 Hello, all good morning. So please type in your name All are attending the session today. So let's give two minutes for the people to join in Hello, good morning. All right, so let's start the session. I'm a audible and Is this screen visible to everybody? So complex numbers. We are going to start a very very important chapter This is the first session that we are going to have You guys in the class. I already Deep to you about what was the need for a complex number, right? so in this particular session, we are going to now talk about important concepts of Representation of complex numbers right and What are the different forms of representation of complex numbers? Why do we need it and what are the various type of operations that we can perform on complex numbers like addition subtraction multiplication division and more importantly, what is going to be the geometrical significance? So we are going to first start with the representation of complex numbers representation of complex numbers Now in representation of complex numbers, there are three types of representation that we normally follow The first being the point form representation in point form representation We represent a complex number z as a plus I be Now first, let me take some time to explain this type of representation Here is called the real part of complex number is called the real part of complex number and It is represented by the symbol RE with a bracket and Z written inside it Okay whereas this number B This is called the Imaginary part of the complex number This is called the imaginary part of the complex number written by I am with a bracket of Z Please note in reality These two complex numbers are these two numbers a and b both are real so a and b both belong to real number and Because of the presence of this I over here B gets the imaginary character So basically I is what I is a kind of a unit for a purely imaginary number Right, just like one is a unit one behaves as a unit For purely real number purely real number I behaves as a unit for Purely imaginary number. Now, what is this I will talk a bit more about this I before we move on to anything else I is basically square root of negative one Okay, so if you find the square root of negative one that expression is named as I I for the first set of the word imaginary So it clearly implies if you square it you are going to get a negative one as the answer So this is the first number that you would have come across We're squaring the number results into a negative number Okay, and this I has a very peculiar property if you multiply I again to it It becomes I cube which is going to be minus I Again if you multiply I to it it becomes I for which is going to be Square of I square which is going to be one Okay, similarly I 5 is going to give you I back again because you're multiplying I to the power 4 into I to get I to the power 5 Similarly I square is going to give you minus 1 I 7 is going to give you minus I I 8 is going to give you 1 again Right I 9 is going to give you I back I 10 is minus 1 I 11 is minus I I 12 is 1 like that a few observations that you can have over here or few properties that you can Think over here you properties of I that is very evident over here one is If you raise I to the power any even number it will either result into a 1 or a minus 1 Okay, if you raise it to a power of 4 it will always result into 1 Okay, if you raise it to a power Which when divided by 4 Leaves the remainder of 1 it will always give you I Like just like I itself I to the power 5 I to the power 9 they are all of the power 4n plus 1 type and If you always raise it to a power Which when divided by 4 leaves the remainder of 2 it will always be minus 1 Like I square I 6 I 10 and Similarly 4n plus 3 will always leave you with minus I and An important property that you will see is that when you add them When you add any 4 consecutive powers, so let's say I these are the 4 consecutive powers if you add them You are going to always get a 0 so I to the power I to the power n I to the power n plus 1 I to the power n plus 2 and I to the power n plus 3 means any 3 Convective integral powers if you add you are always going to get a 0 Yeah, and as Bharat pointed out it repeats itself after every cycle of 4 That's why this power 4n. Is that fine now based on this important property of I I Will take up some questions So let's start with this question first So before I move on to a question There's an important thing which I wanted to highlight over here What is under root of Minus 4 if I ask you this question, what will your response be type in in your chat box, right? It would be 2i because you can read this as under root of 4 Times under root of minus 1 which is actually 2 and this is I right So what formula am I am following over here? I'm following the formula of This could be written as this into this Okay, so basically I'm following under root of AB or I can say under root of A into under root of B is that actually under root of AB right this formula I'm following, okay? In a similar way, what is under root of minus 9? You would say 3i, right? Because you can write it as under root of minus 1 into under root of 9 which is I into 3 which is 3i Okay, and again here. We are following the formula of under root A into under root B is under root AB Okay Now if I ask you this formula If I ask you this answer, what is under root of 4 into under root of minus 9? Okay, so by following the same formula, whatever we have seen in the previous two cases You might like to say that your answer is going to be minus 4 into minus 9, isn't it? Yeah, for root 9 we take the principal root So you'll say your answer is under root of 36 Right and under root of 36 is 6 Now see here, what is the catch? if you realize this was 2i and this was 3i and When you multiply it actually it gives you 6i square Which is negative of 6 idly because I square is minus 1 right now these two answers They are different. They don't match They don't match Right that means somewhere this formula that we were using was not working out Isn't it? The formula that we're using so far under root of A under root of B is equal to under root of AB was not working out Isn't it? So here is the conclusion guys, which is very very important Under root of A into under root of B can be written as under root of AB if at least one of A and B is Positive or zero is I should write it like this it should be Greater than equal to zero, okay? whereas Under root of A into under root of B behaves like minus of under root mod A mod B if Both A and B are negative so please remember this It is very very useful when dealing with complex number problems That under root of A under root of B need not always be under root of AB It is only true when you have at least one of A and B as Greater than equal to zero and When both are negative it will behave as negative under root mod A mod B. Is that fine? Now let's take some questions on Properties of I is there any question from anybody regarding this? Okay, so my first question is what is the summation of I to the power r from r equal to 1 till 4007 that's my one question second question is What is summation from r equal to 0 to 100 factorial? I to sorry, that's not hundred factorial. It's hundred and Here we have I to the power r factorial Okay, just write out in two minutes, then I'll respond to it. Yeah done So I've started seeing the answers already So let's solve this. Let's solve the first one So I've started seeing the answers already. So let's solve this. Let's solve the first one So for the first one it is I to the power 1 I to the power 2 I to the power 3 I to the power 4 like that and it goes on to Uh I to the power 407 I to the power 406 4006 I to the power 4005 I to the power 4004 and so on. Okay Now what will happen in this case as we already discussed the property that Any four consecutive powers of I If you add it it is going to give you zero, right? So we are going to make use of this property So what will happen it can make Groups of four like that here also you'll have four thousand one four thousand two four thousand Two four thousand three So a group of four four if you make it is going to give you all zero zero zero, right So only terms that are going to Give you some result is the last three terms four thousand five four thousand six and four thousand seven And as you already discussed four thousand five is going to give you I Because when you divide this power it leaves the remainder of one 4006 is going to give you I square And this is going to give you I cube So I minus one plus I Sorry minus I Is going to be the result for the first one. So that's minus one. So many of you have answered this with minus one That's correct. What about B B I've only got the answer from Ashutosh I plus 95 to I plus 95 others So when you say summation of I to the power r factorial from r equal to zero to 100 you basically write I to the power zero factorial one factorial two factorial three factorial four factorial five factorial and so on Till you reach hundred factorial Okay now I to the power zero factorial is I to the power one And this is also I to the power one This is I to the power two I to the power six I to the power 24 now the catch is the moment you reach four factorial You realize that Whatever power you get will always be a multiple of four because four has made its appearance now So whatever power you write next Okay till hundred factorial they will all be multiples of four. This is multiple of four. This is multiple of four All these are multiple of four So as a result from the property of I will start getting one one one one for all these answers Okay And before that I will get two I minus one and this is also minus one So altogether how many eyes I would have written how many ones I would have written over here From four to hundred we would have written 97 ones And this is two I minus two So that will give me 95 plus two I so correct Ashutosh Devdas Omkar you're all correct So is this fine how it how the property of I helps you to solve the problem Now we'll come back to the point form representation A plus IB Now in Now in point form representation We represent the complex number like a point on a plane which we call as the argon plane So there is a plane that we draw Whose axes are basically taken as The real z axis And the imaginary z axis Point form representation says you can represent this complex number just like a point A comma B Like how you do in Cartesian coordinates Okay, where A represents the The real part and B represents the imaginary part of the complex number I am again repeating A and B both are actually real in nature for example if I say z is equal to 3 plus 4 I So A is 3 B is 4 so 4 is called the imaginary part and 3 is called the real part However in reality both 3 and 4 are Are real In order to express it in physical world or real world We have to take a real as A and B And we make it imaginary by putting this I next to it. So I is like a you can say A separator of real with the imaginary part Now Few things that we need to note over here is that When you say a complex number is written as a point A comma B It starts behaving or starts taking all the properties that you have learned in coordinate geometry Okay So many of the coordinate geometry fundamentals can be used on the complex number if it is written in If it is written in the expression Just give me one second Many of the complex number. Yeah, many of the complex number properties can be solved by the use of coordinate geometry Guys, am I audible to all Another important concept that is related to a complex number is its modulus So modulus and argument are the two concepts which I am going to discuss next modulus and argument of complex number So let us all understand from this diagram itself. What is the meaning of modulus? modulus is nothing but The distance of the complex number From the origin Okay, this is basically your origin And if you connect this origin to this point A comma B Then this is what we call as the modulus of the complex number Okay So modulus is nothing but The distance of the complex number From origin Okay, it is represented by the symbol Mod Z just like we represent mod in our Equations in real numbers. So this is represented by this Okay Now if I ask you In the present diagram if this is your origin What is going to be the expression for mod Z? So you'll say it's very simple It's the distance of A comma B from zero zero. So use the distance formula So this becomes under root of A square plus B square Okay, so any complex number Z Which is written as A plus BI or A plus IB The modulus of the complex number will be under root of square of A plus square of B And geometrically it represents the distance of the complex number from the origin Is this concept clear? So if I say what is the modulus of 3 plus 4 I What will your answer be? What will your answer be? You would say under root of 3 square plus 4 square which is 5 Is that okay? Now What is the concept of argument? This is a very very important concept. All of you, please listen to this very very carefully. Okay Argument is basically the angle which this line will make with the real z axis Okay, so just like r was representing the modulus Theta would represent your argument Right Now, how do I find this argument? That's a very very important concept which we are going to take next So I'll quickly replicate the Diagram over here Now when you're talking about argument of a complex number We'll talk about two types of arguments One is called The general argument and other would be called as the principal argument However principal argument is the most commonly used. This is more important than the general argument And the word argument is sometimes replaced with the word amplitude. They also sometimes call amplitude Amplitude of a complex number is same as the argument of a complex number Now, what is the general argument as I already discussed with you? In order to represent this angle theta You may choose to write this theta in the interval 0 to 2 pi Correct Then by stating that you are actually stating its general argument. Let me give you some examples If I say your complex number is 1 plus i Right 1 plus i is like a point 1 comma 1 On the argon plane I will come to it Gaurav in some time. Don't worry So 1 plus i if I ask you what is the general argument So you'll see you'll say that this angle is going to be 45 degree, right? So for this your general argument written as arg Is going to be pi by 4 Or 45 degree, okay Normally arg is a short form for writing argument And if I say A complex number minus 1 plus i So minus 1 plus i is somewhere over here That is like minus 1 comma 1 Right, so what is the argument for this complex number You would state this angle From here to here and what is that This angle is going to be 3 pi by 4 Okay If I ask you what is argument of minus 1 minus i So what will you state? In that case your complex number will come over here minus 1 comma minus 1 Right, and if you connect it From origin to this point Then this angle is going to be your general argument. That is pi plus pi by 4 Isn't it? That's going to be how much 5 pi by 4 And in a similar way if I ask you what is the argument of 1 minus i That means your complex number is now here. That is 1 comma minus 1 And when you connect it with the origin This is the angle that you will be stating That is 2 pi minus pi by 4 That is 7 pi by 4 Right, so all your angles would essentially be lying between 0 to pi by 4 in general argument Is that clear? Principal argument is basically more important and it says that Instead of representing your angle between 0 to 2 pi If you represent your angle between Minus pi to pi Minus pi not included pi included Okay, then this would be called as the principal argument Now principal argument is actually defined as Principal argument is actually defined as The shortest angle between The positive real z axis and the line Connecting Z to the origin Okay, so it is essentially the shortest angle that we will be citing over here Okay, for example If I tell you what is the principal argument for Minus 1 comma minus 1 Okay, so let's connect it to the origin Let's connect this to the origin And starting with the real positive real z axis Which is the shortest suit will you take? Is this the shortest suit? Or is this the shortest suit is the white one the shortest suit or the yellow one the shortest suit? Yeah, Bharat general is positive only But principal can be negative also. That's what I'm going to explain next No problem Sameer The white one, right? So white one is going to be your shortest angle now while covering the white one in which direction we are going We are going in clockwise sense And clockwise angles are always negative Are you getting it? So you'll say Principal argument which we normally write with capital ARG Now this is a convention which is followed by international books National books they will write all the arguments with ARG With small ARG But normally capital A is used when you are representing principal argument and small A is used when you are representing general argument But since everywhere they like using principal argument they make no distinction between the notations So in this case your answer will be minus 3 pi by 4 Are you getting it? So You have to go along the shortest path and if that shortest path is along clockwise your answer would be negative angle Right, which implies if you measure along clockwise you will get a negative answer And if you measure in a anti-clockwise your answer would be a positive answer. That's why minus pi to pi interval is given Now why this minus pi is excluded and pi is included is because of A confusion that may arise because of purely negative numbers for example, if I say minus 4 comma 0 So let's say my z is minus 4 plus 0 i I ask you what is the principal argument for this complex number There may be a confusion because this is also the shortest path And this is also the shortest path So if you take this path your answer will be pi If you take this path your answer will be Sorry, if you take this path your answer will be Pi and this path is minus pi So which will be the answer So the convention says that minus pi is not included. So this cannot be your answer. So answer will be pi only So your answer will be pi only So for completely negative real numbers your argument will be pi In the same way for completely positive or you say purely Real positive numbers. Let's say 4 comma 0 Your argument principal argument would be zero This would be zero Okay Now tell me I'll ask you a few questions quick questions All of you please type this on your screen. What is principal argument of Minus half plus i root 3 by 2 Please type in your in your screen. What is the principal argument for minus half plus i root 3 by 2? Give your answer in radiance Bharat are you sure? Bharat Ashutosh Sukirt Sukirt and Niranjan. Okay just joined Yeah, now it's correct guys again make no mistake See when you say minus half plus i root 3 by 2 if you make it on the argon pane Where does it lie? It lies In the second quadrant right minus half comma root 3 by 2 it lies in the second quadrant. Okay Now when you connect it to the origin Which is the shortest suit shortest suit will be this direction Correct. So what is this angle? I have to find out It's very obvious that this angle here would be 60 degree pi by 6 sorry pi by 3 Okay Right you can do root 3 by 2 divided by Mod minus half this will give you pi by 3. So this angle becomes pi minus pi by 3, which is 2 pi by 3 So the answer for the principal argument of this is 2 pi by 3 Okay Let me ask you another question What is principal argument of Minus 1 minus i root 3 Yeah, sure sure no problem Sukirt Yeah, please type in your answer for this Minus 2 pi by 3 is correct. So this is going to be minus 2 pi by 3 Now just like this case Your uh minus 1 minus i root 3 will lie in the third quadrant. So this is minus 1 minus root 3 Okay, so if you connect it with the origin the shortest route is going to be this route And just like the previous case this is going to be pi by 3 So this magnitude y is going to be 2 pi by 3 But a negative sign is attached because you are measuring it in a in a clockwise direction Is that fine? How it works now? Okay, now I'll tell you an algorithm to find The principal argument that we normally follow So if you're given any complex number A plus ib Okay Let's say this is given to you The first step that we follow is We find an angle alpha Which is tan inverse of mod b by mod a Now this is the mod that we use in real terms Both mods are actually the same If you see mod of a complex number and mod of a real number are both the same Because if the real if the complex number becomes purely real Under root of a square plus b square is going to give you mod a only So when you find this Then the second step is We follow this schematic diagram We follow this schematic diagram This diagram says if your complex number is in the first quadrant that means a and b both are positive Okay Then your principal argument would be nothing but the angle alpha that you have found out from step number one if your a is negative and b is positive that means you are in the second quadrant Your principal argument is going to be pi minus alpha Okay If you are in the third quadrant means a is negative b is negative Your principal argument will be alpha minus pi and if you are in the fourth quadrant where a is positive and b is negative Your principal argument will be minus of alpha so this alpha Is uh given different different structures depending upon In which quadrant Your point a comma b will fall Are you getting this point? So if you take this example if you take this example If you solve it by this algorithm you Do it tan inverse mod of minus root three by mod of minus one That's going to give you tan inverse of root three Right that is going to be pi by three Now since it lies in the third quadrant your answer will become alpha minus pi So this one will become your principal argument alpha minus pi which is As all of you rightly mentioned It becomes minus two pi by three Is that clear? How do we find the principal argument through this algorithm? Okay, now we'll take more exercise on it after we learn After we learn something called The polar form representation So let's go on to the polar form representation Of complex numbers. That's what uh gora was talking sometime back Now we all know polar coordinates polar coordinate is basically was invented by newton And uh, it is most useful form of representation of position of a point when it comes to higher mathematics Uh, normally in school level we deal with cartesian coordinates which was given by the french mathematician veni di cata And newton was attributed for Coming up with another type of coordinate system where instead of representing it as distance from two perpendicular lines which we call as the axis is We represent it by Two parameters one is the distance from the origin which we call as the pole So this is what we call as the pole And by representing this angle theta So if the same coordinate If you write it in This form where you represent its distance from the pole and the angle made by the line connecting this point p with the Pole that is the theta Then we are mentioning the polar coordinates. These are called the polar coordinates Okay, this is called the cartesian coordinates Okay Now in polar coordinates How can we use polar coordinates to represent A complex number. It's very simple. We have already learned. What is r, right? If you recall r is nothing but mod of z Right and theta is nothing but argument of z correct If you apply your basic trigonometry, you would realize that Your a is nothing but r cos theta And your b is nothing but r sin theta right So when you start representing it in a point form like this, so instead of writing it like this you could write it in this form correct Where you can pull out a r common and represent it like this Okay, and this is the form that we know is called the Polar form of a complex number Also called as the trigonometric form Also called as the vector form Okay, why it is called a vector form just like in a vector We have the modulus or magnitude of a vector and the direction of a vector in the same way Complex number is also a kind of a position vector Okay Now this property makes uh it very very Uh, you know a useful topic where we can actually apply the concept of vectors also to solve You know questions Later you learn how do we do dot product on complex numbers and cross product on complex numbers also And that has wide wide applications in solving many problems in complex numbers Okay So you could treat this as if you are writing a complex number z as mod z Cos of its argument Plus i sign of its argument Okay, so this remember this So if I ask you a polar form representation, all you need to do is find its modulus and find the principal argument and Exhibit it in this form Okay, so let me ask you this question write polar form for complex number minus half minus i by 2 Yeah, I'll come to that Bharat. That is called the Euler's form. I'll come to that So meanwhile, please give me the answer for this complex number in Polar form. This is the point form Tell me the same complex number in polar form So first you'll find its modulus Right What are the modulus of this complex number? So modulus is going to be under root of half square plus half square Isn't it Which is root 2 times 1 by 2 Which is 1 by root 2 Okay So 1 by root 2 is its modulus And principal argument for this is going to be Where does it lie on the complex argon plane it lies here right minus half minus half So the shortest angle is this angle which is going to be minus of 3 pi by 4 Okay So your answer will be my 1 by root 2 cos minus 3 pi by 4 Plus i sin minus 3 pi by 4 Right, you can also write it as 1 by root 2 Cos 3 pi by 4 minus i sin 3 pi by 4 So this is going to be a Polar form representation for this complex number Is it okay guys you focus on your answer? Please do not look at what others are typing in Because if you look at what others are typing in you will get distracted Okay, again a simple question uh write polar form for Let's say very simple complex number i Next write it for uh, let's say Minus 3 i Let's write it for let's say minus 10 Let's write it for let's say minus half plus i root 3 by 2 Later on will take much complicated examples right now just to begin with i'm taking very simple examples so that You all are aware how to find or how to write Polar form representation for well-known complex numbers So for the first one Yeah, what is your question mod z? Yes many a times, uh, we say It is r sys theta You can mention it like this r sys theta. I can understand this is equivalent to saying r cos theta plus i sin theta This is a short form This is a shortcut which we use to represent a polar form. So if you're typing it on your chat box Uh, if you can write it like r sys with an angle, I will understand that what is your answer So for the first one, what is the answer the answer is going to be This can be represented as 1 cos Pi by 2 plus i sin pi by 2 See i is basically lying on the imaginary axis i means 0 comma 1 So this distance is 1 so modulus is 1 this angle is the shortest angle, which is pi by 2 So if you write it it's 1 sys pi by 2 Now if you want to find it's using formula what happens is When you do tan inverse mod b by mod a you end up getting this Right, which is actually pi by 2 so now It lies neither in the first quadrant nor in the second quadrant But irrespective of whichever formula you use this is the formula normally we follow Irrespective of whichever formula you use out of alpha and pi minus alpha It will always give you pi by 2 as the answer because alpha is pi by 2 And pi minus alpha is also pi by 2 for the second one. What is the answer? 3 cos Minus pi by 2 plus i sin minus pi by 2. Please check your answer For the third one it will be 10 Cos pi plus i sin pi Please note r will always be positive Right r represents the distance from the origin which is the modulus that will always be positive Last one last one Yeah, correct Krishna, you're correct ashish. That's correct for the first one Last one. What is going to be the answer? Please type in Sys 2 pi by 3 that's absolutely correct. So that's going to be 1 Cos 2 pi by 3 plus i sin 2 pi by 3 This is a very important complex number for us which is called omega Which is one of the complex cube roots of unity. We'll talk about it later on in this course Meanwhile guys, it is clear to all of you. What is how do we write polar form to write the polar form? You need to be accurate in finding the modulus and the principal argument Is it clear? Please type in clr on your screen if it is clear to you All right, that's great That's great Now next we are going to talk about the third form of representation of complex number which is called the Euler's form now This form is named after a famous swiss mathematician called Leonardo Euler who later on became blind because of some eye problem, but despite being blind, this fellow came up with 900 publications in mathematics Imagine his Contribution and commitment level that despite being blind. He came up with 900 publication in maths So he's one of the famous mathematician and his work in maths Is very much seen in complex numbers and differential equations Okay Now Euler said that when you represent a complex number in polar form, which is actually r cos theta plus i sin theta This could also be represented as r e to the power i theta Okay So this form is called the Euler's form representation of complex number Now first of all, I would like to prove this. How did he how did he possibly say that these two are same These two are actually the same things Okay So I will tell you two methods to prove it One is a calculus method Now when I saying these two are same that means I have to prove that cos theta plus i sin theta is actually e to the power i theta Okay. Yeah, we can get it from Taylor series as well Now I'll show you a very different way So let us say Let us assume that Z is cos theta plus i sin theta Okay So d z by d theta would be what it would be minus sin theta plus i cos theta Take i common You will get cos theta Plus i sin theta Because i when it comes from this cos theta will remain and this minus one is like i square, right So if you take i common from minus one, it will become i itself. Okay So d z by d theta you can again write it as i into z because this is back to z This term is back to z Okay Now here we get something called A differential equation which Euler was very good at Euler made very remarkable contribution to the field of differential equations Okay So what did Euler do? He separated the variables out like this So z he brought below d z and d theta he brought up Then he integrated both the sides I'm not sure whether you have learned integration of one by x with respect to x so far But the answer for this is ln z log to the base e z And this is i theta plus a c. Okay now In order to find the c Euler said that When z is one Theta will be equal to zero Right because when you put theta as zero z will automatically become one So he substituted this value over here So this is zero. This is zero plus c. So he got c as zero So it becomes ln z is equal to i theta So z becomes e to the power i theta. So finally He came to the expression that he came to the conclusion that Cos theta plus i sin theta could be written as e to the power i theta So instead of representing r into cos theta plus i sin theta You could represent it as r e to the power i theta Is this method understood to to all Please ask me if you have any questions on the chat box If it is clear i'm going to move on to the next method which uh Bharat discuss about Taylor series method Okay, in fact Taylor Maclaurin series method Now what is a Taylor series method Taylor series method basically you need to recall that There are a few expansions that you should all know I've already done this in The chapter limits with you So e to the power x By Taylor Maclaurin series. We all are aware of this expansion We are all aware of expansion of cos theta or cos x Okay We are all aware of the expansion of sine x Correct these expansions should be well known to all of you Now I will try to prove the required Euler form representation through the use of this formula So Euler claimed that e to the power i theta was actually cos theta plus i sin theta, right? Let's start with the left hand side, which is e to the power i theta So if you use this expansion You replace your x with i theta So you will get this i theta square by two factorial i theta cube by three factorial i theta four by four factorial i theta to the power five by five factorial i theta to the power six by six factorial and so on Okay If you group up terms here You will realize that one will contain i and one will not be with one will be without i for example This is without i This is with i this will again be without i because i square is minus one This will be again with i Again the fourth this term would be without ni so basically keep on writing the terms And you realize that alternate terms would have One will not have i one will have i Okay, it'll keep on going like this So let me go on to the next screen We have already covered the screen So i'll rewrite that expression once again. So e to the power i theta will be one plus i theta minus theta square by two factorial minus i theta cube by three factorial plus theta to the power four by four factorial Okay Plus i theta to the power five by five factorial and so on Now if you start grouping up terms Without an i you will write it as one minus theta square by two factorial plus theta to the power four by four factorial minus theta to the power six by six factorial and so on And the terms with i would be theta minus theta cube by three factorial plus theta to the power five by five factorial and so on Now you would realize that this expansion is actually the expansion of cos of theta And this expansion is basically the expansion of sine of theta With a i over here. So this is what Euler found out through the expansions Okay, hence hence You could represent r cos theta plus i sine theta as r e to the power i theta Now every form that you see has a special significance whether it's point form So the use of the various forms point form you use When you require addition right addition or subtraction Or when you want to apply the concept of Coordinate geometry to solve complex number problems Okay Polar form and Euler form Basically are used when you want to do the concept of multiplication and division And when you want to apply the concept of rotation of vectors So these concepts can be easily be applied and realized or appreciated When you look at the complex number in these forms So the reason why we came up with so many forms was It enables us To achieve various objectives like if you want to apply coordinate geometry formula, we'll use point form If you want to apply the concept of vectors Like the concept of rotation, etc. We'll use the polar form or the Euler form representation of complex numbers Is this clear any question so far up till what has been done so far Which expansion? Oh, I'm so sorry. So it's it's alternate plus minus. I'm sorry. So, uh, this is going to be Okay, okay just Slip of the pen Yeah, it's going to be one minus x square by two factorial plus x four by four factorial minus x six by six factorial And so on Thank you for the correction So it's always the even number terms that we write here. We always write the odd number terms Thank you, Ashutosh for correcting that So guys, let's move on now to the operation on complex numbers operation on complex numbers So just like when we learn numbers After that we start adding them subtracting them multiplying them dividing them taking this root of it all types of operations In a similar way in complex number also will learn all type of operations on complex numbers starting with starting with the concept of comparison of complex numbers comparison of complex numbers Now note that when you say A complex number a plus ib is equal to c plus id It is like you are comparing an ordered pair So this is an ordered pair. You would have learned this in your sets chapter, right ordered pair So when two ordered pairs are compared it can only happen when a is equal to c and b is equal to d That's how two complex numbers can be equal okay however a very important thing is You cannot Apply any inequality sign between two complex numbers. For example, I cannot say This complex number is greater or less than or greater than equal to or less than equal to These operations cannot happen These operations cannot happen on complex numbers Because it is absurd You cannot say two plus three i is greater than one plus two i You cannot say that because It's like you're comparing a Coordinate with another coordinate. You're saying the coordinate is greater than another coordinate No, such is such is absurd Yeah, you can say mod of two plus three i is greater than mod of one plus two i because now you are comparing two real numbers Okay, but you cannot put any kind of inequality sign between two complex numbers Is that clear any question with respect to this to this So if I give you a question x minus three plus i y minus two is equal to six plus i Then what is your x and what is your y it becomes a simple question for you to solve type in the answer quickly x will become what y will become what So always Compare real with the real imaginary with the imaginary. So since there's nothing here it is one So x minus three will be six. So x is nine And y minus two will be one. So y will be three right This is simple moving on to the most important concept that is the concept of addition of complex numbers So when we say let's say we have two complex numbers z one and z two Z one is a one plus i b one and z two is a two plus i b two When we add two complex numbers We normally add their real parts together And we add their imaginary parts together In any kind of addition and subtraction operation like terms are always combined with each other You can always add apples to apples. You can always add mangoes to mangoes. You cannot add apples and mangoes, right? Is this clear? Now this doesn't have any meaning for me till I understand it geometrically So I would like you to understand the geometrical interpretation of addition geometrical interpretation of addition This is most important Because if you understand this You will be able to solve many complex number problems just by realizing this geometry Okay, so Let us make an argon plane first imaginary axis. This is my real axis. So if I say there's a complex number z one Right, if you mention it like a coordinate it will be a one comma b one And there is another complex number z two Right, which is nothing but a two comma b two. Can you tell me where does this complex number lie? So where does a one Plus a two comma b one plus b two lie Can anybody tell me the diagrammatic representation or at least type in what do I have to What figure I have to make in order to get to the position of z one plus z two So please help me to locate Please help me to locate z one plus z two Anybody having any clue? Please type it on their screen Exactly exactly very good So guys in order to solve this I will make a Parallelogram Okay now I will not explain in some time how this parallelogram is helping me. Okay So let's say this is complex number zero comma zero By the way, I didn't ask this question. What is the argument of zero plus i zero We have all done arguments of various numbers, right? So what is the argument of zero plus i zero? Can you type it in your on your screen? No shawan it is not zero It is undefined Argument for zero plus i zero is undefined. So that's a complex number whose argument is not defined Okay Its modulus is zero, but argument is not defined It is like asking you what is the degree of the polynomial zero? It's undefined Yeah, it's not defined All right now since this is a parallelogram Since this is a parallelogram We know that in a parallelogram The diagonals bisect each other The diagonals They bisect each other that is This length is equal to this length and this length is equal to this length. Okay. Let me name the points o ab and c Okay, and let's say the middle point is m here correct now if I ask you What is the coordinate of m? You'll say very simple. It is the midpoint of a and b point That is going to be a one plus a two by two comma b one plus b two by two correct That is when you use a m as the midpoint of ab so using midpoint of ab you answered this question Okay, if you use midpoint of OC If you use midpoint of oc Now c is something which we don't know yet. So I'll assume it to be some c comma d Okay, you'll say the midpoint of oc is going to be zero plus c by two and zero plus d by two, right? Okay, now since both of them represent the same thing I can compare x coordinate with x coordinate and y coordinate with y coordinate So if I compare these two and these two You'd realize that c by two becomes a one plus a two by two which means c is a one plus a two In a similar way d is b one plus b two That means this point was actually a one plus a two comma b one plus b two Isn't this what we were trying to locate Right, that means your point c starts behaving as z one plus z two Very much like how vectors are added up. That's why guys, it's it's very important for you to appreciate that Many a times we will use vector concepts to solve complex number problems Okay, especially when it comes to higher concepts We'll be using vectors to solve complex number problems Is this clear to you any question regarding where is z one plus z two located? Okay So What is the modulus of z one and z two over here modulus of z one plus z two is going to be This length o to c So the length of the diagonal oc will be your modulus of z one plus z two What will be the argument of z one plus z two? It's the angle made by this. This is your argument of Z one plus z two Okay, is that okay now having done this They're two important inequalities that come out of this result. So I'll again quickly represent the diagram here So this is zero. This is z one. This is z two This is z one plus z two Okay, it's a point. It's b point and it's the c point Now there are two important inequalities Or any two important inequality properties that we get from this particular diagram Since we know that this length is mod z one Right, we know this is mod z two This will also be mod z two This will be mod z one plus z two Okay We can say that By looking at the triangle oac By looking at the triangle oac. I can say oa Plus ac will always be Greater than equal to oc Right. This is a triangle property, right This is a triangle inequality Okay So in the same way if you write that in complex numbers, I can say mod z one plus mod z two Will always be greater than equal to mod z one mod of z one plus z two Now you would say when will the equality be true? When will the equality exist in case of a triangle when the triangle will collapse? Isn't it when this angle becomes 180 degree When this angle becomes 180 degree, you would realize that equality will start holding true, isn't it? Yes or no so now You tell me in this property When does the equality hold true? when does The equality hold true Anybody any answer when does the equality hold true? What should be the relation between z one and z two? Whether with respect to modulus or whether it's with respect to argument That z one plus z two modulus becomes equal to mod z one plus mod z two Yeah, when they are along the same line, how do you represent that concept that they are along the same line? There should be a mathematical way of saying that right Bharat anybody when you say that modulus of z one Plus modulus of z two is equal to modulus of z one plus z two Ashish is saying z one and z two both should be zero not necessarily Ashish So okay now the answer to this is As you rightly said in a diagrammatical way that they should be in the same line Which is actually mentioned by the fact that argument of z one should be equal to argument of z two If this is true, then only we can say That mod z one plus mod z two will be equal to mod z one plus z two Yes, their argument should be equal absolutely correct Absolutely correct Great second inequality comes from this only That the third side Will always be greater than the difference of the other two sides This is also a triangle property So here, what is the property that the third side is always Third side is smaller than is smaller than The sum of the other two sides right other two sides Whereas this is coming from the property that the third side Is always greater than than the difference Of other two sides Lot of questions are framed on these inequalities. So it's very important that you know them in and out Okay Now again my question here is When does The equality Hold true Now you may choose it choose to answer it now or little later on if you can answer it now I will be very happy, but even if you're not able to answer it now Don't worry. There is some concept which I'll cover after that. You will definitely be able to answer this Is anybody who wants to take a try When will mod z one minus z two be equal to Mod of z one plus z two when will this equal to this Anybody wants to take a try on this? No shawen that is not correct but anyways I'm not expecting you to answer it right now. We'll come back to this. Okay, so we'll Come back to this okay so With this i'm sure you are able to understand in and out of what is addition operation on complex number and how Vectors are actually similar to complex number in this regard Next we are going to talk about Subtraction of complex numbers What do we do is we take a break over here Because it's already one and a half hours of class. So I would expect you, you know, let's have a break And we'll resume in 10 minutes time Okay, let's have a break here. Okay, we'll resume exactly at 10 40 right now. It's 10 32 So we'll resume at 10 40 am Yeah, shushan, you are correct. The difference of the argument should be pi. I'll come to that concept little later on So all of you get a cup of coffee tea ring water Okay, and we'll leave you back at 10 40 All right, so let's start with the process of subtraction. No Bharat. It doesn't hold true when they're at right angles We'll discuss that soon after discussing subtraction with you So, let's say again, we have two complex numbers a1 plus ib1 and a2 plus ib2 and when you subtract them It becomes a1 minus a2 plus ib1 minus b2 And again, this doesn't make any sense to me unless I understand geometrically what is happening When you subtract Two complex numbers. So we'll have geometrical interpretation of subtraction again, let us make a argon pane with z1 vector over here which is a1 plus a1 comma b1 And another vector z2 another complex number z2. Sorry. This is a2 comma b2 Okay Now when we say we are subtracting these two complex numbers treat this as if we are doing Addition of z1 with negative of z2. Okay Now what is negative of z2? Negative of z2 will be a complex number exactly mirror image about origin So if this is This is z2 Then z1 will be exactly At some position like this So this will be the position of negative z2 Isn't it? So it's mirror image about origin And this is your z1 correct Now if I ask you you have to add z1 with Negative z2 then you'll say very simple. I will complete a parallelogram with these two as the adjacent sides Right, so let us complete a parallelogram So when you complete a parallelogram, uh, this is what you will see There is a point that we are getting over here this point Basically, I'm trying to show that now this point will be your z1 minus z2 Okay, in fact, I don't need to show it again because we have already studied that when you're connecting When we are connecting This diagonal From here to here Okay, this point is supposed to be acting as my z1 Plus minus z2 which is nothing but z1 minus z2 point So let me name it o Let's say this is b This is c. This is d Then your d point is acting like your z1 minus z2 point Is that clear? So if I ask you what is the modulus of Z1 minus z2 you will say the distance od will be your modulus of z1 minus z2 Now I'll show you something very interesting over here If you complete this parallelogram If you complete this parallelogram o a b and let's say This is point e Okay, yeah, yeah, shiram. No problem. Okay. Now you would realize one thing here that If you connect this to this a to b The length of a b would be the same as length of od. Isn't it a b and od will be of the same length Right, so these two are of the same length So what what I'm trying to say is that you don't have to make a separate parallelogram o a dc you can actually work with o a e b That is a parallelogram we made over here Let me go back. Do you see this so basically this line a b This line a b over here This line a b over here is actually representing what? z1 minus z2 mod This line was representing z1 plus z2 mod or mod of z1 plus z2 So the idea here that I want to convey is that when you complete a parallelogram One of the diagonals One of the diagonals which is actually in this case I'm showing it with yellow color this yellow color diagonal is mod of z1 plus z2 while your blue color diagonal Is actually mod of z1 minus z2 Is that idea clear Is the idea clear the yellow diagonal Yellow diagonal length Yellow diagonal length represents mod z1 plus z2 Whereas the blue diagonal length blue diagonal length Will represent mod of z1 minus z2. Is that fine? Now next thing that we'd like to discuss over here. This is the modulus that is fine. What is the argument? This is going to be the argument Isn't it So argument of z1 minus z2 would be Angle theta for you and you would realize that If you extend this blue line If you extend this blue line forward like this Okay, this will be the same angle theta Yes or no So you don't need a separate parallelogram For finding the length of z1 minus z2 Neither do you need a separate diagram to find the argument of z1 minus z2 You could have done everything on the on the parallelogram Are you getting it now Many times student asked me sir if this is the argument of z1 minus z2 What is this angle whose argument is this angle? Let's say phi Can anybody answer me Whose argument is phi guys Exactly exactly. No ashutosh Aunkar is correct z2 minus z1 z2 minus z1 That's correct. That's correct. Now see guys There is an important comment which I want to give over here, which is going to be very very helpful When you want To perceive or realize z1 minus z2 Okay Imagine as if Imagine as if There is a vector Whose direction is whose direction is From z2 to z1 Okay, so when I say z1 minus z2, I will imagine this as a vector like this Are you getting this point? Now if I give you a vector like this And if I give you a x y axis and I ask you tell me the angle that it makes With the x axis. What will you say? How do you find the angle of this vector? So you would probably extend this You'll extend this and you'll give me this angle as your answer, right? Right, so angle with the positive direction of x axis will be the angle between These two right lines, isn't it? Yeah, hello Nathan. Are you getting it? So when you say z1 minus z2 imagine it as if you have a vector Whose direction is from z2 to z1 And you want to find the direction of this vector With the x axis Remember you would have learned vectors in your physics as well If you have two vectors like this This is the angle between them, right? For finding the angle between two vectors they must be coming out from a single point Okay, if I ask you what is the angle between this vector and this vector Then you have to make it first like this And then find this angle out You will never say this angle is your answer Right remember this concept in case of In case of vectors The angle is the smallest angle when the vectors are coming out from a single point In a similar way when you say z2 minus z1 Imagine it to be a vector imagine it as a vector Whose direction is from? Whose direction is from? z1 to z2 Okay, so in this case it would be in this direction I am choosing a different color to show that So yellow color direction is the direction of z2 minus z1 So if I ask you what is the angle made by this vector with the x axis What will you say? What will you say? You will pull this backwards and you will say this is the angle So similarly phi is the angle between A phi is the argument of z2 minus z1 Is that okay? Is this understood? If not, please ask me question right now because this is going to be very very important Is it clear? Type clear on the screen Got it? Great Now even in this case we will have some important inequalities coming up So let me just quickly again draw the argon diagram Okay So let's say this is origin This is z1 This is z2 Okay This will be z1 plus z2. That's for sure What is going to be this length? This length is going to be Mod of z1 minus z2. I'm just bothered about length. So I'm not making any arrow on it Okay Now important Important Inequality properties So two of them we have already seen. Now we are going to see two more So we all know that this length is going to be mod z1. This length is going to be mod z2 So first in fact, I should say third Inequality property that we are going to discuss over here Mod of z1 minus z2 will always be less than equal to Mod z1 plus mod z2 Okay It's obvious third side of a triangle should always be lesser than the sum of the other two sides So this is ab is acting as the third side So focus on triangle oab Okay, so ab is lesser than the sum of oa and ob Okay In a similar way, I can say mod of z1 minus z2 will always be greater than equal to the difference of the Other two sides So guys geometrically you should understand when we say When we say mod of z1 minus z2, what does it mean from here? What is the geometrical meaning of this? The geometrical meaning of this is the distance between the points Between the points z1 and z2 That is the meaning of That is the meaning of mod of z1 minus z2 Now some people ask me how I already showed you through the diagram that this length is basically the distance between z1 and z2 Now mathematically also it can be shown if you treat z1 as x1 plus i y1 And z2 as x2 plus i y2 Then what will be your z1 minus z2? It will be simply x1 minus x2 plus i y1 minus y2 Is it and if I say what is mod of that? You are going to say x1 minus x2 square plus y1 minus y2 square which is actually the distance formula Distance formula, which you have learned in your coordinate geometry, isn't it? So that makes it all the more clear that Mod of z1 minus z2 represents the distance between z1 and z2 A special case of it is a special case is mod z If you perceive mod z as mod z minus 0 Then we know that it represents the distance from the origin Isn't it? So what is mod z? If you go back to the definition of mod z and specifically said that it represents the distance of z from the origin So if you perceive it as if you are writing mod z minus 0, it is the distance of z from origin Correct and as a vector you can see as a vector As a vector whose direction is from 0 to z Direction is from 0 to z This will be helpful in finding the argument of the complex number. Is that clear? Does this make sense? What is the geometrical meaning of mod z1 minus z2 because I'm going to ask you a lot of questions Now question number one When will the inequality sign be true? When will mod z1 minus z2 equal to mod z1 plus mod z2? Please type in in the chat box When will be mod z1 minus z2 equal to Mod of z1 Plus mod of z2 guys. I'm waiting for your answer. Okay Shavan is saying argument z1 should be negative of argument z2 No Shavan whatever you are saying that that means z1 and z2 are conjugates of each other which I have not discussed so far I'll come to it. Niranjan is saying angle between them should be 180 degree. Okay Omkar is saying z1 should be equal to z2 See Omkar when you say z1 is equal to z2 mod z1 automatically becomes equal to mod z2 anybody else Kushi, Devdas, Nitin, Ved, what do you think? When will the equality hold true? When will this be holding true? See try to imagine this Try to imagine this You have z1 over here, right? You have z1 over here, correct? Yes or no, and let's say you have z2 right diagonally opposite to it That means Here okay, isn't this length Isn't this length mod z1 and this length mod z2 Right and isn't the entire length mod of z1 minus z2 Correct So do you realize that when they are exactly diagonally opposite to each other? Or in other words the difference of their arguments is pi You would realize that mod of z1 minus z2 becomes equal to mod z1 per z2 So here the answer will be argument of z1 Right minus or you can say difference Of argument of z2 should be pi Many books will write it like this mod of argument z1 Minus argument z2 would should be equal to pi Bharat, is it clear? So Niranjan was correct in this case the angle between those Two complex number lines should be 180 degree That means the difference between the the argument should be 180 degree Is it clear Bharat? Ashutosh Yeah, now tell me by similar thinking or similar logic when will mod z1 minus z2 be equal to The difference in their modulus Now this outside mod is just to make it positive because you don't want it to be equated to a negative quantity. That's why When will this hold true? Right, they'll hold true when their arguments When their arguments are equal Okay Or you can say the difference of their arguments is an even multiple of pi But since we are dealing with principal argument we'll say argument of z1 should be equal to argument of z2 Now we'll go back to the stage where I said I'll come back to the question So, uh, I think it was in slide number 15 Yeah Now can you tell the answer for this? when will mod of z1 minus mod z2 equal to mod of z1 plus z2 now treat this as z1 minus of minus z2 Again, where does 90 degree come from here? Ashutosh Orthogonal is same as 90 degree. How does 90 degree feature in over here? Okay. Let's see this Let's say z1 is here Okay And let's say z2 is diagonally opposite z2 is diagonally opposite to it Let's say here Okay This is z2 Correct Then where will be minus z2 minus z2 will be here then This will be minus z2 correct Now if I say what is the distance between these two you would say mod z1 minus Minus z2 correct, which is nothing but mod z1 plus z2 Isn't it? So this is the distance Right, isn't it also the difference of isn't this distance also the difference of mod z2 and mod z1 So that's what it's trying to say That's what it's trying to say So when is the difference of the distance of z1 and z2 equal to the distance between z1 and minus z2 so read this as distance between z1 and minus z2 When does it become equal to The difference of Distance of z1 from the origin and distance of z2 from the origin Only under this condition So here the answer will be They this inequality will satisfy this equality will satisfy when argument of z1 minus argument of z2 should be So the difference between them should be pi Now it is clear to all of you Especially Bharat Ashutosh who was continuously saying orthogonal and 90 degrees does that make sense to you? We'll come back to many of these questions later on also So we'll now move on We'll now move on to the concept of multiplication of vectors Okay, again, let me take an example of vectors z1 equal to a1 plus ib1 And z2 as a2 plus ib2 Now when you multiply these two vectors You multiply everything with everything. So there's no restriction When I say no restriction means a1 multiplies to a2 a1 multiplies to ib2 Then this multiplies to this this multiplies to this Okay So as a result you get a get this answer a1 a2 minus b1 b2 plus i times a2 b1 plus a1 b2 Again, this result is meaningless to me. I mean, I don't get any geometrical pleasure while I see this result Right, this is completely. I use this result. It is meant for people who are just solving ncrt. Okay So Je has altogether a different Way of thinking about it. They want you to geometrically think about multiplication Before we go into geometrical interpretation of it I would like you to think this in terms of polar form Let's see the same complex number z1. I write it as r cos theta 1 plus i sin theta 1 And this complex number. I want you to take it as r2 cos theta 2 plus i sin theta 2 okay now All of you multiply z1 and z2 That is these two terms and tell me what result do you get? What result do you get? All of you, please tell me this Now multiply it into multiply this it in the same way as I have done this multiplication But tell me the result in terms of cos something plus i sin something Tell me that something That's absolutely correct How about others? Don't look at the chat box to just solve you solve yourself and just Plug in the answer Shushant, Andrew, Ashutosh, Niranjan Sameer, Omkar So Ashutosh got the same answer. Okay, great You will realize your trigonometric compound angle formulas are getting formed right so when you do this when you do this activity You get cos theta 1 cos theta 2 minus sin theta 1 sin theta 2 right Which is actually cos theta 1 plus theta 2 so this will become cos theta 1 plus theta 2 When you do this activity you get sin theta 1 sorry cos theta 2 sin theta 1 plus cos theta 1 sin theta 2 which is nothing but sin of theta 1 plus theta 2 Right Now what is the conclusion over here? It means that when two complex numbers are getting added up. Sorry multiplied up You get such a complex number Whose modulus is the product of the moduli of z1 and z2 because r1 is mod z1 and r2 is mod z2 and Whose argument is some of the arguments of z1 and z2 Isn't it? This is very very important right so If you look at it very very, you know carefully in a geometry wise Let's say I start with the complex number z1. Let's take an example Let's say 2 2 cos Okay, pi by 6 plus i sin pi by 6 And this is a complex number z2 which is 3 cos Pi by 3 plus i sin pi by 3 Right, so when you multiply z1 with z2 What is happening your result is becoming 6 cos The sum of these two arguments which is pi by 3 plus pi by Pi by 6 plus pi by 3, right? Yes or no Which is actually 6 cos pi by 2 plus i sin pi by 2 That means it is actually 6 i if i'm not wrong Isn't it? So look carefully what is happening in the argon plane Let me first draw z1 So let's say this is z1 z1 is located at a distance of 2 So this distance is 2 And its argument is pi by 6 means this angle is 30 degree Okay Okay Now when this complex number z1 got multiplied with z2 What did it do to this complex number z1 first of all it increased its modulus by a factor of 3 Means it extended its modulus by a factor of 3 Are you getting it? So this got extended This got extended by three times Correct so from two it became six From two it became six And not only that it gave it a rotation of 60 degree It gave it a rotation of 60 degree in this direction Are you getting this point? Exactly it is scaling and rotating that's the very right word you use Bharat So multiplication of z1 with z2 means you scale the modulus of z1 by a factor Which is given by modulus of z2 And you rotate the argument of z1 by Argument of z2 in an anticlockwise sense So this is anticlockwise rotation that is given to it This is something which Je wants you to appreciate what is happening when you're multiplying two complex number Are you getting this fact? So let me ask you this question If z1 is 3 e to the power i Pi by 6 Okay And z2 is Let's say just e to the power i Pi by 2 Then when you multiply z1 into z2 what is happening? What is happening? Don't tell me the result I know the result But what is actually happening that is what I want to know When I write nothing it is basically one over here How is z2 influencing z1? Right it gets scaled by a factor of 1 means its distance from Its distance from origin doesn't change So let's say it was initially at a distance of 3 At an angle of 30 degree So this distance doesn't change And it just gets rotated by Pi by 2 So it gets rotated by an angle of Pi by 2 that means it now comes here So this is 3 and this is more 90 degree added to it So altogether the argument becomes how much This angle will become how much This angle will become 120 degree Getting a point So my answer will directly you can see from the diagram and tell that your answer will become 3 e to the power i 2 Pi by 3 Yes it is always anti-clockwise Had I done this had I said z2 was e to the power minus i Pi by 2 Then what will be z1 into z2 It would rotate it by 90 degree in a clockwise so it will now become this Are you getting it? So it will become something like 3 e to the power Minus i Pi by 3 as your answer Isn't it so treat the concept of multiplication with the concept of This is synonymous with the concept of scaling And rotating scaling and scaling and rotating Okay, what you did not understand Shushant So I'll give you one more example Let's take another example Let's say z1 is 2 And z2 is 5 Let me write it in polar form Cos Pi by 3 plus i sin Pi by 3 Okay, when you multiply z1 into z2 Okay, what happens? I'll show you geometrically what is happening So 2 means 2 comma 0 right So we are talking about this line Correct Now when z2 multiplies with z1 First of all it scales up the magnitude This is the magnitude right This is the mod of z1 It scales up by a factor of 5 So 2 will become 10 now Correct And secondly Its argument is actually 0 right When you say 2 it is 2 cos 0 plus i sin 0 right So its argument is 0 As you can see it is completely lying on the real z axis Positive real z axis So what it will do? It will change its argument by Pi by 3 anti-clockwise So it will rotate it anti-clockwise Pi by 3 And make this length as 10 So this will be 10 and this angle will now be Pi by 3 So product This will be the product of z1 and z2 Is it not clear? So it is scaling up by a factor of 5 It is rotating anti-clockwise by Pi by 3 So wherever you reach that would be your z1 into z2 complex number Have I made sense now Shushant? Shushant is it clear to you? Okay Now without doing any kind of multiplication Tell me what is the final result for this? So let z1 be 3 z2 be 2 e to the power i Pi by 2 Okay And z3 be 1 by 4 e to the power i Pi by 6 What is z1, z2, z3? Where will you reach? What is the location of this? Tell me the location on Arganed plane Without doing any kind of multiplication Right Ashutosh It is just like multiplying the magnitude with magnitude and rotating anti-clockwise So first start with 3 Where will 3 lie? Then see what will z2 do to it? Then again see what will z3 do to it? Do stepwise That's app Okay I will not reveal the answer let others tell First quadrant 1.5 at an angle of 2 Pi by 3 How can we first quadrant? 2 Pi by 3 is in second quadrant Oh you are saying about the position of the first Okay Okay Let's start with this z1 which is 3 3 is let's say here 3 is like 3 comma 0 Making an angle 0 Okay Now multiplied with this vector means You are making 3 as 6 and rotating it 90 degrees So you will reach here So you will reach at 0 comma 6 Okay So this has now become 6 And the angle has changed by 90 degrees Okay Next is you make the magnitude 1 fourth Means 6 1 fourth will become 3 by 2 So 3 by 2 is let's say 1.5 like here And further you add 30 degrees to it So further you rotate here by 30 degrees So this is 30 degree more So this is your final z1 z2 z3 And if you write it in a polar form or let's say Euler form You would write it as 3 by 2 e to the power i2 Pi by 3 Or in Polar form it is 3 by 2 cis 2 Pi by 3 Is it understood now? What does multiplication do? Okay Now if this understanding is clear I would like to pose a question to you Which is purely a geometry based question Let's say this is z1 and let's say this is z2 Okay Can you locate z1 into and z2 onto this Tell me a systematic diagram or you know Methodology that you will adopt to locate z1 into z2 You are only provided with Compass scale pencil Okay So I'll tell you a method And you would need to explain me why that method works Okay So first I will make a point Let me call it as o a b Okay Now I'll make a Point 1 comma 0 over here Let me call it as p Okay Let's say this length This length is r1 Okay And this angle is theta 1 Okay And let's say this length is This length is Let me choose a different color pen so that This is r2 And this angle is theta 2 Okay Let me connect a to p Let me connect a to p Fine And at the same time let me make a triangle Like this Now this triangle Let me name it OBC triangle Okay I have made this triangle in such a way that this angle here This angle here is also theta 1 Okay And not only that Triangle OAP is similar to triangle OCB Now can you show that this point is actually your z1 into z2 Can you show that This length OC will actually be your mod z1 mod z2 That means it will be r1 r2 Can you show this I've already made a diagram in such a way That I purposely made triangle OAP similar to triangle OCB Okay So if you say that I've already made this construction by construction This is by construction So by this construction you can say OA by OP is equal to OC by OB Because you're saying triangle OAP is similar to triangle OCB Now what is OA? OA is r1 What is OP? OP is 1 OC is not known to me right now I have to prove that it is r1 into r2 So let's say I'm writing it as OC only And OB would be nothing but r2 So if you cross multiply You get OC as r1 into r2 Right That means it is proved So this length O2C is nothing but It is representative of the modulus of z1 into z2 And by construction you can see this combined angle This combined angle from hair to hair Green one This is theta1 plus theta2 Which all the more makes it evident that C is basically the position of z1 into z2 Are you getting it So this is the geometrical location of z1 into z2 Please remember it Now apart from whatever we did so far Two important properties come out from The multiplication of complex numbers One property that came out was If you multiply two complex numbers It's modulus will be same as the product of modulus of z1 and z2 Yeah that's what big angle is theta1 plus theta2 That's what is the argument of Z1 into z2 Right Do you remember it's actually argument of z1 plus argument of z2 We had figured it out That's actually theta1 plus theta2 Away Is that clear Oh yeah my mac is low on charge Yes give me one second And not only that it can be scaled up We can generalize this And say that Mod of z1 z2 z3 till any number of complex numbers you take It is equal to the product of the respective moduli This is one important property which we learned from this exercise That when you multiply Many complex numbers you simply multiply their moduli Right And the argument of I'll first take the very specific case And the argument of z1 into z2 is the sum of the arguments of z1 and z2 Okay Which we can generalize And say that Argument of z1 z2 z3 till zn Is actually sum of the argument of Each one of them individually found out Is that fine Is that okay These are two important things which come out from this exercise The best way to realize The best way to realize Product of two complex numbers is when you use Euler form Euler form gives you a better idea That if z1 is r1 e to the power i theta1 And z2 is r2 e to the power i theta2 When you multiply it You don't have to break your head r1 r2 gets multiplied And the angles get added up So this is a clear indication that The moduli gets multiplied Get multiplied And arguments get added Okay So that's why When I was dealing with Euler form I told you The best use of Euler form is when you Multiply or divide complex numbers So with this we are going to move on to the concept of Division of complex number I will again start with point form representation Which I know is a useless thing Because it doesn't give us any information About what geometrically is happening with the vectors Okay So this is what you do But if you want to express it as Capital A plus IB form We need to do some operation Okay Now what is this operation Listen to this very very carefully In order to convert it to this form We normally multiply the denominator with its conjugate Now what is the conjugate As of now it is very confusing term for you But just understand that conjugate means You are just changing the sign of i with minus i That is called conjugate Yeah, class is only till 12 o'clock Okay What is the conjugate I will talk about it little later on As a separate you know section As of now understand that I am multiplying the denominator And numerator with the conjugate of the denominator Now what is the benefit of doing this The benefit of doing this is When you multiply A2 plus IB2 With A2 minus IB2 You realize your result will become A2 square plus B2 square Check it out Okay, you can do it And check it out that it becomes this Correct So your denominator becomes purely real quantity like this Whereas in the numerator you will get A1 A2 plus B1 B2 plus i times A2 B1 minus A1 B2 Is that okay So you can write it as A1 A2 plus B1 B2 by I forgot A2 here A2 square B2 square And i times A2 B1 minus A1 B2 by A2 square plus B2 square Just to express it as A this is going to be A plus i This is going to be your B Okay again this operation is completely useless When it comes to J This is just meant for your school level Simplification of complex numbers Because it is important for some of the problems So you should be knowing it What J is more interested in Is in knowing what is geometrically happening When you divide This is more important with respect to J So let me treat this by using polar coordinates first Let's say Z1 is this Z2 is this Now I would request you to tell me if I divide Of course I will get R1 by R2 I will get what over here as an angle Just tell me these angles You can type it in the chat box What is what is going to be those angles The best way to figure it out is When you write it in the Euler form Euler form is so convenient It tells you in one shot That when you divide you are going to get R1 by R2 e to the power i Right theta 1 minus theta 2 So these things that I ask you over here These are going to be Theta 1 minus theta 2 What do you understand from this What do we understand from this Can somebody explain me by typing it on the screen What do you understand looking at this What does it convey What does this convey What happens when you divide one complex number by another Yes good Ashish Right Niranjan Good Andrew You are all correct So what happens when you divide a complex number by another complex number Very good and what happens to the magnitude Niranjan It gets scaled down by a factor of R2 So this means The modulus of Z1 Is scaled down by factor of Modulus of Z2 And what does this represent Argument of Z1 Is rotated by Argument of Z2 in clockwise direction That is what I want you to interpret So as an example if I say Z1 is 6 cos pi by 2 plus i sin pi by 2 And Z2 is let's say 3 cos pi by 6 plus i sin pi by 6 Without doing any division tell me what will be Z1 by Z2 I just want you to interpret and give me the result in a geometrical way Don't do any kind of no mathematical operation What will be the answer You can use cis and all to give me the answer Fast guys it's not that difficult absolutely correct So it becomes 2 cis pi by 3 or you can say 6 by 6 by 3 cos pi by 2 minus pi by 6 plus i sin pi by 2 minus pi by 6 Which is same as this Does it make sense Great great great So let's now go into geometrical Location of Z1 by Z2 So in a similar way I will not you know take much of your time I'll just quickly convey This idea let's say this is my Z1 this is my Z2 Okay, and let's say this this length is r1 this angle is theta 1 And this length is r2 and this angle is theta 2 Okay Again in a similar way Let's take a point over here P which is 1 comma 0 Okay, let me connect this to this Let me make a similar triangle over here Okay, let me call this point as point C Now when I say I've made a triangle OAC by construction by construction Triangle OAC is similar to triangle OBC OBP Okay, so these two triangles are similar Okay Now let's see what is your how does your point C behave? That means what is your OC distance And what is this angle by the way By construction you can see if this is your theta 2 Then this angle from here to here automatically becomes theta 1 minus theta 2 Okay, so at least I know what is the argument of a complex number Let's say for the time being I call it as Z3 So we know that argument of Z3 is theta 1 minus theta 2 So half the work is done Let us clarify on the length of OC So can I say by this similarity condition OA by OC Will be equal to OP by OB by OP Correct So OA is what? OA is R1 OC is unknown OP is R2 OP is 1 So this makes OC as R1 by R2 That means mod of Z3 is R1 by R2 That is mod Z1 by mod Z2 Which further confirms or reinforces the fact that your Z3 was the location of Z1 by Z2 Right, so these ideas should be clear in your mind Apart from this I also learned two important properties What are they? I learned that when two complex numbers are divided And you take the modulus of that complex number It is as good as modulus of Z1 by modulus of Z2 That means separately we have to take their individual modulus So why I am telling you this question is because In school exams let's say a question like this comes Find the modulus of 2 plus 3i divided by 3 plus 4i Okay, you know what happens In school exam many people waste a lot of time They waste time converting 2 plus 3i divided by 3 plus 4i as sum A plus B i Right, and after that they find the modulus of this as under root of A square plus B square You know it's a such a wastage of time Don't do this when you are solving problems in J Of course this is good for your school But in J e it is simply this can be simply solved in one step It is simply modulus of 2 plus 3i divided by modulus of 3 plus 4i That's going to be a root of 13 by 5 That's it done nothing else you have to do Does it make sense? So don't use this approach in J e at least Don't do this and all then this It is going to waste too much time Second thing which I learned from this exercise was argument of z1 by z2 is basically Argument of z1 minus argument of z2 Okay, why I'm telling you this question is let's say I ask you this question Find the argument of 1 minus i divided by minus half plus i root 3 by 2 Okay, can you give me the answer for this fast? Within 20 seconds. Can you give me? Yeah, Ashutosh don't worry But use the methods which school teachers tell you that's what I I told you in the ptm also School has a different way of representation of answer however concepts are the same But you have to go down to a granular level and write everything so that you know You don't lose marks in any any step Yes guys fast fast. What is the argument for this? You know many people in school what they will do they'll convert this to a plus ib form And then they apply the formula that this equal to tan inverse mod b by mod a Then figure out the quadrant Right and use this This is such a waste of time waste of time Okay So how do you do it in a very smart way? I should have got the answer by now guys Is my internet slow or like you people are slow Niranjan says minus 11 pi by 12 Devdas says 5 pi by 3 Others just a calculation Where is Bharat gone? Bharat Sukirt Ashutosh says pi by 3 Pi by 6 now he changed his answer Omkar says minus 11 pi by 12. Now guys, it's a very super simple question. It's basically argument of 1 minus i minus argument of minus half plus i root 3 by 2 Okay, argument of this is going to be See this is this is in which quadrant. This is basically in your fourth quadrant, right? It's here So this is minus pi by 4. So this is minus pi by 4, correct? Where is this? This is in the second quadrant So this is going to be 2 pi by 3 So pi by 4 minus 2 pi by 3 That's going to be minus 11 pi by 12 done How much time it takes? It hardly took me 15 seconds. Is that clear guys? So how it works? Would you like to take up one more example? Let's try this out find the argument of find the argument of half minus half i divided by minus 1 minus root 3 i If done, please type in your answer So give this back 5 pi by 12 Why I'm getting so many answers guys, that means There's some problem in understanding of the concept. So three people have replied and all of them have given different answers Oh my god, one more 13 pi by 12 Okay, adhvayat wants to change it to 5 pi by 12. Okay Shushant minus pi by 12 Devdas wants to change the answer now Okay guys simple This is just going to be argument of half minus half i minus argument of minus 1 minus root 3 i So this lies in This lies in again fourth quadrant correct And this angle is going to be minus pi by 4 So this is minus pi by 4 Whereas this lies in the third quadrant And this angle will be minus 2 pi by 3 So minus of minus 2 pi by 3 that will come over here, which is 2 pi by 3 Minus pi by 4 which is going to be 5 5 by 12 that is going to be your answer right shirish What is going on? Okay, so guys, uh This is pretty much for today because our next concept would be involved taking a little bit more time That is the concept of conjugates So next session when we meet which will be again, I think a holiday time only for us So next session we'll talk about You know a lot of things like conjugate We'll talk about the uh, we'll do much more problem solving with whatever concepts we have done so far Uh, then we'll talk about the concept of uh Locusts we'll talk about the demyverous theorem We'll talk about nth roots of unity We'll talk about how complex numbers can be uh treated like vectors And what all operations just like vectors has dot product and cost product How does dot product and cost product work on complex numbers and what is their significance? right And most importantly, we are going to talk about, uh, kony rotation formula Which is the most important concept in complex number kony rotation method Okay, if you could do some prior reading before you come for the next session, it will also be helpful for me Uh to you know keep the class a little fast today. I'll appreciate you all were very very quick in your response So I hope today's concept was clear about representation of complex numbers in various forms And uh operation on complex numbers I've still not done completely with the modulus Properties, there are a lot of modulus properties, which I'm going to talk in next session Okay, now you have understood a lot of things so it will make very easy for me to scale this concept up Read about all these things which I'm mentioning nirangin Like kony rotation formula locus concepts and all those things So guys, uh, thank you very much for coming online Over and out from this side And enjoy your diwali. Don't burn yourself. Okay. Take care. Bye. Bye