus a two right now how. We all know that in a parallelogram, the diagonals bisect each other. Correct. So if I ask you what is the midpoint of AB, what is the midpoint of AB. Let's say I call this M. So this is your A one comma B one this is your A two comma B two. So midpoint of AB you'll say sir A one plus a two by two comma B one plus B two by two. That would be nothing but A one plus a two by two comma B one plus B two by two. Correct. Yes. Now let us say let us say hypothetically speaking, the coordinates of C was C comedy. Can I say the midpoint of the midpoint of OC should also be the same point M. Yes or no. It should also be the midpoint of OC because the construction is that of a parallelogram. Correct. So midpoint of OC, midpoint of OC will be zero plus C by two comma zero plus D by two. Okay. And since both of them represent the same coordinates, it's very clear that A one plus a two by two is equal to C by two and B one plus B two by two is equal to D by two. In short, your C comes out to be A one plus a two and your D comes out to be B one plus B two. Thereby, my justification that this point will be your Z one plus a two point is absolutely right. Isn't it. Okay. Is this understood. Okay. So somebody asks you that this is it one this is it to where is it one positive. What will you do simple. So set the parallelogram with OZ one and OZ two as the adjacent sites, just like how you basically start and learn adding of two vectors using parallelogram law of addition. And the opposite, a vertex over here opposite to that O will be your Z one plus Z. Okay. So this is with respect to the positioning on the argon page. Now a lot of things come out from this diagram itself. Let us try to figure it out. The first thing that comes in the diagram here is look at the triangle. OAC. So I will write it down in triangle OAC. Okay. So all of you focus on this triangle OAC. All right. Now, as of how you see a triangle being formed here OAC, right. So in a triangle, can we say, can we say the following things. The third side is always lesser than the sum of the other two sites. Isn't it? So can I say OAC will always be lesser than OA plus AC. That means OAC will always be lesser than OA plus O. What is, what is AC? OAC is OB actually, right? Because this side length and this side length will be equal. Correct. In short, what you're trying to say is that OAC is nothing but mod Z one plus Z two. Remember, what did I say about modulus when I started complex number after the last class? Modulus of a complex number represents the distance of that complex number point on the argon plane from origin. Please remember this. Okay. This itself will be, you know, the core principle behind solving many questions. Okay. Right. So don't, don't just know that, okay, complex number. We add the real part and we add the binary part. That is not the only thing that you're supposed to know. Maybe for school, it will work fine. But as a competitive level exam aspirants, you are expected to know a little bit more also. Okay. So here I'm claiming that Z one plus Z two modulus will be less than mod Z one plus mod Z two. But with a small modification here, I say it is less than equal to. Right. Now, why isn't it equal to? Because see, it is not necessary that OAC will definitely form a triangle. It may be like flat also. Isn't it? So what happens if you are Z one and Z two are in the same line with O, then there is no triangle getting formed. Right. So as you can say this parallelogram will become flat. Right. So equality can also hold true. Right. So please note down this is one of the triangle inequalities. I'll call it as the number one. So this is a triangle inequality which comes directly from your geometrical understanding of Z one plus Z two operation. Okay. Now, I mean, this is a question which I already answered, but still I'm asking you officially. When do you think the equality will hold true? Anybody? Anybody? When do you think mod Z one plus Z two mod will be equal to mod Z one plus Z two? When? When dash filling the blanks. First of all, there's no Y axis shall believe. Welcome to complex numbers. No. I want more generic answer. They lie on the same line. Any two complex number will also lie in the same line. They just see much, much more refined than what you said. Okay. Okay. Nikhil, sorry. I missed out your answer. A1 by B1 is equal to A2 by B2. What is the complex number way of saying the same thing? You're on the right track. Just say it in a complex number language. Their arguments are same. Brilliant. Awesome, Nikhil. Okay. So when their arguments are same, then the equality will exist. Please note down this itself has come as a question in competitive exam. Note this down. Note this down. Very, very important. When two complex numbers are such that Z1, Z2 and origin are in the same line. That means the argument of Z1 and argument of Z2 are same. Then you would realize that modulus of Z1 plus Z2. Modulus of Z1 plus Z2 will be equal to modulus of Z1 plus modulus of Z2. That means in that case, your triangle will become flat. And if your triangle becomes flat, the third side will be equal to the sum of the other two sides. Isn't it? Okay. Simple. So this will come as a question like this. Let us say Z1 and Z2 are two complex numbers such that mod of Z1 plus Z2 is equal to mod of Z1 plus mod of Z2. Then which of the following conditions are true? And one of the options will be argument of Z1 is equal to argument of Z2. So you have to mark that option. Make sense? Clear. Why did it? Why is it less than? In triangle, don't we know that the third side is always less than the sum of the other two sides? That is why it is less than, but it could be equal to also, but it will never exceed it for sure. Is it clear? Is it clear, Situ now? Yeah. Okay. Now, if this is obvious, then one more. Okay. I've already written one over there. So I don't want to write one, one, two times. The second inequality that you will be understanding here is that the third side will always be greater than the difference of the other two sides. Isn't it? This is also a triangle inequality, right? In a triangle, the third side, by the way, once we have known that it is less than equal to, can I make that correction over here and here as well? Okay. Because the triangle may be flat also. So here, can I say the third side will always be greater than equal to the difference of the other two sides? Okay. That means mod of z1 plus z2 will always be greater than equal to mod of z1 minus mod of z2 mod. Okay. Many times we write it as mod z1, tilde symbol mod z2. Tilda symbol just signifies the difference. Okay. It takes the absolute difference between the two quantities. Okay. So note this down. This is your second in triangle inequality, which is very, very important and asked in competitive exams. Now, one small question I would like to ask you if you want to answer, you can answer here. When do you think this will be equal to? When do you think mod z1 plus z2 will be equal to mod of the difference of the modulus of z1 and z2? When? Anybody? This symbol, this symbol you're talking about Z2. This is a tilde symbol. On your laptop, you will see over, I think one of the keys on the number bar, you will see that this is symbol. Right? This is a different symbol. This is to be read as a difference. Difference means it is just going to give you an absolute difference. Arguments are same, but opposite sign. Then how they are same? Nikhil, this is an Alzheimer on you. Same, but different sign. Okay. Good try. As of now, I will leave this blank here. When I will come back and answer this. Okay. And in fact, you will only answer it maybe after some time. Okay. So let us leave this as of now. We will come back again to this question and try to answer it. Okay. But we'll try it, Nikhil and whoever try to answer this question. So we'll come back and answer this question. Now let's move on to the concept of subtraction of two complex numbers. Any other question? Apart from when is this equality going to hold to in this page? This I will answer in some time. Don't worry. In fact, you will only answer it. Okay. Now, few things that you can note down over here is number one. I mean, I've already made it in the diagram, but I'll still write it. C represents the C point basically represents the position of Z 1, Z 2. O C represents your modulus of Z 1 per Z 2. Okay. And, and this angle, this angle, let's say angle theta, this angle theta represents your argument of Z 1 per Z 2. Okay. By the way, many people ask me this question. Sir, is there any relationship between argument of Z 1 and the argument of Z 2 with argument of Z 1 per Z 2? No such relation because the parallelogram can be of various shapes and sizes. Of course, whatever relation, you know, between the angles possible for a, you know, parallelogram, the same set of relations will hold true, but there's no fixed relation like that. Okay. So as of now, I will say no such relation exists. Very good, Nikhil. You're very close to the answer in fact. Okay. Angle between Z 1 and Z 2 is 120 degrees. No say two, but good track. We'll come back to that. Don't worry. Okay. Don't worry. I'm not going to leave this question and address because it's an important part of our theory. We'll come back again. So meanwhile, I will just go to the next operation, which is subtraction of two complex numbers. So subtraction of, of complex, complex numbers. So let's say we have Z 1 as a 1 plus IB 1 and Z 2 as a 2 plus IB 2, and I want to subtract it. See subtraction is same as saying you're doing something like this. Okay. So it is actually an addition of one complex number with the negative of the other. That's it. Okay. So as mathematical operation, it is very easy and this is what we, you know, mathematically can see, but what is important is not this. What is important is the geometrical interpretation. Where is Z 1 minus Z 2 located? What is the modulus of it? What is the argument of it? Right. So all those things we will try to now see in this particular discussion perpendicular to each other. No, no, no, but good track. Okay. The answer really coming now. Yes. Okay. Let's discuss it guys. Angles. Okay. So let's, let's, let's make it. Okay. And let's say this is my, this is my Z 1. And let's say this is my Z 2. I mean, where should I make it? Okay. Anyways, Z 1, Z 2. Okay. So this is your point A 1, B 1. And this is your point A 2, B 2. Okay. Let me name it. This is origin. This is point A. This is point B. Okay. Now, very simple question. Where do you, where do you think is minus Z 2 located? Where do you think is minus Z 2 located? All right. Reflect Z 2 about origin. Exactly. So if you reflect this guy about the origin. Let me make. Like this. Exactly mirror image about origin. You'll end up getting a point. Let's say I call this point C. This point C is your negative Z 2. Okay. And as a matter of coordinate, it will be minus A 2, minus B 2. Okay. Now my, my job is to what? My job is to locate which point A 1, minus A 2, B 1, minus B 2. That means I want to locate this Z 1, minus Z 2 as a point on this argument thing. Okay. In short, I want to locate the coordinate A 1, minus A 2, comma B 1, minus B 2. Okay.  I'm going to do the same thing. I'm going to do the same thing. Okay. Now what I'm going to do is I'm going to do the same activity as what I did to add to complex numbers. So I'm going to make. I'm going to make a parallelogram by using. O A and O C as the adjacent sites. Okay. So please bear with me or bear with my diagram. I think good enough. No, it's looking like a parallelogram. Right. I did a decent job. Okay. So this point D. I'm claiming that this point D is your Z 1, minus Z 2. Do you all agree with me on this? You'll say yes, sir. We agree because if you have made a parallelogram, that means the midpoint of AC and the midpoint of O D should be the same. And the midpoint of AC. Okay. Let's say I call this point to be for the time being C, comma D. So the midpoint of O D. The midpoint of O D will be C by two comma D by two. Correct. And the midpoint of AC would be a one minus a two by two comma B one minus B two by two. Now, since both of them have the same midpoint, same point, because it is a property of a parallelogram that the diagonals bisect each other. So we can say a one minus a two by two is equal to C by two. That means C is a one minus a two. And similarly, B one minus B two by two is equal to D by two. That means D is B one minus B two. So this point C comma D that you have written, this is actually a one minus a two comma B one minus B two. The one which you wanted to look at. Okay. Any questions, any concerns? So let me officially write it over here. The point D represents the point D represents your Z one minus Z two. The length OD represents the modulus of Z one minus Z two. And this angle, everybody pay attention. This angle, this small angle here that I'm showing that angle theta represents the argument of Z one minus Z two. Okay. So please make a note of this. Okay. Note it down. Now many people ask me sir, in order to know the modulus of Z one minus Z two and in order to know the argument of Z one minus Z two, do we always have to make Z one minus Z two, you know, complete that parallelogram and figure out all those things. The answer to that is actually no, there's a shorter way to achieve the information about the modulus and the argument. Okay. So let me make a diagram which I've already made in the previous slide once again. Okay. So let me just, let me just make a diagram for the same location of Z one there too. So I'll try to copy this diagram over here. So let us say Z one is here. Z two is here. Okay. I've kept my Z one there to same location. Okay. Now please understand this fact very, very important fact. When you talk about, when you talk about OD length. Okay. So OD length and let me just complete the diagram over here. We'll discuss it once again. OD length is actually here. Yeah. So this point was actually Z one plus Z two. Right. We had already discussed it in our previous slide. Correct. Now what I claim here is that this length OD, I hope everybody can see here, OD length. OD length is actually equal to AB length over here. Okay. So if you join A and B, this length AB length is actually the same as OD length. So you don't have to draw actually any kind of a parallelogram to know the length of, or to know the modulus of Z one minus Z two. So modulus of Z one minus Z two is nothing but this AB length, which is the second diagonal. So the first diagonal we used to know the length of Z one plus Z two, right? Or modulus of Z one plus Z two. The other diagonal gives you the modulus of Z one minus Z two. So please make a note of this. So this gives you more of Z one plus Z two. And this gives you more of Z one minus Z two. Okay. So from the same diagram, you can know modulus of Z one plus Z two also modulus of Z one minus Z two also separate parallelogram is required. Okay. Now, many people ask me, sir, can, how do you know the argument of Z one minus Z two from this diagram? Now there's a trick for that also to know the argument of, so note this down, I would request everybody to write this down actually to know the argument of Z one minus Z two. Basically, you find out the angle between the vectors or the vectors, the vectors Z two Z one. Or in this case, the vector, I would write it as B a vector. Okay. So B a vector. Now, give it a shape of a vector. So that is why complex number chapter is so, so important that it uses the concept of coordinate geometry vectors, of course, signometry is also involved. So this chapter is multifaceted. It has got so many other verticals of maths coming in. So the angle between the vector B a and the positive real Z axis. Okay. That will help you to get the argument of Z one minus Z two. Now this is something which many people don't know. So if you think as if there are two vectors, one is your positive real Z axis and other is your B a vector. Let's say B a vector is like this. Okay. What is the angle between them? That angle gives you the argument of argument of Z one minus Z two. I hope everybody here knows how to find angle between two vectors. Do you know that? Are you sure? Should I ask you a question? Yes. Are you all aware of finding the angle between vectors? Yes, I would have taught you in the bridge course. Okay. I want to ask you a question now. Okay. Okay. So I'll just give you a question. So this is a vector. Okay. And there's one vector which is basically coming like this. Okay. Fine. This angle is 30 degree. Fine. Let me name these vectors a vector B vector. Find the angle between AB. Find the angle. Between. Vector and the. Yeah, sure. Sure. Very good. I think. Very good. Excellent.  You know, under the strict. See, very good. So when you find the angle between two vectors, you first make them co initial. Right. So here, your vector B, your vector B and vector A must be made co initial first. So A is like this. And B should be made like this. Okay. So what are you doing here? You are going to make them co initial. Correct. Co initial means having the same initial point. So something like this. And then find the shortest angle between them. The shortest angle between them here is 150. So 150 is the angle between. Angle between A and B vectors. Very good. Okay. In the same way, if I ask you what is the angle between BA vector and the positive real Z axis, make them co initial. And once you make them co initial, the angle between them is the argument of Z one minus. Okay. So for that, you don't have to make a separate diagram or a separate parallelogram kind of thing. Is this fine? Any questions, any concerns up till now with respect to representation of Z one minus it to modulus of Z one minus it to argument of Z one minus it. Right now, time for some triangle inequalities. By the way, say to you want me to go to the previous slide for a second or are you, are you done with that? Should I go to the previous slide? Yeah. Give me a second. Yes. This, this position. Okay. Done. Okay. Great. All right. Let's get all these scenarios that we have discussed so far. And I think now I have given you enough resources or enough tools you can say to answer that question of the previous slide. Sorry. I changed the slide by mistake. Yeah. This question, which is still an address. I think one of you have almost answered it. Now can you come back and answer this question mark? When do you think, when do you think modulus of Z one plus Z two is equal to the difference of the distance of Z one and Z two from the origin? Think, think. No. And Nikhil, you're correct. Yeah. See here, everybody please pay attention. Now what is the way, what is the way that you should be looking at this scenario? Okay. This is basically read it as mod Z one plus Z two. Read this like this. Mod of Z one minus of minus Z two. Okay. In geometrical sense, you should read it as the distance between.  See, what is this? This is nothing, but it is the distance between modulus of Z one minus Z two is the distance between the two complex numbers and one in that to represent it as a point or as points. Isn't it? So what is mod Z one minus Z two? Mod Z one minus Z two is nothing, but it's the distance between Z one and Z two simple as that. Right. When you wrote more, do you remember what you, what we basically discussed about it is the distance of Z from the origin. Right. So you read it like this, you'll automatically come get the same idea. So it's the distance between Z and origin. Okay. In the same way, this is a more generalized view of the same thing that represents the distance between Z one and Z two. Right. If this is kept in mind, you will be able to answer the previous question here. The question here is, when do you think the distance between, between Z one and minus Z two is equal to, is equal to the difference of the distances of the complex numbers from the origin. So when do you think it will be, let's say I call this point as point A and point B. So I'll just make a diagram out of it. So let's say this is Z one. Okay. And I'm just asking out of, you know, I'm just keeping this point somewhere over here. Okay. I don't know the exact position. That is something which you have to tell me. When do you think this distance, sorry. This is minus Z two. When do you think this distance is equal to difference of this distance and this distance still not getting it. Okay. Now let me make the actual situation. Then probably you'll be able to answer it. Let us say, let us say Z one is here. Okay. Z two is diagonally opposite over here. Let's say this is Z two. So diagonally means, diagonally means origin A and B are collinear and it is on the B and AR on the opposite sides of the origin. Okay. Like this. That means this angle is 180 degree. So as to say this angle is 180 degree. Okay. Now here, let us try to see whether this condition is getting met or not. This condition is getting met or not. See Z two is here. So minus Z two will be here. This will be minus Z two. Let's say I call this point as C point. Can I say that the distance AC from this diagram, this distance is OA minus OC. Correct. So distance AC is OA minus OC. OA is mod Z one. OC is mod minus Z two, which is same as mod Z two because modulus of a complex number and the modulus of the negative of the complex number is the same both are at the same distances from the origin. So here, what did you realize that Z one minus of minus Z two modulus is same as modulus of Z one minus modulus of Z two. That means here the situation must be such that the difference of the argument. So this can only happen when the difference of the argument. So see this is if let's say I call this as theta, then the argument of the argument of the other complex number, in this case, it is this. Okay. This difference. This difference should be 180 degrees. Now many people will say, sir, how is the difference because five is a negative quantity as per the diagram. Okay. So please note down, note down that this situation can only arise when the argument of Z one and argument of Z two, they differ by 180 degrees. They differ by 180 degrees. Okay. That means Z one origin and Z two must be in the same straight line. But the Z one Z two point must be on the opposite sides, not on the same side. If they are on the same side, then argument Z one will become equal to argument Z two that I don't want. Okay. Is this clear? Is this clear? This question has come. I don't know. I've lost count how many times it has come. So many times it has come in the comparative example. Is it clear? Make sense. Any question you have, please do let me know. Now, why do I write that difference? See, because even if you are Z one is on this side, Z two is on this side, it will still hold to. So I can't say, you know, argument of Z one is more an argument of Z two is less. Either can happen. Argument of Z two can be more an argument of Z one can be less. But what will be holding definitely true here is that the difference of their arguments will be 180 degree. For example, in this case, if you take argument of Z one to be 30 degrees and argument of Z two, you can all figure out from the diagram that is minus 150 degrees. So if you take the difference here, if you just take argument Z one difference argument Z two, that will come out to be 30 minus minus 30 minus minus 150. That will come out to be 180 degrees. Okay. Statement in the cloud statement in the cloud is ha see basically say to I'm sorry basically Manu. I'm trying to create a scenario where the distance between A and C is equal to the difference of OA and OC because I want this condition to be met. No, this condition should be met. What does this represent distance between Z one and negative Z two, right? So this is your negative Z two. This is your Z one. So difference between Z one and negative Z two. That is your AC distance is equal to OA that is modulus Z one minus modulus of Z two. If this condition is to be met, that means this condition is to be met. And if this condition is to be met, the scenario here must be basically applied. Then only this condition will be met. So your OA and B that is your Z one origin and Z two must be collinear and Z two and Z one must be on the opposite sides of the origin. Getting it now. No, it makes sense. Another modulus is because the other modulus is because you want this quantity to stay positive. Because if it is not positive, let's say mod Z one was three and mod Z two was five. Three minus five will become negative two. But more of any complex somewhere, more of Z one per Z two, that should remain a positive quantity. So here we are talking about the absolute differences. Okay. So in order to make it absolute, we are basically putting an extra modulus sign. That extra modulus sign is just to make it positive. That's what I wrote. No. Oh, I think minus sign. Yeah, correct. Z one minus minus it clear in the left diagram. Why did you mark that as theta shouldn't be. Real Z. Where where a mark theta. Here. Next page. Okay. Let me go to the next page. We'll talk about it. It should be with the positive real Z axis. This is from positive real Z axis. Okay. Now in the subtraction also we will have some triangle law of inequality. So let me do the third triangle law of inequality here. Now everybody focus on everybody focus on which diagram, which diagram, which diagram. Listen to this diagram itself. Okay. Let me write it here itself. This three I will write here. So all of you now focus on this diagram in triangle or AB. Okay. Just focus on that diagram. Can I say AB length will be lesser than the sum of the other two sides. In fact, you can say less than equal to also because it need it didn't be always in a form of a triangle. Okay. So can I say mod Z1 minus Z2 will always be less than mod Z1 plus mod Z2. This is third triangle law of inequality, which is going to be tested. A lot of questions are framed on this as well. So please note this down. And as usual, now I'm going to ask you. When do you think the equality will hold true? When do you think this will be equal to this? When filling the blanks. When dash mod Z1 minus Z2 will be equal to mod Z1 plus mod Z2. Think now you should be able to answer it. Read it like this. When do you think is the distance between Z1 and Z2 would be equal to some of the distances of Z1 and Z2 from the origin? When do you think the distance between Z1 and Z2? Right, Nikhil. Let's say Z1 is here, Z2 is here. When do you think, when do you think from the diagram itself you will be getting your answer? When do you think the distance between Z1 and Z2 be some of the distances of Z1 and Z2 from the origin? You say, sir, you have actually drawn it. So the answer is there in the figure only. Okay. So when your difference of argument of Z1 and Z2 is here, please note this down very, very important. Is this okay? Now, fourth triangle inequality that comes from here is that the third side of a triangle is always greater than the difference of the other two sides. Okay. That means mod Z1 minus Z2 will always be greater than equal to mod Z1 minus mod Z2 whole mod. Is it fine? Now, here also a question will come. When do you think the equality will hold true? When do you think this will be equal? Now, read this question like this. You will automatically get your answer and imagine it. Maybe you can close your eyes. When is the distance between Z1 and Z2 equal to the difference of the distances of Z1 and Z2 from the origin? The moment you imagine this, you'll get your answer. There is no rocket science involved. When? When exactly? I think you have got the crux of the situation. That's why you're getting all the answers. So when their arguments are equal, that means they must be Z1, Z2 must be in the same line and Z1, Z2 must be on the same side of the origin. Means their arguments are equal. Okay. So when their arguments are equal, let me just quickly draw a small diagram to illustrate that. So when let's say Z1, Z2 is like this. Z1, Z2. Okay. So you'll automatically get your answer that the distance between Z1 and Z2 will be, let's say I call this as A and B. So the distance between, the distance between Z1 and Z2 will be OA minus OB. Okay. Now I'm just talking about difference. I'm not claiming which is more than the other. Absolute difference. Okay. So here when the arguments, so basically both Z1 and Z2 have the same argument, then only this condition is going to be holding true. So let me summarize this triangle inequalities because they are going to be very, very important and we'll take one question based on that. Please note down everything and do let me know if you have any concerns and queries. What about the holidays? Do you have any holiday coming up for Ganesh Chaturthi? If yes, which day? The Friday. Friday? Okay. This is for DPS Friday. Friday is already. Okay. Okay. September 1st week. I think we're sitting in September 2nd week. No, the share is in October 1st. Even my knowledge about festival seasons is very less. One day I logged into my MS team and I was waiting for the class to join in. Then I got a call from the school admin, said you have logged in today's holiday. No, madam. Sentom doesn't give holiday. We in fact study more on a holiday. Yes, as a student, we should see holiday as an opportunity. Today I will complete this. Today, you know, they solve backlogs. So no, no holiday from Sentom. Sentom holiday will now come in the Dashayra two days. I think Naomi and Dashmi and of course Diwali day. And then it will come 31st and 1st. New year. So these are the only. We don't give holidays. Maybe a national important 15th August. If it comes. Then all then we give 26 January. We give. What sir? You don't like us to enjoy holidays. Okay. Hello. Say two as a question certain figure. Why is a BZ1-Z2 a B is not Z1-Z2. This is the distance between Z1 and Z2. Oh, A is more to Z1. Oh, what happened to my writing. And OB is more to Z2. That's how this condition is met. Huh. A B. Again, AB is what distance between Z1 and Z2. No. Correct. And isn't it equal to this, which is more to Z1 plus this, which is more to Z2. That's how this condition is met. This condition is going to hold true under this situation, which I have drawn here, which gives you the fact that, oh, looking at this, I know the difference of the argument is pie. That is why this condition is holding true. So basically it's a realization of the same concept through diagram. Okay. Now let me quickly summarize these four triangle inequalities. Quickly. Summary. Summary. Of. The four. Triangle. Inequalities. Okay. So first we learned that. The modulus of Z1. Z2 will always be lesser than equal to. Modulus of Z1. Plus modulus of Z1. Plus modulus of Z2. Modulus of Z1 plus modulus of Z2. And. Here equality holds. I'll write it in brackets and equality holds. When. Argument of Z1 is equal to argument of Z2. Okay. This is number one. Number two. Modulus of Z1 plus Z2 will always be greater than difference of. Modular. Plural of modulus is moduli. Of Z1 and Z2. And here the equality will hold. Here the equality will hold. When. The difference of the argument. Of Z1 and Z2 is a pie. The third inequality that we saw. While we were doing subtraction. Modular Z1 minus Z2 will always be less than. Modulus of Z1 plus Z2. And here also equality will hold true. When. When. Argument of Z1. Difference argument Z2 is a pie. Okay. And the last one, which I just now took recently. Modulus of Z1 minus Z2 will always be greater than difference of modulus of Z1 minus Z2. And here. And here the equality will hold. When. They are having the same argument. Argument or amplitude. By the way, another word for argument is amplitude. Okay. So if you read amplitude, then, okay, don't start thinking in like, in terms of physics, simple harmonic motion amplitude. No, amplitude means complex number means argument. Okay. And unless central stated, we are always talking about the principal argument. Is this fine? Any questions, any concerns, please note this down. Now we are going to take up some questions. In fact, one question will take up not. So if they give up. Then your third Saturday will be working in school. No free lunches. One off they will give and one class they will give. No, normally third Saturday is off in NPS. Even I am a teacher in NPS. I mean, one of the NPS school. So. Third Saturday would be working there. I mean, I'm just assuming that the same fund will apply to all the NPS. Maybe I may be wrong. Okay. Should we take questions now done? No, no, no, no, you didn't get the point argument of Z1 plus argument of Z2 is not 180 degrees. See, see, see, see, again. Diagram. Let's look at the diagram. See, if let's say one complex number is there, let's say two. Okay. Another complex number is such a way that it is exactly mirror image. I mean, let's say somewhere over here. Yeah. Okay. Now, here you are saying argument of Z1 plus argument of Z2 is 180 degree. No, that is wrong. Argument of Z1 minus argument of Z2 is 180 degree. How? See. This is argument of Z1. This is argument of Z2. But this argument is a negative angle. Because it is clockwise. So let us say, I'm just giving some rough example. If this is 160, then this is actually minus 120. So when you add them, like what you are saying, you will not get 180 degree. Are you getting it? When you subtract them, you will get 180 degree. So when you do 60, sorry, 60 minus minus 120, then you will get 180 degree. That means you do theta minus 5. Then you get 180 degree. Getting it. That's the mistake which many people do. See, it could have been like this also. I'm not denying that it's always like Z1 in the first. It could have been like Z1 is here. And Z2 is here. Okay. And let us say this angle is minus 60 degree. Then this would have been 120 degree. So here now the difference of 120 minus minus 60 is 180 degree. That is why I wrote that tilde sign. That means difference depending upon whichever is more that will come to the left, whichever is less, that will come to the right. Got it. Sir, in the third one shouldn't be where have I miswritten anything? Third one, no third one. I'm talking about mod Z1 minus Z2 only. Did I write anything? I've written properly only where I've written minus. This one. Third one. So the third side is always lesser than the sum of the other two sides. Correct? No. Yes. It is equal when this condition is met. Equality. This inequality I made equality in the next step here. And this equality will hold true when this condition is met. What is the problem? I didn't understand your question. Why not Z1 plus Z2? That I've already taken. No, they just went here. One, two, three, four. There are four inequalities. What you're talking about is already taken in one and two. Check. Three and four are different. There are four inequalities here. Mod Z1 plus Z2 is also there. Don't worry. But Mod Z1 minus Z2 is also there. All right. I hope I have given enough time for you to copy. Let's take some questions. Let us take some questions. Questions. Where is the right question for this? Okay. Let's take this one. If Z1 is any complex number such that modulus of Z plus four is less than equal to three. Find the greatest value. Find the greatest value of mod Z plus one. I would request you to give me a response on the chat box. Okay. Okay. I'm going to go to Manu.  See, you can solve this question in any of the two ways which you have discussed. One is your geometrical idea that itself is like, you know, a very strong tool to solve it. Other is you can use your triangle inequality. See triangle inequality itself has come from the geometrical idea. Okay. So any of the two approaches which you feel convenient with, you can adopt that way. Very good. So three people have responded so far, Manu, Nikhil and others. So let us take this. So I'll solve it in two ways. One is by using your regular inequality. See what do we want to find out the greatest value of mod Z plus one, the greatest value. Okay. So when greatest value thing comes basically any quality comes that this should be less than equal to some number. So if I say number X is less than equal to let's say five. Then what is the greatest value of X? Five. But for that you need to, you know, remember the less than equal to sign in your mind. Similarly, if somebody asks you, what is the least value of four X, you'll say four, because it's always greater than equal to that number. So when somebody asked me the greatest value, I think of less than equal to inequality. Right. Now, wherever there's a less than equal to inequality, I will try to implement that. Now, let us try to see mod Z plus one. I want to write it less than equal to some number. What is this number? If I am able to find out my job is done. Yes or no. But what is given to me, I have to use that only to achieve it. So what is given to you is a complex number Z plus four. And they have given that the modulus of that number is less than equal to three. Now see what I will do.  Can I write it like this? Correct. Right. Now, since this, the information about this complex number is already given and three is anyways, you know, a well-known number to us. Okay. Can I use the fact that, can I use the fact that if this is Z one and this is Z two, Z one plus Z two modulus is less than equal to mod Z one plus mod Z two because I had to use less than equal to symbol. So can I say this is less than equal to mod Z plus four plus mod negative three. Yes or no. Correct. In short, what are you trying to say? You're trying to say, let me write it in brackets here. This is not a part of the solving. This is just a formula that we are implementing or the inequality that we're implementing. So you're trying to say mod Z plus one is less than equal to mod Z plus four plus mod of negative three. In short, you're trying to say that this is less than three already. Correct. And this is as good as a three. So you're trying to say a mod of Z plus one is less than equal to six, which means the greatest value, the greatest value of mod Z plus one is equal to six. So six is the answer to this question. I think Nikhil and Vaishna have got it well done. But personally, if you ask my opinion, if I were to solve this question as a student, I will not take this approach. I will take the geometrical approach because that is more closer to the basic understanding. So I will give you another method, method number two, which I'm sure most of you would like it. Meanwhile, copy this if you want to and do let me know if you have any questions. Guys, most of you, I know you will be misusing the modulus thing. Many people say mod Z1 plus Z2 is equal to mod Z1 plus mod Z2. Please remember that will be two only under a certain condition. Treat complex number like vectors. Don't apply scalar laws on it. Like how you apply it to normal scalar quantities. So complex number is what a position vector. So how you deal with the position vector the same way you can deal with it. All right. I'll show you method number two. See, method number two is basically based on the geometrical idea. What is the geometrical idea? Let us try to understand first of all, what is the meaning of this geometrically speaking? Z is some complex number. Let's say some moving complex number. It is dancing around. But it is dancing around in such a way that it is satisfying this condition. What can you interpret from this? Okay. Should I give you a better reformed version of this expression? What can you interpret from this? Geometrically speaking. Geometrically speaking. Excellent. Excellent. Excellent. Very good. See, yesterday it means that Z is such a point whose distance from minus four. That means minus four. Let's say it's a complex number minus four comma zero. So Z is such a point whose distance from minus four always is less than equal to three. That means whatever is this distance, whatever is this distance, that distance is always less than equal to three. Then where should Z be lying? Any idea? Nikhil has already given the answer. Others? Tell me. Where should Z be lying?  Correct. That means your Z can be moving anywhere on or inside the circle. Okay. So Z is like free to move, but it is moving in such a way that its distance from minus four, which is actually a minus four comma zero complex number should always be less than equal to three. So it should be within that circle of radius three, having the center at minus four comma zero. It can be on the circle also. Okay. If this is understood, then you'll be able to solve the inner requirement of the question. So the question is asking, what is the greatest value of mod Z plus one? Now see, mod Z plus one is what? Mod Z plus one is mod Z minus minus one. Correct. Basically it represents the distance of Z from minus one comma zero minus one comma zero is here. Let me show you with the yellow color. This is minus one comma zero. Why? Because this is already three, right? This radius is already three. Correct. So this is minus one comma zero. Now which complex number point in this zone, which I've shown by dotted line. It is which complex number out of these old dots, which I have made is at the greatest distance from minus one comma zero. Which one? Which one? Which one? Out of all the dots which I have made inside or on the circle, which one is at the great diametrically opposite one? This guy, yes or no? And this guy is what? This guy is minus seven comma zero. Correct. So what is this distance? You say sir, simple that distance is six units. So the greatest distance of Z from minus one comma zero is going to be six units. That is the greatest value of mod Z plus one is six. Isn't it a nice and interesting way to get the same thing, right? So if you're not using any kind of an inequality and all those mathematical jargons, is it fine? Any questions? Okay. Now, depending upon the scenario, you can use any of the methods. Of course, both the methods have their own advantages. Many times you realize that triangle inequality will work. Dramatical interpretation may be difficult. Okay. And vice versa, wherever you feel geometrical interpretation works fine. Go for it. Okay. So spend enough time on addition and subtraction. We'll talk about multiplication and division also. They are pending from my side. So yes, can I move on to the next slide? We'll take more questions as you know, we solve more and more concepts as we learn more and more concepts. Our questions will incorporate all these. Okay. As of now, we'll have to cover a lot of ground actually. Last year, I remember we had to cover up this chapter. In the summers also, sorry, in the break between your 11th and 12th also, it was so big actually. So we'll come back. Don't worry. I'll be taking more questions. Okay. Meanwhile, I'll talk about multiplication of complex numbers. Okay. Let's talk about multiplication of complex numbers. See, when you have two complex numbers, A1 plus IB1 and A2 plus IB2. Okay. And you multiply them. So multiplication doesn't have any restriction. That means just like addition and subtraction hand. Multiplication doesn't have any restriction. You can multiply anything with anything. That means real can multiply with real. Real can multiply with imaginary. Imaginary can multiply with imaginary and so on. So when you do that operation, you get A1, A2. I'll first write down the real parts, whatever I get A1, A2 and I square B1, B2. I square B1, B2 is minus B1, B2. Okay. Then I will write the imaginary part IA2, B1 and A1, B2. Okay. So Z1, Z2 is going to be this. Now this is a very dry expression. I mean, it doesn't give us any insight. Okay. It just tells you what is the process of multiplying. But beyond that, does it help you to locate Z1 into Z2? Where is Z1 into Z2 located on the argument? Nothing is evident from it. It is dry. I'm not happy looking at this somehow. It did not give me any insights. So in the point form, you realize that the point form importance is only in subtraction and addition. When it comes to multiplication, division and raising it to any power, polar form and Euler form, they are the kings there. I mean, you will understand a lot of things from the polar form and the Euler form rotation. So I will now do a similar operation, but now assuming two complex numbers in polar form or Euler form. And then you realize that looking at the answer itself, you realize that, oh, this is what is happening when you are multiplying two complex numbers. So let us create that moment. Okay. So now this is a point form operation, which I found it very dry. I'm sure most of you also found it dry. So I will give you some better view of the same thing. So let's now look at the polar or Euler form representation. Okay. We will do the same operation in polar and Euler form. So let us say my complex number is r1, cis theta1. Sir, don't write cis and all. Write fully. Okay. And in the same thing, Euler form, you can write it like this. I hope everybody knows Euler form, r e to the power i theta, theta should be in radians. So what do you write as r cos theta plus i sin theta? Same thing can be written in Euler form as r e to the power i theta. Okay. So now when you multiply these two complex numbers, let us see what happens. Of course, Euler form will give you a straight cut, you know, understanding, but even if in the polar form, if you do, you'll get r1, r2. Okay. Now see, I'll just follow this expression. A1, A2 minus B1, B2.  Okay. And A2 B1 plus A1 B2. A2 B1 is sin theta1 cos. Let me write it. Yeah. A2 B1 is cos theta2 sin theta1. And A1 B2 is sin theta2 cos theta1. Correct. Now if you look at this expression, you'll say, oh, I can see this happening. So your product basically shows some important, your product shows some important characteristic here. What is the characteristic that I see? First thing I see is what happens to the modulus. And then I see what happens to the argument. So from here, two very interesting insights come. Number one, if you multiply two complex number, you get another complex number whose modulus is r1, r2, which means it is the product of the moduli of the two complex number. Correct. So when two complex numbers multiply, the resultant coming out from it, or the product coming out from it, product is the right word to use, the product coming out of it, the modulus of that complex number will be the product of the moduli of the two complex numbers which were multiplied. That is insight number one. Now many people ask me, sir, can we generalize it also? Can we say that, mod z1, z2, d-d-d-d-d-d, till zn if I multiply, its modulus will be product of the moduli of all these complex numbers? Can I say that? The answer is yes. We can say that. So in general, please note down that when you take the modulus of the product of n complex numbers multiplied, it is as good as the product of the moduli of the complex numbers. Okay, this is number one. Number two, you'll also see that the argument of the product is actually the sum of the arguments of z1 and z2. Something very similar to what happens in log operation, isn't it? So when you add, sorry, when you multiply two complex numbers, the resultant complex number or the product that comes out, the argument of that product is the sum of the arguments of the two complex numbers which were multiplied. Right? Now subject to the fact that it is between minus pi to pi. Of course, because we are always talking about the principle argument. So please, please ensure that you keep the result within minus pi to pi. Now many people say, sir, what if it exceeds pi? See, you already know your trigonometry very well, correct? Write a coterminal angle, which is basically having the same answer as that given angle. I'll just take an example here. Let's say, let's say argument of z1 was pi by 2, let's say. And argument of z2 was, let's say, 3 pi by 4, right? Now I ask you, hey, tell me the principle argument of z1, z2. What will your answer be? What will your answer be? Of course, as per the property, you will add them, correct? But when you add them, you would realize your answer will overshoot. In fact, in this case, it will become 8 plus 6, sorry, 4 plus 6. 10 pi by 8, if I'm not wrong, correct? 10 pi by it is 5 pi by 4. But 5 pi by 4 doesn't belong to minus pi to pi, which is supposed to be the range of our principle argument, right? So it doesn't belong to minus pi to pi, correct? So what do you do for this case? See, simple. 5 pi by 4 is this angle, right? Correct? But when I ask what, let's say, this is your product, z1, z2. So when I ask what is the principle argument, you must state this answer. Are you getting my point? You must be stating this answer. Correct? So what is this answer? That is what I want to know. What is this answer? This yellow angle. What is that yellow angle? This is 5 pi by 4, right? The yellow angle is minus 3 pi by 4. So you should state your answer as the answer to this question will be minus 3 pi by 4, because this is a general argument. You should not state general argument. You should always state the principle argument, which is this game, okay? So please ensure whenever somebody asks you the argument, as a, you know, a convention, we always give the person the principle argument, okay?  Don't want general argument. I always want the principle argument. How do you conclude the third result? Oh, no, yellow. Yeah, white one is a part of the first one. You're talking about the yellow, the last one, right? Yeah, I can't see it. Yeah, this is theta 1, this is theta 2. Hey, read this. This is like r cos phi. I sign phi. So what is phi? Theta 1 plus theta 2. And phi is supposed to be the argument of the product, whatever complex number has come out as a multiplication and result. So argument of z1 into z2 is phi and phi is theta 1 plus theta 2. That's how this result comes. No problem, Sethu, no problem. I think it was an oversight. Is this fine? Any questions? Any other questions? No, here people ask me, sir, can we further generalize here? Can we say that if we have z1, z2, zn multiplied, the argument of it will be, I should write it in white actually because sorry, I should have written it in white. So it will be rhyming with the previous one. Yeah, so argument of the product of z1, z2, z3, zn. Can I say it is the sum of the arguments of z1? Yes, you can very much say this. But again, please keep your answer limited to. That means restrict your answer to minus pi to pi interval. That means keep your answer in the principal argument. Is it fine? Now, this is something which of course comes out from that multiplication result. And one more thing I would like to add here. Euler gives this in straight one shot. Euler is such a strong representation that when you multiply these two, it's you directly get r1, r2, e to the power i5, 1 plus pi 2. Sorry, theta 1 plus theta 2. Sorry for saying phi 1. So Euler is basically, he was gem of a mathematician. He basically figured out that your resulting complex number or your product will basically have an argument, a modulus, which is product of the moduli of z1, z2. And the argument of the product will be the sum of the arguments of z1 and z2. Straight away, one shot. Thanks to Euler. Anyways, so this is the mathematical way of looking at the, you can say, the product of two complex numbers. Now, what is happening geometrically? Let us try to understand that. Before that, you want to copy anything, you want to write anything, please do so. Okay. Geometrically, what is happening? Geometrically speaking, when you multiply a complex number, let's say I start with z1. Okay. And let's say Euler form or any form you want to write it doesn't make a difference. Let's say z1 is a complex number, which is r1 e to the power i theta 1. Okay. So on the argon plane, where is it located? You'll say, sir, at a distance of r1 from the origin and having an angle of theta 1 with the positive real real z axis. Okay. So this is your location of z1 point. Now, when you multiply this complex number with another complex number whose argument, sorry whose modulus is r2 and argument is theta 2. Right. Let's say you multiply z1 into z2 or z1 z2 like this. Okay. Then what is the location of this guy? See what is going to happen. This complex number z1 you rotate it anti-clockwise by theta 2 angle. So you just rotate this further by theta 2 angle. Okay. And you scale the modulus by a factor of r2. That means let's say r2 you multiply it to this. So it becomes r1 r2. So here is your location of z1 z2. Are you getting my point? So what are you doing here? Listen to this concept. Let's say z1 was at a distance r1 from origin and at an angle theta 1. You multiplied z1 with z2. z2 is another complex number whose modulus is r2 and argument is theta 2. So how do you locate the product? This is the way to locate the product. Just rotate z1 complex number by further theta 2 anti-clockwise and scale the modulus by factor of r2. Scale up or down that depends on r2. I cannot comment about up or down but scale it by a factor of r2. So whatever you do in fact in the diagram it is shown to be longer but it could be shorter also in reality. It depends on r2. So this is how you locate your z1 into z2. So this is the concept which we call as the rotation concept in complex number. So multiplying one complex number by the other creates a basically a rotation kind of an event. And of course there would be a scaling up or down of the length. So let me write this phenomena in English so that you can refer to your notes later on. So when z1 is multiplied to z2 when z1 is multiplied to z2 the following things happen. Rotate z1 by argument of z2 anticlockwise anticlockwise and scale modulus of z1 I mean so much English is also not good and scale modulus of z1. I could have used this notation by factor of modulus z2 and by doing this z1 z2 is located and thereby obtain location of z1 z2. This is very very important. If you understand this many of the future concepts will be easy for you to understand. Okay. Let's take a small quick example also on the same. Let's say I had a complex number 2 e to the power i pi by 6 okay. I mean I can write it in polar form I can write it in point form but this is just a you know Euler form which makes us draw the complex number very easily. So let us say it is located at a distance of 2 from the origin and pi by 6 is this angle so 30 degrees. I take another complex number let's say 3 i pi by 3 okay and I am multiplying these 2 complex numbers so can you tell me where will be z1 into z2 located don't multiply it literally just tell me looking at this figures looking at the figures that I have written in these 2 complex numbers what will happen to obtain or what should you do to obtain z1 into z2 location so first you will say sir rotate this guy z1 by how many by how many degrees see by argument of z2 z2 by how many degrees see by argument of z2 here is how much pi by 3 so rotate it pi by 3 and d clockwise means 60 degrees by the way when you rotate it 60 degrees you will come on this line okay exactly on the imaginary z axis okay this is an imaginary z axis and what do you do you change the modulus of z1 by a factor of modulus of z2 is 3 so 2 into 3 6 times so this length will become 6 so just choose a point which is at a distance of 6 okay so this distance is 6 okay and on the imaginary z axis so by the way you have reached a point which is 6 I by the way okay so product of these 2 will actually give you a 6 I you can check it out also by actually multiplying it so 6 e to the power I pi by 3 plus pi by 2 sorry pi by 6 plus pi by 3 that is going to be 6 I pi by 2 6 I pi by 2 is actually 6 I is it fine any questions now coming back to this figure another important thing we will be discussing there first note down anything that you want to if you ask that question I will ask your counter question how do you write this in polar form how what is 6 I in polar form that means if you write it as R cos theta I sin theta what is theta and what is R in short I am asking you what is modulus of 6 I and what is argument of 6 I R is 6 okay the same thing you have actually answered your own question the same thing if you write want to write in a polar from what do you write R e to the power I argument that's what was here so it became 6 I means I think they can you know your conversion between polar to Euler is not that clear as of now which you need to do nothing you know the modulus you know the argument right so R e to the power I theta that's the Euler form representation but of course write your theta and radiance don't write I 90 that will be wrong okay there is some you know convergence scenario that is going to be should be satisfied I don't want to get into that now see everybody let's go to this diagram this is very important phenomena that I would like to point out let me make two triangles over here one is your triangle let me name it oh okay let's say this is z2 and let me name it as B okay so this is your a point z2 is your B point and this is your C point okay if I make triangles like this all of you please watch out for these triangles if I make triangles like this this is your theta 2 by the way or let me make a fresh diagram because this figure is slightly cluttered hmm where should I make it okay let me make make the diagram here this is your are you this is your R one this is your theta one this is your R2 and this is your theta 2 okay so basically I have shown you the location of z1 let's call it as A and z2 let's call it as B okay and I had also drawn for you let me use gray line okay this this length is R1 R2 this whole angle is theta 1 plus theta 2 by the way all of you please if this whole angle is theta 1 plus theta 2 and this is already theta 2 so can I say only this part if I make this will be theta 1 correct this was your location of z1 z2 let me call it as C point okay now let me construct a triangle over here one is this triangle the triangle O C B and another triangle is this triangle O AP now P is the point which is actually 1 comma 0 okay now comment upon the nature of these two triangles how is triangle O C B and triangle O AP O C B O C B O C B as you can see my cursor is dancing on it O C B and O AP how are these two triangles related to each other can somebody tell me looking at it only lot of things will be clear they are similar right correct how can you say it is similar sir very easily we can say if this is theta 1 as you can see yellow angle here theta 1 and this is also theta 1 so these two angles are same first of all and if you take the ratio of let's say O C by O B okay this ratio will come out to be R1 R2 by R2 which is R1 this is same as the ratio between O A by O P which is R1 by 1 same that means the sides are also proportional so when sides are proportional and this angle is basically common angle that is theta 1 theta 1 both the places that means these two triangles will be similar to each other and this itself has been asked as a question in the comparative exam right so they will say this is the location of Z1 this is the location of Z2 this is the location of Z1 Z2 this is the location of 1 comma 0 you make a triangle like this like this like this what is the relationship between these two triangles one of the option will be they are similar to each other okay and you have to mark that option you have to choose that option so please note this down if you make two triangles one connecting O Z1 and the complex number one and the second triangle by connecting O Z1 Z2 and Z2 the two triangles that will be formed will be similar to each other is it fine any questions any concerns so so deeply they go to the concept of multiplication you see that they made triangles they've asked you the nature of the you know triangles also so it is not just about learning these operations in a dry fashion you have to analyze things okay you have to analyze and challenge your learning now let's say a J at once aspirin they can make a question like okay what will be the ratio of the areas of the two triangle something like that can be framed right so all those things tricky things can be framed is this fine any problem with respect to addition sorry multiplication of complex numbers out of all the things which I have discussed I would like you to take this thing very very seriously okay very very important the concept of rotation in in complex numbers is very heavily asked so if you're done with this can I now move on to the division of complex numbers so we have to cover division we have to cover the concept of conjugate we have to cover the concept of logarithms we have to cover the concept of powers so everything has to be done by today so let's see whether we are able to complete so I'm now moving on to the next slide where I'll be talking about division of complex numbers division of division of complex numbers or quotient of complex numbers so when you divide one complex number let's say even plus IB1 by another complex number A2 plus IB2 what happens okay so when you divide one complex number by another complex number you end up getting another complex number let's say that complex number is A3 plus IB3 now normally when we divide complex numbers and we get a complex number we leave the answer as A3 plus IB3 form rather than leaving it in a very raw stage like this so many times the question itself comes what is A3 and B3 in terms of A1 B1 A2 B2 okay so in order to obtain that you need to do a small operation that operation is you have to multiply your denominator and numerator with a term which is very similar to the term in the denominator just that I is written as a minus I okay by the way many people use the wrong word here so sir you mean to say you are rationalizing the denominator no idea don't use the word rationalization here this is not an irrational term that I am making it as a rational rationalization word means you are making an irrational term into a rational term by multiplying it with something I am not converting an irrational to a rational okay in school also I have seen many teachers and many trainers use the long word you are rationalizing the denominator this is not an irrational term this is a complex number the right word is you are realizing the denominator realizing the denominator what is this sir yes you are actually making it real see how if you multiply with this number in the denominator you will see you get A2 square minus IB2 square correct if you expand it you get A2 square minus A2 square I square is minus 1 itself B2 square which is actually A2 square plus B2 square which is completely real term so you can see though you can use the word you are realizing the denominator English word realizing that is what we are using here you are realizing it we are making it real okay so here by multiplying it with this complex number you are realizing the denominator by the way this complex number is actually called you would have already learned in the school this is called the conjugate of A2 plus IB2 so you are multiplying the denominator with its conjugate now there is a separate subtopic of conjugate that I will talk in sometime not to worry alright meanwhile having taken care of my denominator let us look into our numerator term numerator term is basically a plain and simple I would say a dry multiplication operation let us do that A1 A2 IB1 minus IB2 will give you plus IB1 B2 and the imaginary parts would be if I am not mistaken A2 B1 minus A1 okay now your denominator is already I will rewrite this as denominator is already denominator is already A2 square plus B2 square so divide it individually so if you divide it individually you get A1 A2 plus B1 B2 upon A2 square B2 square I A2 B1 minus A1 B2 upon A2 square B2 square okay so this becomes your A3 and this becomes your B3 okay so if somebody says write down this division as a complex number then this is what you need to do in order to write it as a A3 plus IB3 or let us say A plus IB form but this is good for an operation point of view it does not give you any insight it is a dry operation I mean it is just you can say mindless operation that you do not apply any mind just doing it and getting your complex number what we are interested in or what competitive exams are interested in knowing whether you are geometrically able to see what is happening when one complex number is divided by another complex number okay so we will start that analysis in some time but before that you please make a note of this any question anybody I think somebody's mic was on okay so this is just an operation point of view this is good but this does not give us any insight dry operation alright so let us now try to understand the same operation when you write the two complex number in either polar form or Euler form so when you write it in polar form okay or let's say Euler form then you will understand the real meaning of division what is actually happening when you are dividing it in fact I will not waste too much time doing the same process I will just use my Euler form notation Euler form notation is super fast so Euler form says that hey if you are dividing one complex number by another complex number this is what you are going to see okay that means if you write the same result in a polar form you are going to see something like this now looking at this I get lot of ideas this is insightful this tells me two things what are those two things I would like to hear from you what are the two insights that we carry from here when you divide one complex by another you get another complex number whose modulus is r1 by r2 which is modulus of z1 by modulus of z2 now this is a very important property which says that when you divide one complex number by the other and you are finding the modulus of the quotient it is as good as quotient of the moduli of the two complex numbers now there was a case where this question was asked in school I don't know exactly which of the n-pace it was asked the question was asked find the modulus of 3 plus 4i divided by 5 minus 12i and the student wasted I would not like to name a long back I think 2 or 3 years back he wasted 2-3 minutes converting it into a complex number as a plus ib form and then he started finding the modulus of that no need to do that don't waste your time finding this as a single complex number right you can directly use this result and say oh the answer to this will be just modulus of this by modulus of this over that's it 5 by 13 finish it off end of the game don't unnecessarily convert this to a single complex number and then find it's modulus by doing under root a square plus b square don't waste your time time is very important especially in competitive exams are you getting my point same is true for z1 into z2 modulus so z1 into z2 if you are multiplying it don't waste time if you want to find out the modulus of the product just take the moduli individually multiply it over game is over don't convert it to a single complex number and then find it's you know models are you getting my point is it clear the second take away from here is that argument of z1 by z2 as you can see from here okay is theta1 minus theta2 which is nothing but the difference of the arguments of z1 and z2 of course difference here is not a exact word here it's argument z1 minus argument that means whatever is on the numerator that argument should be taken first minus whatever is the denominator that argument should be taken next okay of course please ensure that the result that you get is between minus pi to pi because you always want to state up a simple argument of any complex number is it fine any questions here any questions please note down now we'll also see geometrically what is happening when you divide one complex by the other by the way looking at these two expressions you must have got an idea for that as well done anything that you would like to ask or know from this page do let me know okay so geometrically what is happening see when you have a complex number let me make a diagram let's say z1 z1 is here and z1 is let's say I'll just write it over here z1 is let's say r1 e to the power i theta1 that means this length this length is r1 and this angle is theta1 okay you are dividing it by another complex number whose modulus is r2 and argument is theta2 then where is z1 by z2 located then how would you locate it okay now when you look at the final expression it is r1 by r2 cos theta1 minus theta2 plus i sin theta1 minus theta2 that means what is happening here is that you take this complex number z1 rotate it clockwise by how much theta2 correct so take that complex number rotate it by how much by theta2 which is the argument of z2 and then scale the modulus of that by a factor of 1 by r2 okay so what are you doing you are first rotating this guy by an angle of theta2 where is the same r coming from yeah so rotate it by an angle of theta2 and then you change your then you change your modulus by a factor of r2 that means this will be r1 by r2 right are you getting my point so if you obtain this point this point will be your z1 divided by z2 are you getting what are you doing so you are rotating just exactly opposite of what you used to do in multiplication in multiplication you used to rotate it theta2 clockwise correct and then you used to scale the modulus by a factor of r2 here what are you doing to do you are rotating it by theta2 clockwise scaling it by a factor of 1 by r2 make sense should I write that down also okay so for this complex number rotate rotate z1 by by argument of z2 clockwise and scale modulus of z1 by factor of 1 by modulus z2 when you do that you end up locating your z1 by z2 complex number is it clear so this is also like a kind of a rotation but of course the direction and the scaling factors are different okay so this is very very important now one more geometrical interpretation we will be talking about once you have copied this diagram read the scenario and then we will take some questions based on the same followed by a break now another interesting insight that we should all have here maybe I will draw it to the of course length of any complex number length means the distance of that complex number from the origin will be mod of that complex number so if you are talking about this length it is r1 by r2 only r1 by r2 is mod now see very important let me make the diagram once again let's say this was your z1 location and this is the modulus of z1 this is the argument of z okay and let us say this is z2 location this length is r2 this angle is theta2 okay and let us say this is the location of let me use a blue color this is the location of z1 by z2 okay let me call it as o and c okay now if you complete a triangle if you make two triangles like this one is like this another is like this okay where this point is your point 1 comma 0 okay now please note that while rotating it you rotated it by an angle of theta2 okay so this is theta2 so what can you comment about triangle oac and triangle obp they are they are similar yet again okay and you can check it out how checking is very easy this is theta2 this is also theta2 as per the diagram and if you do oa by oc or if you do oa by oc you will end up getting oa oa is r1 oc is r1 by r2 so r2 will go up so oa by oc is giving you r2 similarly if you do ob by op ob by op you will get r2 by 1 which is the same as r2 okay so both will give you the same result that means not only the angle but even the sides are proportion angles are equal and sides are proportion which means these two triangles are similar and you may be asked a question related to this in the comparative exam see why is taken as 1 comma 0 is because in order to make this as a similar triangle to this you have to have this length as a unity that is why we have taken 1 comma 0 many students ask me this question so why all of a sudden you have taken a 0.1 comma 0 because then only triangle obp and oac will be similar to each other so if you want to create a similar triangle to oac you have to take the point obp p being 1 comma 0 so think in a other way direction so let us say had I asked you the question that make a triangle which is similar to oac and having one of the sides as ob then you will definitely choose another side as ob and make a triangle like the triangle obp so construction wise you have to choose the point to be like that then only you can make a you do a reverse activity see let me just write it down here if I say if I say make a triangle make a triangle obp which is similar to triangle oac there is already theta 2 here correct and there is already theta 2 here so you need to choose this point so the point has to be on the real z axis let us say I do not know this point for the time being I will call it as a comma 0 correct so as for the proportionality thing you will have ob by ob equal to oa by oc ob is r2 ob I do not know as of now so I will put an a oa is r1 oc is r1 by r2 so r1 by r2 r2 will go on top so r2 by a is r2 so what should be your a 1 only so this point should have been 1 comma 0 so indirectly you think you will get your answer what should be your point b that is why while I was drawing the diagram I already told you that if I take that point to be 1 comma 0 then the triangles would be similar or vice versa if the triangles are to be similar then that point has to be 1 comma 0 so any way you want to read the same statement it's your call okay good so let's take some questions let's take some questions okay I have a very simple question over here maybe all of you would have done lot of these questions in school exam but still let us do it again after this question setu I will go back to the figure once again if you have drawn a figure on your notebook just read the figure once again when you divide what do you do you rotate it clockwise by theta 2 only you know as per my operation I came to know that argument is theta 1 minus theta 2 so theta 1 has to be diminished by theta 2 so there would be a clockwise rotation of theta 2 from there it became theta 2 see geometrically has come from that operation only when you did that operation z1 by z2 what is the end result you got r1 by r2 cos theta 1 minus theta 2 plus i sin theta 1 minus theta 2 so argument became theta 1 minus theta 2 so in order to locate my r z1 by z2 I had to reduce theta 1 by how much theta 2 so that reduction is like taken as a anti clockwise motion by theta 2 got it so when I did that I realized on the diagram that basically that angle becomes theta 2 in there alright so please solve this question and let me know your values for x and y okay Nikhil Prashim is getting a different answer anybody else I don't think so you should be finding these kind of problems difficult to solve see as I told you the process you have to follow multiply with the conjugate of the denominator okay so in the first expression which is this expression I have done the same do a similar activity with the other one as well multiply this with the conjugate so okay so basically realize the denominator here for both of these expressions so on the denominator here it will give you 10 see don't waste time calculating this this will be 10 for both the cases it will be a square plus b square a is 3 b is 1 okay so what do you have on the numerator is I will just write it down in real terms so 3x plus 3 this will be 3 x minus 3 and minus I x minus 3 so 3x minus 3 and minus xx minus 3 and this will be 3 y minus 3 plus I y minus 3 okay this is equal to I in short basically you end up getting something like this I have taken the LCM of 10 and 10 I have send it to the right side and this is what we see now please pay attention here here you have to use your comparison of complex number so there are two complex numbers okay this is one complex number and this is another complex number by the way 10 I I will write it in a slightly fancy way 0 plus 10 I so this is another complex number so when you are comparing two complex numbers real will be compared with the real that means this will be compared to 0 and this operation which is supposedly giving me only y minus x this will be compared to 10 okay so from this operation I get x plus y is equal to 6 and from this operation I get minus x plus y is equal to 10 add it so y is 8 anybody who said y is 8 absolutely correct and if y is 8 x is minus 2 so your answer to this question is correct Seethu correct I think the first one to get this right was Nikhil got it right so x is minus 2 y is 8 is it fine any questions simple I mean this is a typical school level complex number question so this is how your question will be framed in school exams can I move on to the next one okay we will take this question oh I think I have not done conjugate sorry I have not done conjugate with you right if mod z1 is equal to mod z2 an argument of z1 by z2 is 5 find the value of z1 by z2 very good Noil if you read this expression as geometrical interpretation this question is just 10 second question 10 seconds not more than that so these two scenarios just imagine in your mind right Nikhil absolutely right just imagine this in your mind you will get your answer like this click off the finger correct Fischer see let me just take a scenario okay I will take up two complex numbers which meet both the situations one mod z1 is equal to mod z2 which means the distance of both the complex numbers from the origin is equal second the argument of z1 minus argument of z2 please note that argument of z1 by z2 as you have already seen is differences argument of z1 by z2 is seen as a difference of argument of z1 minus argument of z2 and this is 5 so let us say if this angle let's say this is z1 and this angle is the argument of z1 then z2 is first of all at a difference of 5 from here that means this will be r2 of course and this distance is same that means this distance and this distance are same right they are asking you what is the sum of these two complex numbers by the way if this is your a,b then this point must be minus a minus b so when you add two complex numbers z1 z2 here you will end up getting 0 nothing else a1 minus a1 and b1 minus b1 they will cancel each other out okay it's a simple you know question if you look at it from geometrical point of view so wherever possible I would encourage you all to look at it from geometrical angle now see if let's say geometrically you don't want to solve it for let's say a school level question so this is how you solve it so this is your geometrical way of thinking of it geometrically I would say non geometrically non geometrically if you want to solve it let's say a model of z1 and model of z2 are same and this is equal to r okay and let argument of z1 be theta okay let's say now as per this solution argument z1 by z2 which is argument of z1 minus argument of z2 okay if this is theta and let's say this is any unknown angle let's say phi let's say argument z2 is phi then you have been given that phi is phi minus or theta minus correct so from here we can interpret that phi is theta minus correct now let us say I want to construct my z2 so I will require modulus which is r itself and I would require this argument which is phi as per my assumption but phi is what? phi is theta minus phi if you write it down properly this is as good as cos phi minus theta cos phi minus theta is minus cos theta and this is minus i sin pi minus theta which is actually sin theta in short you have written negative of r cos theta plus i sin theta isn't it? isn't this negative of z1 because z1 is having modulus of r and argument as theta so it is negative z1 so you are saying z2 is negative z1 so what should be z1 plus z2? 0 only no isn't this a longer way to solve the question when you can solve it in quick time by using your geometrical idea right anyways so I have given you both the options you can solve it non-geometrically whichever you feel suits you and depends on situation also many a times geometrically may not be very nice way to solve it or may not be a efficient way to solve it so both the options you should keep open whichever is lesser time taking that approach we will take a break here right now right now 6.17 is the time as per my watch we will meet exactly at 6.32pm okay see you on the other side of the break so the next topic that we are going to talk about in fact sub topic that we are going to talk about is actually the conjugate of a complex number conjugate of a complex number okay conjugate of a complex number you can actually call it as a type in operation only okay so you can actually find the conjugate of any complex number let's say a plus i b by reversing the sign of i so i will become minus i in this case and this complex number is called the conjugate of the complex number in fact both are conjugates of each other both are conjugates of each other conjugates of each other okay in many books in fact in international books they will write conjugate as z star in fact i b they will write it as z star okay so how do you obtain a conjugate of a complex number just by changing the sign of i with a minus i that's it okay so change change i with a minus i okay that's how you end up getting conjugate of a complex number few examples to cite over here let's say if you have 3 plus 4 i its conjugate will be i don't minus 4 i absolutely if you have a complex number let's say 3 i plus 2 what is the conjugate of this write down write down in the chat box anybody minus 3 i plus 2 i was actually waiting to see somebody writing 3 i minus 2 so many people think finding conjugate is just reversing the sign in between no it is not reversing the sign in between it is changing i with a minus i now let's see geometrically where is the conjugate of a complex number actually located so if let's say a complex number is a plus i b okay its conjugate is a minus i b means as a point if you write it it should be at a comma minus b location so if somebody asks if you want to make a point a comma b as a comma minus b you will get that point you will say sir simple i would reflect that point about the real z axis okay if you reflect it about the real z axis you will end up getting the conjugate which is a comma minus b location okay now in additional information i am giving you if you reflect it about the imaginary z axis you will get a negative of z conjugate okay please note that so if you want to change the sign of x you have to do negative z conjugate okay that will give you the reflection about the imaginary z axis is this fine so what is the conjugate of a complex number it is clear so if the complex number is located in the second quadrant its conjugate will be in the third if it is located in the third quadrant its conjugate will be in the second if it is located in the fourth quadrant its conjugate will be in the first if it is located in the first quadrant the conjugate will be in the fourth it is located, you just have to reflect that particular point about the real z axis that will give you the location of its conjugate. Now, certain properties of conjugate that is very, very important. Let's talk about it. The first property is if you conjugate a complex number twice, it will give you the same complex number back, which is very obvious, isn't it? If you reflect a point about x axis and then again, sorry, real z axis and again reflected about the real z axis back, you will get the same point, isn't it? Second thing that you will observe here is that modulus of the conjugate and modulus of the complex number will be the same. That means the distance of z from origin, okay, whatever is the distance and the distance of z conjugate from the origin, that means this distance, they will be the same, correct? Of course, mirror image congruent triangles will be formed, okay? If you connect, if you directly connect this, okay? So please note this down and also you can look at it from your expression point of you also. So if z is this, its modulus will be under root a square plus b square. If z conjugate is this, its modulus will be again under root a square plus b square. Okay, both are same. So please note this down. Third thing is argument of a conjugate is negative of the argument of that complex number. So if you see this is, if this angle is theta, this angle will be negative of that, okay? And both will be negatives of each other. So the argument of a complex number and its conjugate, I'm talking about principal argument and always I'll be talking about principal argument unless until stated otherwise. So argument of a complex number and its conjugate will always be opposite in sign, no matter whichever quadrant the complex number lies on, okay? Even if it doesn't lie on a quadrant, means if it lies on real z axis or imaginary z axis, still this property is going to be true, okay? No doubt. Next, if you do the operation of some words difference of two complex numbers and you take its conjugate, it is as good as doing the same operation on the conjugates themselves, okay? So if you add two complex numbers or subtract two complex numbers and take the conjugate of the result, it is as good as you're doing the same operation on the conjugates, okay? Same goes with product, okay? And same goes with quotient, okay? So conjugate of z1 into z2 is same as z1 conjugate into z2 conjugate and z1 by z2 whole conjugate is same as z1 conjugate by z2 conjugate. Next property. I hope you have all copied till 6th. I hope it is visible, yeah. Next property. If you add a complex number to its conjugate, it'll always give you twice the real part of z. This is very obvious because if you add a plus ib and a minus ib, don't you get 2a, okay? 2a is what? Two real part of z only, no? Okay? Now, there is a very interesting corollary which is related to this. The corollary says that if you add two complex numbers and it gives you zero, that means z plus z conjugate gives you zero. That means the complex number is purely imaginary, okay? So when a complex number added to its conjugate gives you a zero. It means the complex number z was purely imaginary. It means it did not have any real part, okay? It directly comes from this property itself that two real z is zero means real z is zero. That means there's no real part. Only imaginary part is there. That means i lambda kind of a complex number that number would be, okay? Similarly, if z minus z conjugate is done, it will give you i twice imaginary part of z.  Obviously, a plus ib minus a minus ib will give you two ib or you can say i twice of b, which is i twice of imaginary part of z. b is the imaginary part of z. Remember, a is the real part of z. Okay? So a corollary also here comes up that if z plus z conjugate, sorry, z minus z conjugate is zero. That means z is equal to z conjugate. That means the complex number is purely real. This is a very, these two, you know, you can say inferences that we have drawn from it. They are very, very useful in solving many questions. Please make a note of this, okay? Last but not the least, z multiplied with its conjugate will always give you mod of z square, will always give you mod of z square, not z square. Please don't get me wrong here. I'm not saying z square, z square, mod z square are two different things. Now, last class I was having with Raja Ji Nagar. There was a student who thought that z square and mod z square means the same thing. No, definitely not. Unless until z is purely real. Okay, so they're different things. z square, when you say, let's say z is a plus i b, let us say z is a plus i b.  But mod z square is just a square plus b square. They are different things, very different. One is a complex number while the other is, I mean, one is a complex number may have, may have imaginary part to it, but the other is actually purely real. This is purely real. This is a complex number. Okay, please do not confuse between z square and mod z square, very, very important. Let me make a cloud. Okay, this is a very important property and you will see it used in many, many questions. Okay, a lot of questions have been framed on this property. So I hope all of you have noted down all these nine properties and we'll be applying these properties in solving many questions, especially the ninth one. As I told you, last but not the least, it's the most important property which comes handy while solving many questions. Done. Should I move on to the questions now? Okay, let's start with this question. Okay, let's take this question. If z1 and z2 are two complex numbers such that z1 minus 2z2 upon 2 minus z1 into z2 conjugate is unimodular. Now what is unimodular? Unimodular means modulus is 1. That is to say, if you take the modulus of this complex number, this is 1. Okay, while z2 is not unimodular, that means z2 is not unimodular, you have to find what is modulus of z1. Okay, so you need to find out modulus z1 from it. I would request you to solve this question and give me a response on the chat box. Okay, let me also help you out with this. See, this is as good as saying modulus of z1 minus 2z2 divided by modulus of 2 minus z1 z2 conjugate is equal to 1. Okay, that means modulus z1 minus 2z2 is equal to modulus of 2 minus z1 z2 conjugate. Okay, now what do I do with this? I mean this is just a dead end, right? So what I'm going to do is I'm going to now square both the sides. Okay, now everybody recall the last property which we did. Please recall in the last property I had said mod z2 is nothing but z into its conjugate, isn't it? So can I say a similar situation has arisen over here where you have squared the modulus of some complex number? So can I say the left-hand side expression I can write it as z1 minus 2z2 times z1 minus 2z2 conjugate. Similarly, sorry, forward the square here. Similarly here also it's 2 minus z1 z2 conjugate times 2 minus z1 z2 conjugate, whole conjugate. Now just expand it. In fact, conjugate property I can use further to write it as z1 conjugate minus 2z2 conjugate. This is 2 minus z1 z2 conjugate and this is nothing but 2 minus. Now remember if you do a conjugate on 2, it will remain a 2 because 2 is purely real, correct? So when a complex number is purely real, z and z conjugate are same things. So there's no point doing a conjugate on 2 because it will remain a 2 whereas conjugate of this will give you this. So it will give you z1 conjugate and z2 conjugate conjugate which is z2 itself. Here also if you see I did not write a conjugate over 2 because 2 is a purely real number. Now let us expand it. So if you expand it, you get z1 z1 conjugate which is nothing but mod z1 square. You'll end up getting minus 2z1 conjugate z2 minus 2z1 z2 conjugate plus 4 mod z2 square. Here also you get 4 plus mod z1 square mod z2 square and minus 2z1 z2 conjugate minus 2z1 z1 conjugate z2. Okay, just multiply it. A lot of terms I think will be cancelled off. I think this gets cancelled with this. This gets cancelled with this. Is this fine? Let's take everything to the left hand side. So mod z1 square minus mod z1 square mod z2 square plus 4 mod z2 square minus 1. In fact, I can take minus z1 square common from here also. So that will give me mod z2 square minus 1. Correct me if I'm wrong. Yeah. Take mod z2 square minus 1 common and you'll end up getting 4 minus mod z1 square. Okay. So this gives you two possibilities. Either your mod z2 square is 1 which means mod z2 equal to 1 but this is not possible because the question setter itself has mentioned that z2 is not unimodular. Isn't it? Yes or no? So this is not possible. What is possible? The second possibility that 4 minus mod z1 square is 0. That means z1 square is 4. In short, oh, sorry. In short, mod z1 is 2. In short, mod z1 is 2. This is what you wanted to find out. This is your answer. And one important thing. In fact, it may sound very trivial. Please do not write plus minus 2 in such cases because modulus of a complex number cannot be negative 2. Okay. Please do not write plus minus 2. It is always positive. Is this fine? So as you can see here, a lot of properties of conjugate came into being and we use all those conjugate properties to solve it. Please note this down and do let me know if you have any concerns about any part of the solution. Good enough. Any questions? Can we take one more question? Okay. Let's take one more question. Okay. Let's take this one. This is a very interesting type of question which is asked in various shapes and sizes. So here this question says modulus of z1 is 1. Modulus of z2 is 2. Modulus of z3 is 3. And modulus of 9 z1 z2 plus 4 z3 z1 plus z2 z3 is equal to 6. Find the modulus of z1 plus z2 plus z3. Yes. Anybody with any success? Okay, Nikhil. Good try. Anybody else? Okay. Let's try this out. See, I have been provided with this information and I need to reach out to this information. How will I reach out to this information? Okay, now let's see. Can I do one thing over here? Can I pull out within this modulus symbol z1 z2 z3 out? So when I pull out z1 z2 z3 out from each of these terms, you will see that you will be left with 9 by z3, 4 by z2, 1 by z1. Correct? This is like two complex numbers multiplied, right? So can I write it separately as modulus of this and modulus of this? Okay. Now here I am slightly stuck. What to do next? Okay, that would be an obvious concern. What to do next from here on? Now, all of you please pay attention. We have not utilized the fact that we have been given the modulus of each of these complex numbers. So mod z1 is given to us as 1, which means mod z1 square is also 1. Okay? Mod z2 is given to me as 2. So mod z2 square is given to me as a 4. So normally these numbers 1, 4, and of course 9 as well. Can I connect it to something? You say obviously we can write this as zz1 conjugate. This is z2 zz2 z2 conjugate. This is z3 z3 conjugate by the very property. So I can say 1 by z1 is z1 conjugate, 4 by z2 is z2 conjugate, and 9 by z3 is z3 conjugate. So can I say, can I say this term 9 by z3 is actually z3 conjugate. This term 4 by z2 is actually z2 conjugate. And this 1 by z1 is actually z1 conjugate. Okay? This is given to me as 6. Now let us try to use our modulus and conjugate properties in combination. This term, I can write it as mod z1, mod z2, mod z3. This is conjugate of this whole thing. Am I right? Remember I told you when you add two complex numbers or subtract two complex numbers and take a conjugate of the whole, it is as good as adding their conjugates or subtracting their conjugates and depends upon the operation. Now recall here one more property that modulus of a complex number and modulus of its conjugate are same thing. So can I not write this as modulus of just z1 z2 z3? Now out of this, this is 1, this is 2, this is 3 and this is unknown. So I have to find this out. This is my requirement. So from here I can say modulus of z1 z2 z3 is going to be 6 by 6 which is a 1. So your answer in this question, the answer to this question is 1. Nikhil got it. I think Nikhil was the only person to get this right. This is a very, very commonly asked question in CET, Comet K, Bitsat, Manipal, Jee Main. Jee Main actually it is easy for Jee Main standard but in fact many school also will ask these questions. Is it fine? Any question? Any concerns? Please have a look at it and let me know if you want me to explain any part of this solution once again. How did I remove conjugate? See, we have already known this property, CETU. This and this means the same thing, isn't it? So this has a single complex number, the conjugate of it modulus and the same complex number modulus should give me the same result. That's how I removed it. Clear? All right, everybody's convinced? Okay, so next operation that we are going to talk about is how to find square roots of a complex number. This is also a very interesting operation and this is asked in school many a times. So a square root of a complex number gives you another complex number but there will be two answers coming out. So two answers will come out from it. Now, unlike in case of real numbers, when you had under root of a real number, you only state the real solution but in case of complex numbers, we state two solutions. So both the square roots will be mentioned. There's nothing like principal square root and non-principle square root for complex numbers. Okay, so CETU, this is the only place where you can write plus minus both. Okay, so we'll be talking about how to find out these square roots. Square roots. See the name of the topic, square roots of complex numbers. Okay, now this is something which I would like to explain with an example. Okay, let's say I want to find out square root of 4 plus 3i. Okay, and let's say the square root of this complex number is x plus iy. Now, by the way, many people asked me, sir, you said two answers will come out but you only assumed it to be one x plus iy. See, don't worry, x and y can have changing values or different values and hence you will get multiple answers. Don't worry about that. Okay, there's no point taking two separate values when both the values will come from the same assumption. Okay, so let's say under root of 4 plus 3i is x plus iy. Right, now see the steps. The steps are very important. This is what you need to follow when you're solving questions. Square both the sides. So when you square both the sides, this will become x square. Now I've already squared a complex number a little while ago for you when I was trying to say that z square and mod z square are different things. So there you get x square plus iy whole square, iy whole square is minus y square plus 2x iy, which is i into 2x square. Now here it is a case where you are comparing these two complex numbers. So when you compare two complex numbers, remember the very first thing that we did today, the real parts will be equal and their imaginary parts will be equal. So you can say 4 is x square minus y square and 3 is 2xy. Okay, now from here I can say I have to find my x and y, how to find my x and y. Now there are various roots. One root is many people write y in terms of x and they substitute in the first one. They get a y quadratic in x and they find their values of x on there and hence y value. But that root is slightly tedious many a time. So there's an interesting methodology which we adopt to make this process simple. Recall in your childhood days, you would have done this property a plus b the whole square is a minus b the whole square plus 4ab. Have you all done this property before? Identity before? Yes sir, I think in class 7th or 8th only you would have done it, isn't it? So I am planning to use the same over here by taking my a as an x square and b as a y square. So see what I am going to do. Can I say x square plus y square whole square is x square minus y square whole square plus 4x square y square. In short, you are trying to say that x square plus y square the whole square is x square minus y square whole square plus 2xy the whole square, isn't it? Now, from the given expression over here, can I say x square minus y square will be 4 square and 2xy the whole square is 3 square? Right? In short, it is 25. So what is your x square plus y square value? Can somebody write this down on the chat box? What is x square plus y square value? If x square plus y square whole square is 25. Simple question. LKG level. Montessori level question. Arun Nithi, correct. It's just 5. Don't say plus minus 5. Why am I not plus minus 5? Why not minus 5? How can square of two real numbers give you minus 5? It can never happen. Remember x and y that you are using in your expression, they are ultimately real numbers. Square of two real numbers cannot give you negative 5. Now, having got this, your life is pretty easy because using these two equations. So I'll just write it down on the side. So using this equation that is the x square minus y square is equal to 4 and x square plus y square is equal to 5. Let's solve for x and y. Add them. So x becomes plus minus 3 by root 2. Subtract them. So y becomes plus minus 1 by root 2. Now, two values of x and two values of y comes out from here. So if I have to make a complex number, I will get four possibilities. How four possibilities? When you take a positive x, positive y, negative x, positive y, positive x, negative y, and negative x, negative y. So four possibilities will arise because there are two each of x and two each of y. So you can have four permutation combinations. So plus root 3 by 2, plus 1 by root 2 i, minus root 3 by 2, plus 1 by root 2 i, plus root 3 by 2, minus 1 by root 2 i, and minus root 3 by 2, minus root i. So four possibilities comes out. But I know there should be only two answers and I'm getting four. That means two of them are frivolous. Two of them are false roots. Correct. How do I know which is false root and which is the correct root? Which of the two is the correct answer and which of the two is the incorrect root? Which is usly, which is knuckly? How do you figure out? Any idea? Okay. The answer to this dilemma lies in this equation, very, very important equation. Whichever satisfies 2xy equal to 3, that will be your answer. Or those will be your answer. The rest of them will be neglected. So does this satisfy 2xy equal to 3? 2 root 2 into 3 by root 2 into 1 by root 2. Does it give you a 3? Yes. So this is a root. Correct. But does the second one give you the 2xy equal to 3 condition? No, it gives you actually minus 3. So this is not our answer. Similarly, this will also be not our answer. But this will be your answer. So in short, the answers out of these two, the square roots that you will get will only be 3 by root 2. In fact, you can write it in one step plus minus 3 by root 2 plus i times 1 by root 2. Is it fine? Any questions? Any concerns? I'm sure in the school while you were doing complex numbers, square root would have been taken up. It's an important concept for school as well. But what is not told in the school is actually the geometrical positioning of the square roots. What, sir? Again, you started geometrical interpretation. See, again, my duty is to tell you from each and every perspective. You didn't do your Ramakrishna. Why? In complex number, what did you do all? Addition of complex number, multiplication, division, a little bit of conjugate. That's what you did. That's insufficient, guys and girls. That is not going to help you crack any exam. There's a big mismatch, at least in NCRT and what is the requirement for the cooperative exams? Anyways, let's not go into some other direction. So now I want to show you if you have a complex number, let's say z. And you want to find out the square roots of that complex number and you want to locate it directly on the argon thing. Now, everybody please pay attention. Let's say I have been provided with a complex number z whose modulus and argument is known. If I want to locate the square roots of this complex number, let me write it down. Let's say the square root of z is z1 and z2. z1 and z2 will be located like this. z1 will be located at half the argument. Anyway, z1 or z2 are whichever. I should say one of the roots will be located at half the argument. Let's say z1 and at a distance of root r. And the other complex number is a negative of this. Negative of this means it should be located exactly mirror image about origin. In short, it should be located here. Again, at an angle, at a distance of root r from the origin and this gap, this gap between or you can say this argument difference should be 180 degrees. So, this is how your locations of the roots, the square roots of a complex number can be shown on the argon diagram or the argon plane. The mutants now imagine they deleted the three laws. I hope it's a joke. Seriously. Sir, for complex numbers, they deleted the imaginary and the real part, sir. Okay. So, say this is a lie. Tell me that it's a lie. Okay, let's have a question. Let's have a question which will be done by you, not by me. Up till now, what I've been observing that I am only giving question and I am only solving it. Okay. So, this time you'll solve it. And of course, Nikhil, my good friend, he is making an honest attempt to solve most of the questions. That's everybody is absolutely quite. What UT and all are going on? UT, UT, UT, UT, UT, UT, UT. Any UT is going on? Come on, guys. When will you live life from UT to UT? Let's do our duty. Let's not live in UT. Let's take some questions. Okay. I'll answer that question in some time. But let me just first give you a question. Yes, I would like you all. I think first one we already did now. So, skip this. Maybe we can try third one. Please do this question. Do it now. School marks matters of class 11. Actually speaking, no, it doesn't. But if let's say board exam doesn't happen, okay, whose probability is there, then maybe they will average out your school 11th and 12th marks to find your final grades. Second thing is your school marks will automatically increase if you are able to solve these kind of questions because we are going much beyond school. School is like the seed of a mango. If you're taking care of the entire mango, seed is also taken care of. So, core is your school. Okay. You're going much beyond the core. So, school marks will automatically increase. I mean, I have never seen a person who has got into GE Advanced and has done badly in school. Other way round is definitely seen. Okay. Done very well in school but did not get into any exam. But other way round is rarest of rare. Yeah, I know that. I don't want to blame anybody here but they teach only what NCRT prescribes. NCRT will not prescribe a lot, doesn't prescribe a lot of things. At least in maths, they don't. Physics and chemistry, I say they are good. Physics, chemistry, NCRT is good. Maths, NCRT is not that great. Give me a response on the chat box please. I'll have some Neer, Svalpanneer. I'll come back with Svalpanneer. Yes, any response? Anybody with the square root? Okay. Let's discuss it out. Oh, okay. You want some time? Okay, fine.  See, unlimitedly, you should solve it, right? You are the one who is going to face it. Okay, done. Okay, great. So, all right. So, the same process will be repeated once again but this time I'll be slightly faster because I've already discussed with you the nitty gritties of the operations. Okay. So, I expanded it and I compared the real and the imaginary parts with each other. Okay. So, the process which I discussed, I'll be adopting that process to solve this question. So, I'll be using this formula. That means x square plus y square whole square is equal to minus 8 square which is 64 and 2xy the whole square which is 225 which is 289. So, x square plus y square is equal to 17. Only 17 is possible. No plus minus please. Only 17. So, now that you know x square minus y square is negative 8 and x square plus y square is 17. What is wrong with my 7? Yeah, 17. Let's solve for x and y from here. Add it. 2x square is 9. That means x is plus minus 3 by root 2 and subtract it. 2y square is 25. So, y is plus minus 5 by root 2. Okay. So, there are four possibilities for x plus iy. 3 by root 2 plus i5 by root 2. 3 by root 2 minus i5 by root 2 minus 3 by root 2 plus i5 by root 2 and finally minus 3 by root 2 minus i5 by root 2. Okay. Now, out of these four, the ones which satisfy the second equation that is 2xy is equal to minus 15. That will be respected. Rest will be ignored. So, 2xy is minus 15. No. 2xy is minus 15. Yes. 2xy is minus 15. Yes. 2xy minus 15. No.  Is this fine? Any questions? Any concerns? Excellent. Excellent. Very good, Situ. Is it fine? One question for sure will come in your school exams also if at all your school teacher has taught you this. Else, we'll move on to the next operation, which is logarithms of complex numbers. How to find log of complex numbers and what kind of question related to log of complex number will be asked. Let us take that into picture. So, logarithms of complex numbers, logarithm of complex numbers. Now, see here, for logarithm of complex number, the best way is your Euler's notation. Euler's form is the very, very, you can say handy way when you're dealing with logarithms of complex number. So, let us say if you have a complex number, which is Re to the power i theta. Okay. And somebody is asking you to write the log of that complex number to the base of e or ln z.  So, this is nothing new that I'm talking over here. So, this is ln r and this is i theta. So, basically, it gives you another complex number.  That is to say, this complex number real part is your log of the modulus and your imaginary part is the argument of that complex number. So, if you take a log of a complex number, you get another complex number whose real part is ln mod z and imaginary part is argument of that complex number. So, if somebody asks you a casual question here, what is ln of 1 plus i? What is ln of 1 plus i? So, if you remember this isn't directly, you can say it is going to be ln root 2 plus i pi by 4. Is it clear? Any questions here? So, it is a plus ib a being ln mod z and b being argument of that complex number. So, mod of this complex number is root 2 argument is pi by 4. This is your answer. Clear? Any questions? Any questions? Okay. If you don't have any questions, I have a question to ask. Okay. So, my question is, find the value of sin of ln of i to the power i. Why did I write a 1 here? Okay. A single question. Find ln of, sorry, find sin of ln. ln means log to the base of i to the power i. Excellent, Sethu. Very good. Okay. Now, before you solve this question, I have another question related to this. i to the power i is which of the following? Is it purely imaginary? It's purely real. None of the above. Okay. Let's say I ask this as a supplementary question. So, which of the following is correct for i to the power i? Right.  Yes. How? Let's see. See, what is i? In Euler's notation, i is 1 e to the power i pi by 2. This is the Euler's notation for i. Remember, r e to the power i theta and r is 1 for i and pi by 2 is your argument. Okay. So, if I do i to the power i, basically you're doing e to the power i pi by 2. By the way, let's not write 1 and waste our time. I mean, this whole is to the power of i, which is nothing but e to the power i square pi by 2, which is actually e to the power minus pi by 2. So, basically it's a purely real number. Okay. Please note this down. This itself can come as a question. i to the power i, is it purely real or not? Okay. Now, see, the question sitter has asked you, sign of ln of i to the power i, which you just not figured was e to the power minus pi by 2. So, ln e to the power minus pi by 2 is actually minus pi by 2. I hope you all know your log properties by now and sign minus pi by 2 is negative 1. So, the answer to this question is negative 1. Is it clear? Any questions? Any concerns? Do let me know. Okay. So, I would like you to solve one more question related to this concept and then we can call it off. Done, everybody? Okay. Let's take another question. Last question for the day.  In short, find your a and b. In short, the question is asking you to find a and b values or a and b expressions, whatever you get. Okay. So, this is an interesting question. I will also, you know, help you out. See, 1 plus i, if you write it in the Euler form, this is your 1 plus i expression. Correct? No, no, no, the answer is much more complicated. So, if I raise it to the power of minus i, which means you are raising this to the power of minus i, which means you are doing root 2 to the power minus i into e to the power of minus i square pi by 4, correct? Which is root 2 to the power of minus i into e to the power pi by 4. Okay. So, this is your expression for this. Okay.  You call this as a plus i b as per your given requirement of the question. Not really Nikhil, but good try, good try. Now, take a log of both the sides of the base e. Okay. So, when I take log of both the sides of the base e, this will give you ln root 2 to the power minus i plus ln of e to the power pi by 4. By the way, ln of e to the power pi by 4 is just a pi by 4. So, it's something like this, pi by 4 plus. This minus will come outside here. Okay. This is equal to minus i ln root 2. Am I right? Or you can say something like this. Pi by 4 plus i ln 1 by root 2. Okay. So, this is your ln of a plus i b.  Imaginary part is the argument of that complex number. Correct. So, a plus i b ln is giving you this. What does it tell you?  So, let this be r. That means you're trying to say ln r is pi by 4 this term. So, this term is your this term. So, ln of r or ln of mod a plus i b is pi by 4. That means r is e to the power pi by 4. Correct. Second thing you're saying argument of z, that means your theta is ln of 1 by root 2. Correct. Yes or no? Now, you know r and theta of a complex number. What is the real part a r cos theta and what is the imaginary part r sin theta? So, your answer to a is e to the power pi by 4 cos of ln of 1 by root 2 and imaginary part will be e to the power pi by 4 sin of ln of 1 by root 2. Is this fine? So, the answer is not as simple as what you people thought it to be.  Is it clear? Any questions? Any questions? Any concerns here? Okay. So, with this, we end our today's discussion. There are many more things to be covered up. Okay.