 So complex number is what we are going to start today Okay, this is actually a new chapter to most of you complex numbers Right complex number is a chapter which is not taught in any of the boards early I don't think so even in IG curriculum. I think mother will be the best person to tell me on that Why are you exposed to are you exposed to complex numbers? No Right, so this is a new concept for all of us and It is one of the most I would say feared chapter for a 11th grader Now for various reasons number one. Yeah, of course, you do not have any prior exposure to it, right? You might not have ever studied complex number in your junior classes. Okay Second thing is it's a multifaceted concept multifaceted means it will have a lot of aspects Which come from other chapters? For example, it will have some touches of trigonometry It will have some touches of coordinate geometry. It will have some touches of vectors So because so many concepts help you to solve a complex number problem. Sometimes it makes the concept tricky also Some somebody says that it should make our life easy because you can use the multiple concepts from other vertical of mass to Solve complex number problem. I agree. So both pros and cons are there. It makes life easy. Sometimes it makes life simple difficult also sometimes Now first of all, why do we need a complex number? Okay, what was the need for a complex number? So as you all know Invention basically is an outcome of a necessity Right necessity is an invention is the mother of all invention, right? Hello everyone. Hello. Good evening Okay class started at six, sorry three forty five So when we when we were hunters or when we were primitive man when we were food food gatherers or seed gatherers or hunters We only required natural numbers, right? So our quest for numbers were You know taken care by natural numbers, right? One two three four By the way, natural number itself is defined in different ways in different country You would be surprised to know that in some countries zero is also a part of natural number, okay? Anyway, so why does call natural number because it came very naturally to man, right? We use our fingers You know, we use sticks we use straight standing lines to count, you know objects around us So even a small child a small baby who starts counting He or she will start counting from natural number. He will never say one point two five one point three six He will never count those kind of numbers But slowly man started doing some kind of barter system, okay? So from food gatherers we started living the life of you know small traders So, let's say I am a trader. I gave let's say Anurag five mangoes, okay? So I will say I have five he will use a natural number to denote that pipe But let's say I take away that five mangoes from him. How will he denote? How many mangoes he's left with if you're using natural numbers at that point of time You will not have any number to denote that you are left with no mangoes or zero mangoes So zero was put into the system and then zero was put into the system people thought that their quest for Numbers was complete or they had a whole set of numbers with them. That's why they started calling it as whole numbers The word whole was because they thought they do not need any number at all They have whole numbers all the numbers I have in my world, okay? But soon they were proven wrong when they started doing a little bit more business when lending and borrowing started happening So let's say I gave Anurag five mangoes and I demanded from him seven mangoes Okay, of course Anurag has to borrow from someone. Let's say a person who has you know mangoes to lend to an Iraq So he will borrow two mangoes and give it to me along with the five he already has So, how do you show the concept of? Oh, I can't So I can't show the more mangoes up. So let's say I know I borrow two mangoes from Akash and he gives it to me. So how does Anurag show the fact that he has borrowed two mangoes? Okay, so it's your negative two and because it's negative two basically man needed integers So whole numbers again, it was bettered by the use of integers. Okay. I'm just drawing another circle Okay Okay Now man started doing farming. Okay, so man went into farming. So let's say Anurag again our Hero of the situation. So let's say he had a farm whose dimension was Let's say two meter square Hello, what's this small piece of land? Two meter square is not even the size of the bed. I'm sitting on So, let's say two meter square land he had and Anurag had three sons, okay So when I know I was becoming old these three sons demanded that you should divide the land equally among us Right because we are your natural hair of your land. So how will he divide or how much? Land should he give to each of his son? Of course, now, you know the answer now You know that he should give two-third meter square to each one of each of his sons But do you think when integers were known this number two-third was no Correct. So man needed a number like this and therefore he started talking about Rational numbers. So let's make another circle like this. Sorry for a crooked circle. Okay So he required rational numbers like q stands for quotient because you are taking one number divided by the other and The word rational also has come from the word ratio ratio ratio of P by Q. So as you can see, it's a P by Q type of a number Okay, now, let's say the same piece of land Anurag wants to protect firms trespasses, right? He doesn't want anybody to enter his or encroach upon his piece of land So he goes to a person who sells barbed wires. So how much barbed wire will he demand to you know, cover up the boundaries of his Cover up the boundaries of his let's say agricultural land How much barbed wire he will need? Now, you know, the answer is four root two Because root two etc is known to us now that time when let's say irrational numbers were not into picture third was not into picture How else would he you know explain how much barbed wire he needed? So he needed something called irrational numbers. So there was another let's say, you know circle I'll draw a small circle, which is Q compliment Q compliment means other than other than rational numbers Now when he got all these numbers, he started calling all these numbers as real numbers Why real because He could signify any real quantity around him by the use of these numbers whether it was the farm size Whether it was the size of the, you know barbed wire he needed any length any dimension He could signify by the use of these numbers. So they were used to Signify real things which he had around us. Okay, but so he ends, you know, gradually man started solving equations and One day Anurag While he was solving equation, he stumbled upon this. Okay, which made him, you know Think very hard that how can how can we have a number? Especially a real number whose square is going to give me a minus one. So he was unable to solve such questions. Okay I'm just giving a funny example. However, this was not the real, you know in Incident that happened it basically happened with the Italian guy called Gary Lamo Cardano Cardano was a guy who was attempting to solve cubic equation Just like quadratic equation, there's a formula He was trying to come up with a cubic equation formula and in the attempt to solve a cubic equation Cardano started, you know stumbled upon these kind of scenario. Okay, not a similar one But not a same one, but a similar one. Okay, so there he started pondering that I need to have a number Whose square will give me a negative answer and there and there he felt the need of Imaginary numbers there he felt the need of imaginary numbers now this word imaginary did not come very you can say on an ad hoc basis Imaginary the word I basically was chosen by Euler Euler was a Swiss mathematician who basically gave this word But before that it was called latent numbers and all those stuff before they finally Decided that they are going to choose the word imaginary. Okay, so there was a need for such things and These numbers came into existence Now most of you would be knowing that we have already started talking about Two-dimensional complex numbers which are called quaternions Okay, so when one of the English mathematician Hamilton, he has introduced quaternions into the system Lot of you know cryptographic work in the field of computer science has been done on quaternions and who knows 10 years down the line or 15 years down the line your children or grandchildren would be studying this as one of the chapters in Class 11th and 12th Jee may be famous questions on quaternions, right? So what I want to convey is the field of maths is dynamic, right? It is not like You know something which has been already decided and you have to work only with it There are a lot of debates a lot of changes are happening in the field of maths number theory Lot of research work is still happening. Okay Yeah, you read about that quaternions Sorry, I don't want to waste time talking about things which are not relevant as of now for our JEE preparation So this is a brief history why complex numbers came into picture. Now. What is complex numbers complex numbers are basically a bigger set which encompasses Real as well as imaginary numbers. Okay, so this is called a complex number We normally write it with a C with a slight stroke over it. Okay So real numbers are actually complex numbers Imaginary numbers also complex numbers and a combination of this is also a complex number So complex number is the broader set as of now. It is more broader than your real number set Okay So before I start going into more details about what's the complex number? Okay, and what kind of operations we can do it do on it. I would like just to give you a brief idea about The content of our study So overview of our chapter What are we going to study under this chapter? The first thing that we are going to study under this chapter is representation of complex numbers representation of complex numbers Okay There are three types of representation, which I'll be discussing with you point form polar form and Euler's form and We'll see how One form is helpful vis-a-vis the other in certain type of cases We'll be also talking about operations on complex numbers Operations on complex numbers. This is going to be the elephant in the room because It's going to take a lot of time because we have operations of addition Before that comparison addition subtraction multiplication division log Square root nth power They're raising one complex number to the other all those operations will be taken care So I think operation itself would be around two and a half hours of you know Subject matter for us under complex numbers Third we are going to talk about third. You are going to talk about the D Moavres theorem D Moavres this the pronunciation is D Moavre D Moavres theorem Okay, so the help of D Moavres theorem. We are going to talk about a lot of things Like finding the finding the n-th n-th power of Of our complex number Okay, and how this helps us in studying the n-th Roots of Unity in particular Okay, which also includes your cube root of unity including your cube root of unity including Cube root of unity roots of unity Okay Next thing that we're going to talk about is your rotation theorem a rotation theorem and This is something where we'll be introducing a new type of formula to you called the Kony's rotation formula Kony rotation formula Okay, this is a slightly tricky concept of all the four that we have discussed so far. It requires a bit of practice We'll do that through questions not to worry next we'll talk about Next we'll talk about the application of complex numbers application of complex numbers in Geometry numbers in geometry Okay, and when I say that I basically mean how it is used to find find Locusts or you can say solving Locusts questions Solving Locusts questions Locust problems Okay The last part that I have written for you This is a favorite of Jay Jay loves this like anything, okay? So they want you to understand all the operations that you are going to come across in complex numbers To the idea of Locusts Locusts is also a subject matter of coordinate geometry Right, so you would understand that how important Locusts becomes overall for you So this is I innocent in a nutshell. This is what we are going to cover in this chapter Looks to be simple only five concepts are there, but each of these concepts are pretty long, right? Representation of complex number itself is going to take us one one and a half hours today Okay, and some of the operations we are going to introduce today in today's class. So this is the overall Content of your complex number chapter Now how many classes will take for this? I Think three classes minimum It may go beyond I just some of you have asked me about the the share a break. Sorry, not the share a break Christmas break Christmas break. Yeah, so Christmas break is basically not there We are giving how we're off son 25th because that's the actual Christmas day so 25th if at all you have a class that will not happen and 31st and first Being the last in the first day of the year and the next year that will be off But the bad thing is or the good thing for most of you would be even those classes will be compensated somehow before the Actual holidays, so you may have to you know come on a non-centum day for a class. Okay, we'll put those dates In advance to you. Don't worry about it Mother. Yes, you did not miss a lot of things. We just started talking about the overview of the chapter So you can treat this to be the very first slide Okay All right So what is a complex number a complex number is actually a two-dimensional number? Okay, it is a two-dimensional number a plus B. I Okay, now here you are seeing something very you know interesting. I what is I? Okay, now I is basically The first word are the first letter of the word imaginary Okay, many people call it as iota also. Okay, it is under root of minus one Now what is this under root of minus one doing over here? Why do we need it? See as I already discussed with you Complex number is a two-dimensional number. It has got a real dimension to it and it has gotten imaginary dimension to it Let me explain this with the help of vectors. I'm sure you all have done 2d vectors in physics Correct. You have all done 2d vectors. Okay. Let's say I have a vector vector P Okay, if I write the vector P Or if my let's say vector P says that I'm just giving an example to you Let's say this vector P isn't such a direction that in order to reach from initial to the terminal position You have to go three units to the right and you have to go four units to the top Right, how do you signify such a vector? You will say sir. Beerah sir has taught us to write it as 3i cap plus 4j cap Correct. Yes or no, right Absolutely correct. I'm not denying it. Absolutely correct. Now why 3i cap and 4j cap? Why didn't you just write 3 plus 4? You'll say sir for two reasons. First of all, how would I know that this is a vector quantity? Second thing, how would I know how much I have to travel along x-axis and how much I have to travel along y-axis in order to You know go from the initial to the terminal position, right? So this i cap and j cap are basically unit vectors which tell you that Okay, you are going three unit Along the x-axis that means you are covering that i unit three times And you are covering this j unit four times in order to you know cover the entire vector, isn't it? In the same way when you write any real number, let's say you write a real number five You know actually you write five into one Because five means this unit one has been written five times So this unit is very important when you are signifying any type of real quantity around you or any kind of a physical quantity around you five mangoes If I just say five, how will you come to know which quantity I'm talking about? So one mango is like a unit and that unit is five times and hence, you know, okay He's talking about five mangoes. That's almost you know, two kgs of mangoes if you talk about the bigger ones, right? So unit plays a very important role Right. So when we write any number There is a hidden unit called one there when I'm talking about real numbers In the same way the unit for imaginary numbers is I So if I just write a plus B and let's say A and B are real quantities You will not be able to understand what is the real part of it and what is the imaginary part of it? Just like in this vector if I just write three plus four You will not know how much I component I have covered how much the component I've covered So when I say a plus bi I actually mean a into one plus B into I So basically I acts like a unit For signifying how much imaginary character that complex number has Are you getting my one acts like a unit to show how much real character that complex number has so it has both the characters So I acts like you can say distinguisher Or you can say a separator through which you can understand that okay This complex number has B amount of imaginary character to it Are you getting my point? Okay, so hence this I was chosen and since they wanted to it to resonate very closely with one unity That's why they chose negative one under root because this could be this could act like a very safe unit or signifying complex numbers Now many people may ask me sir. Can you write it like a comma B also because no people can understand that okay the first Element of the ordered pair is your real part second element of the ordered pairs of imaginary part Yes, we definitely follow this in practice right we also write complex number like a point Okay, that is something which I'll talk about in some time but is it clear that when you represent a complex number as a plus I be a Into one will behave as the real B into I is the imaginary of That complex number However, one thing I would like to emphasize over here that both a and B Even though a is called the real part of the complex number. I should not write belongs to I should say equal to and B is called the imaginary part of the complex number Even though this is what we name it but in reality a and B are real numbers only Yes, a and B are real numbers only I have seen people making a lot of mistakes while you know explaining it when I say Tell me the imaginary part of this complex number Watch out listen my question carefully. What is the imaginary part of this complex number many people say bi? No Imaginary part of the complex number is just be What is the real part of the complex number a? Are you getting my point? For example, if I say what is the j component of this vector? You'll say for don't say for j What is the real part a what is the imaginary part B? Now along with I it becomes an imaginary number For example here for along with j becomes the the j component. I mean you can say the j You know the component of the vector along the j direction Okay, so here, please be very careful when you are using the word Now that kvpy has done away with the interview But if that's a if you're appearing for a kvpy interview such wrong choice of words will give you a negative in your points Is this fine? So a is the real part B is the imaginary part both a and b are actually real numbers As you can see let's say I take an example of 2 plus 3 I Okay, 2 and 3 both are real numbers 2 and 3 both are real numbers, but we'll say real part is 2 imaginary part is 3 imaginary part is 3 Okay, don't use 3 I 3 is the imaginary part. Okay, so please be careful about the use of your terms Is this fine? now this I Will talk about it a little bit more because we are going to speak We are going to speak not only I as a Separator, but also I as a complex number or you can say imaginary number in itself So we'll talk about I so we'll talk about Properties of I the first property That you would observe is that It is such a number which when squared will give you a negative answer Okay, so I value is under root of Minus one so if you square it you will get a negative one right so Let's not be surprised anymore because now we'll be going now We are going to see such kind of activities quite a lot that square of a number can give me a negative answer Okay second thing is if you see If I just raise some powers to I let's say Okay, just keep watching these powers Okay, you would realize that it is following a cyclic nature That means after every change of four in the power it is repeating its previous value You see this Okay, as you can see I to the power one I to the power five I to the power nine They're all II each Okay, so after every change in the power of four it is regaining its previous value Okay, I to the power to I to the power six I to the part and I to the power 14 and so on and so forth they will all be minus one Okay, so this is following a cyclic kind of a nature. So this is following a cyclic kind of a nature Okay, third thing that you would observe from here is that if you raise I to any power which is a multiple of four and Being some integer you will always get a one check out this one I to the power four I to the power eight I to the part 12 I to the power 16. Okay, so if somebody asks you hey, how much is I to the power four thousand and four Without much waste of time you can easily say it's one because this is a multiple of four So I raised to any multiple of four will give you one Along with it you can also observe that I to the power four n plus one is going to be I I to the power four n plus two is going to be minus one and I to the power four n plus three is going to be Minus I Okay, so please make a note of this So if I is raised to any power which is leaving a remainder of one when divided by four That will give you an I eventually I Race to the power any number or any integer which is leaving a remainder of two when divided by four will give you a minus one eventually and I raised to the power any integral power which leaves a remainder of three when divided by four will give you a minus Fourth property that you should note down if you if you add any four consecutive powers of I If you add any four consecutive powers of I it is going to give you a zero I to the power zero is one Because it comes under this category. No zero is a multiple of four mother. So I to the power zero will be one Okay, so this is very important. Please note down any four consecutive powers of I any four consecutive or consecutive You know powers you can say integral powers also, but it can work for any any case power of I Will always add up to give you a zero What are the proof for this proof is very simple? Taken I to the power n common It'll give you one plus I plus I square plus IQ Okay, and I square we know is minus one IQ we know is minus I So one minus one cancels I minus I cancels giving you a zero Okay, so these are the four important properties that we need to keep in our mind While solving questions based on I Okay, any any doubt so far any doubt so far in whatever we have done. Okay. Let's take questions Let's take questions Sorry, I think this is complex them vector question go back to the desktop complex numbers So many questions that I need to start with the relevant question. Okay, I think it's sad so many Okay, let's do these simple questions Let me start with the first one Evaluate this that means I write it in a simpler form Evaluate this Or let me give you this question if you have to write this as a plus bi what will be a and b What will be a and b? Let's start with the first one Five or more consecutive term It'll depend upon which is your fifth term because till four it will be zero. No first one done Very good. Okay. So first one was I Plus I square now I square is minus one I to the power of four is one In the denominator, you'll have one again a minus one and This will be again a plus one. So this will leave you with an eye. I means zero plus one eye So your a becomes zero and b becomes one very good People have got it, right? Let's do the third one third one is quite easy Okay, actually one more thing I would like to highlight away when you say negative six under root It is basically written as root six I It is written as root six I Okay, so when you're doing negative of Negative six it is basically negative root six I now go ahead and solve this question You have to write it in a simpler form Okay, but now very good. So let's cube it if you cube it you get minus six root six I cube Remember I cube is minus I so it'll be six root six I So you can write it as zero plus six root six I so a is going to be zero and B is going to be six root six Provided I want to write it as a plus B. I Okay Next one We'll do the third one. I think fourth one we can take it for the later or we can do fourth one also no issues Yeah, third one. What's the answer? Express this as a plus B. I form Express this as a plus B. I form Now, please put the question number so that I know you are answering for which of the above questions Okay, very good. Aditya Pranav shares Very good Okay, now you have summation I to the power P from P equal to zero to 300 So when you're writing it You write a Pete I to the power zero I to the power one I to the power two I to the power three I to the power four Let's put this in brackets. I to the power five I to the power six I to the power seven I to the power eight Let's put this in brackets And I'm sure you must be realizing why I'm putting those in packets Okay, so the last few numbers would be 300 I think 298 is uh No, uh 96 300 is also multiple so I can have I to the power 300 I to the power two 99 I to the power two 98 I to the power two 97 Okay Now if you see the reason why I have put them into brackets is because Four consecutive will add up to zero So this will be zero. This will be zero and all these numbers that you'll get will be zero each Which property have I used? Which property have I used over here? The fact that some of four consecutive powers of five will add up to give you zero Ultimately your answer will be I to the power zero, which is one Okay, so you can write it as one plus zero I One plus zero I Is this fine? Okay Let's do the fourth one Let's do the fourth one one plus I By one minus I square. Okay. Now before we square it Let's try to write it in a simpler form Now, uh, we have not officially done the concept of conjugate of a complex number But whenever somebody sees a problem like this The natural feeling that he or she will get is let's multiply it with This okay the same feeling of rationalization will come in your mind Correct Okay, so let's do this So when you uh multiply one minus I with one plus I you'll get one minus I square Okay, and on the numerator you'll have one plus I to the power of two Now what I wanted to discuss with you here is that Normally when you have such kind of powers or such kind of you know terms coming up We use our normal identity that we have learned a plus b the whole square. There's no difference between uh, you know Applying those identities to real number and applying those identities to a Non-real number or imaginary number the identities won't change. Don't don't worry about that So when you write one plus I the whole square you can write it as one square I square plus two into one into I the normal formula a plus b the whole square formula And in the denominator you have one minus of minus one Okay, now let us write down the values One square is one I square is minus one This is two I divided by two one plus one is two I minus I cancels two and two cancels so that will give you an I So the very first term that you have over here. This is actually I square. This is actually I square okay, so Without wasting much time Without wasting much time if this is I square The other term has to be reciprocal of I square correct So if this is I square Then this is one by I square Because it is just exactly reciprocal of this term So I square is minus one I square Again is a minus one. So this will give you a minus two Is this fine Any problem in this Okay, one interesting thing which I forgot to discuss with you How would he get the same answer if you squared it at the start If you squared it at the start Oh, you can do that no difference Multiple ways to solve the question mother So you may decide you may start you may square the numerator. You may square the denominator. Yes That is also going to give you the same result No difference no difference at all Now one very interesting fact When very interesting fact in fact, it's a myth. It's a it's a wrong notion that I would like to you to you know understand When we say under root of a into under root of b Okay, many people construe this as under root of ab Okay, please note that this formula works if at least If at least One of a and b is greater than equal to zero That means if both of them are negative this formula will fail Okay, how this formula will fail? So let me show you that So let's say I take a as negative four B as negative nine Okay under root of a would be under root of negative four, which is actually two i Under root of b under root of b will be under root of negative nine, which is three i Okay, so when you multiply them You will get two i into three i which is six i square Six i square is minus six Okay, now if you claim that this is equal to under root of ab Under root of ab in this case would give you a six Okay, and these two things are not equal That means this formula will not work under those circumstances So here is something very important for you to note down Under root of a into under root of b would become negative under root mod a mod b If your a and b both are negatives If your a and b both are negatives Okay, please make a note of this So people who blindly use this formula for everything Here is a word of caution. Here's a word of warning. Do not use this if a and b both are negative quantities Here a and b both are negative quantities. So the formula will slightly change Now why does the formula slightly change because this convention has been derived from the concept of geometric mean Which will study in the chapter sequence series and progression Now the moment somebody hears the word mean Or the word mean itself was coined to show something in between So whether you talk about arithmetic mean geometric mean harmonic mean It basically is a quantity which is between two numbers Right, so if somebody says What is the geometric mean of minus four and minus nine? So basically, I mean some of you would be already knowing it geometric mean is under root of ab, correct So under root of ab just like arithmetic mean is a plus b by two Geometric mean is defined as under root of ab Under root of full ab product Now we haven't done that chapter yet. So most of you would not be aware of it But those who are aware they would know that it is written as under root ab So under root of ab should give you an answer which is between minus four and minus nine When I say between not exactly midway, but somewhere between it. So it must be a negative quantity, right? So how can My answer becomes six. So this cannot be accepted as our right answer So it should be somewhere in between minus four and minus nine And hence this change in the formula has happened because of that Okay, these are just conventions which are made by you know, uh, no mathematicians to make To give uniformity across the field of mathematics Okay, so very interesting fact note down I've seen even teachers making this mistake Okay, so one question I would like to take up with you Let's pull out one from our I'll put the poll on. I'm not taking the attendance so if at all You have left the call now. You only marked as absent shake So the question as you can see on your screen is a sequence which goes from i To i square three i cube all the way till hundred terms And this expression simplifies to which of the following Two of you have answered so far good nice Last 30 seconds Last 30 seconds and I'll close this all right five four three two one Cool. All right 50 of those who have voted have said option number a Okay Next highest vote has gone to c. Let's check See, uh, this is a special type of series which you will learn sooner or later in the chapter sequence series and progression Which is called an agp agp agp. It stands for arithmetic o geometric progression Uh, there is a particular way to solve these kind of questions Which i'm doubtful that most of you would be knowing So let me take this as an opportunity to tell you that So if I write till hundred terms for this It will be this right Now in an agp like this first of all why the name agp was given because as you can see This kind of a series exhibits two types of uh progression characteristics One is an arithmetic progression as you can see the number one two three four Da-da-da-da-da-da-da So this is showing you an arithmetic progression characteristic And the second type is like it shows you the geometric progression characteristics i i square i cube i to the power four Da-da-da-da-da till i to the power hundred. Okay So when a progression, sorry when a series shows both types of uh, you can say characteristic We call those as an ag Arithmetic geometric series. Okay Arithmetic geometric progression if you don't write a plus in between So how do we uh solve such kind of uh series? What do we do is we first see what is the common ratio of the gp involved over here As you can all see the common ratio of the gp involved here is i isn't it Don't worry about the ap involved. Just worry about the gp involved So the gp involved has a common ratio i correct So multiply s with i So when you're multiplying it Write it one shifted to the right. For example, if you multiply this with an i right underneath this If you multiply second term with an i right underneath this If you multiply third term with an i right underneath this Okay, and so on so Ultimately, this will have 99 i to the power hundred and it will Just shoot this particular last figure and become 100 i to the power hundred and one 100 correct Let's subtract Let's subtract So s minus is will be like s one minus i take s common Uh, consider a zero over here if nothing is written and consider a zero over here if nothing is written So i minus zero will be i Two i square minus i square will be i square This will be i cube. This will be i to the power four and so on This will be again i to the power hundred And this will be sorry minus hundred i to the power hundred and one Okay, now here if you see there will be 25 such you can say No four terms I'll just show you one more So if you start grouping it, there'll be 25 such groups like this that will be created along with hundred Okay, so So they will all be zero zero each isn't it? Because we have all seen that We have all seen that Four consecutive powers of i will add up to give you zero correct So i one i two i three i four zero i five i six i seven i eight zero i nine i ten i eleven i two zero like that So if you go to i let's say 97 98 99 100 They'll add up to give you zero. Okay, so nothing is left other than Other than minus hundred one minus i on the right side Oh, sorry my mind hundred i to the power hundred and one sorry and that also is hundred i to the power hundred and one is like i to the power hundred Into i to the power one so i to the power hundred will become a one Because hundred is a multiple of four correct So ultimately what you see is Minus hundred i on the right side Okay, so our answer here would be Minus hundred i divided by one minus i If you look at the options None of the options so far resemble it So what i'm going to do is i'm going to slightly simplify this by multiplying with one one plus i So on the numerator i will get this on the denominator i'll get a one minus i square which is two So two and hundred will get cancelled 50 And if you multiply i inside will give you i square which is minus one And plus i So this will give you 50 times one minus i which i think is Option number So janta was absolutely correct. A is the right option for this Is that clear any questions any concerns? All right, let's take one more question Anybody has any question with respect to this Sir in the previous page What would have happened if one of the numbers were negative? Okay, you're asking me the previous page doubt One of the numbers were negative only one of the numbers were negative, right? If only one where negative other is positive or zero you'll follow the first formula If both are negative both the formulas Are you getting my point? Is that your question mother? Or are you talking about geometric mean when one is positive one is negative? geometric mean When one is positive another is negative is not defined because both are real numbers and your answer will come out to be imaginary That i will talk when i do geometric mean with you Sir, i just checked the first four terms pattern. Can we do it? For any sir sequence. Yes. Are you talking about the approach that we followed? The approach that we followed multiplying with the common ratio and subtracting it. Yes, that is a Widely followed approach. It can solve any a g s problem a g s means arithmetical geometric series problem by the use of that approach Yes, if one of them is negative other is positive you can use under root a under root b is equal to under root a b No problem with that For example, if you do under root of four into under root of nine Okay, it is same as under root of four into nine. So this will give you two y This will give you three this will give you under root of 36 Which is equal to six i and this is also equal to six i both are correct So it is matching with our formula Got it mother. Is this what you want us wanted me to explain? all right, so next question that I have for you is if x is equal to minus five plus two root minus four simplify Simplify x to the power four nine x cube Plus 35 x square minus x plus four Simplify means write it in a simpler format Or you get or I can say write it as some a plus i b Okay, uh, how many of you are actually solving this question? Uh, by the way, this is four i if I'm not mistaken Yeah, root root of negative four is four two i two into two, uh, four i How many if you're actually literally putting x value in all these places and trying to simplify I'm sure most of you are doing that. Isn't it? Now, let me cut you short on that if you are trying to do that because it is going to take a lot of time Right, you are not dealing with simpler powers raising it to a power of four cube, etc. That is going to take away considerable amount of your time Okay, so here is an approach that I would like you to understand Let us try to play with this expression So let's take the minus five to the other side Let's square both the sides So if you square both the sides, this is what you end up seeing Oh, it was four i my bad So you'll end up seeing something like this x square plus 10 x plus 41 equal to zero correct now All of you please pay attention. I'm going to tell you a very interesting and shortcut way This term you treat it as a dividend and this term you treat it as a divisor So this is your divisor. This is your Dividend and divide it by using your long division method By using your long division method I'll tell you why am I doing this just be with me for Few seconds So when you divide you get x square, this is x to the power of four 10 x cube 41 x square When you subtract you get minus x cube Minus six x square so put a minus x Again, when you subtract you get four x square. Let me copy this You get 40 x Plus four Just put a four So four x square plus 40 x plus 164 And subtract it We'll end up getting minus 160 as your remainder, right? So what I'm trying to say here is This term which you were actually trying to figure out. What is this expression? It could be written in an alternative way as This Into this right By Euclid's division algorithm. I can write this expression as quotient Times divisor plus remainder Right now you must be wondering why at all I'm doing this is because when you're putting x as five Sorry minus five plus four i here. That means you're putting x as minus five plus four i here also But this guy this guy is zero when you're putting x as minus five plus four i because this came from This fact if this is zero that means zero into something whatever it is It will be zero and you'll be left with minus 160 only So your answer will be minus 160 Right, so you don't have to put a minus five plus four i in every x and try to simplify that that is going to take at least five minutes of your time easily But this was just a case of a long division method which will take one minute or so and then you are done with your answer First of all the word remember itself is A misnomer never remember anything This method is basically if you incorporate it while solving questions, that is what I can say is the right use of word See whether what I did was I'm actually finding out the expression of this polynomial when you're putting x as minus five plus four i correct So this term I wrote it like this and finally I realized that When you put minus five plus four i into this you will end up getting a zero as the answer correct So what I did was I wrote my dividend which is your this term As this times quotient plus the remainder Okay, why did I do like that is because I know that this guy is going to give me a zero So whatever is the you know quotient I don't care everything will become a zero and whatever is the remainder Actually in that I have to put my value of x and get the answer. But since there was no x here Let us say hypothetically speaking. Let's say if there was some x term remaining Then you only have to evaluate that remainder part You don't have to worry about the other terms because they are anyways getting multiplied to This term which is zero And hence everything will become zero there. So remainder only will give you the desired answer Getting it whatever what I'm trying to say Make sense Okay all right So with this we close the i chapter of course we have done a lot of properties of i and We are ready to talk about now the next part of the chapter