Complete Complex Numbers Course For Electrical Engineering | Ahmed Mahdy | Skillshare

Complete Complex Numbers Course For Electrical Engineering

Ahmed Mahdy, Electrical Power Engineer

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14 Lessons (2h 9m)
    • 1. Why are Complex Numbers Important ?

      3:25
    • 2. What is an Imaginary Number ?

      5:39
    • 3. Properties Of Imaginary Numbers

      5:11
    • 4. What is A Complex Number ?

      5:01
    • 5. Operation and Properties of Complex Numbers

      10:04
    • 6. Solved Examples on Complex Numbers

      10:27
    • 7. Graphical Representation and Polar Form of Complex Numbers

      12:50
    • 8. Solved Examples on Trignometric and Algebric Form

      12:06
    • 9. Operations on Complex Numbers in Trigonometric Form

      10:26
    • 10. Solved Examples on Operations of Trigonometric Form

      16:05
    • 11. Exponential Form of Complex Numbers

      1:17
    • 12. Solved Examples on Exponential Form

      10:20
    • 13. Cubic Roots of Unity

      10:42
    • 14. Solved Examples on Cubic Roots of Unity

      15:52

About This Class

Welcome to my own course for complex numbers:

This course is designed for complete beginners and newbies who wants to learn everything about complex numbers.

In this course we will understand the meaning of Imaginary Numbers and their Properties, then we will discuss the Complex numbers , their Operations and their properties.

we will talk about the Graphical representation of Complex Numbers,Polar form, Trigonometric Form and Exponential Form, we will also have solved step by step examples on each of these Topics.

we will discuss the cubic roots of Unity and have various solved examples on them.

in this course i use whiteboard to help you understand each single step of the equations because My Mission is to let you 100% confident and Understand every aspect of Complex Numbers.

See you in my Course !

Transcripts

1. Why are Complex Numbers Important ?: Hi, everyone, In this lecture, we're going to discuss the importance off the complex numbers. So before we learn about with the complex and numbers, we need to know why complex and numbers are really important in our life. So why are we using their complex numbers? Okay, complex numbers are used in many fields. Number one, if we discussed is the electrical engineering department, for example, in power engineering, when we are using on talking about an easy supply and easily supply, which have that sign a soldier we form like this, that voltage off the supply is varying with time. So our disease, our supply or a cease employ. If we are talking about a three phase system which you are using in electrical engineering , we will have a V A and B B and V, C or R, and is anti the three face system. So if we are talking about it, you will find that we're If we're talking about the phase number A, it has the maximum sine omega T, and we can write it in the complex, a form which is the maximum the magnitude off the wave and their angle of the wave Okay, so this can be described it boy complex for another one, Which is that face number being which is the maximum and an angle with negative 120. This can be divided into three maximum go Zion 120 plus V maximum sign 120 J or oil Jay. This number this fall is called a complex A number. So using the complex number, we can describe our three face system. So it is just a foreigner or knowledge why you are realizing that complex number. You don't need to learn in this course this reefer system or just knowing having some knowledge about the three for systems that we use complex in, um, another single. We have there representation off some electrical elements. You know that our bar system consists off induct and sick Avastin's conducted since even resistance and so on. So with the usage off the complex number, we can describe that in doctors and in this form J. O Miguel, where Jeanne is imaginary part which represents also a complex number another things that contestants in the power system is one over Jim O Me guessing, which is also represented, boy and complex the number. We can also use it in that quantum mechanics in orderto describe their electromagnetic waves. So that's that's why our course is really important. The complex number you will see that it is in every aspect off our life. So in the next picture we're going to discuss their imaginary numbers and then more goingto talk about the complex numbers, some roles and some examples. 2. What is an Imaginary Number ?: in this video are going to discuss the meaning off imaginary numbers. So before we get toe the complex numbers that we have to understand what is mental boy? Imaginary number. So what is an imaginary number? Okay, so, somebody as we know that if we have or if we would like to get is that square root off number four? So, what is the square root off number for it? Assembly Quito plus on my nest. Oh, okay. If one legs a square root office 36 it would vehicle plus or minus six. What if I told you to get their square road off a negative for? So what is that? The problem now is that we have here a negative number. The square root. It means that we need a tow numbers. The mart implications off them will give us for two multiplied boy toe will be equal toe for six month applied by six with equal 36. But now I would like to numbers. We must apply to give us an negative four. So we learned that we cannot get the square wrote off negative for So now we learn it and using what is this in public, The numbers or the imaginary numbers we can get the square wrote off negative for So what is the square root off negative for it is plus or minus do. Oh, so what is? All I know is that imaginary number. It is something which does not exist in rail life. It is called imaginary since we don't have a square root off Negative four. Okay, so the I is represented boy toy assembly A wrote negative one The square root off negative one in elected get engineering. We use Z instead. Off we are talking about O j. Sarriegi and so instead off using their oy. Why? Because in electrical engineering I represent is the current inside the circuit. So that's why we need toe Understand that there are two. The difference between Jay and Ali in mathematics we use is that term. OK, but in electrical engineering, we use the term G. So that is the difference between them. Okay, So what is the difference between I and the real number or the imaginary number and is a real number? The real number off course toe ring for and so on? The imaginary number is something like Oy, oy for oy and so on. So an example off the imaginary number if we would like to get their square road off. Negative, sir. Physics. So how we can get the square root off any six, we can divide it and toe wrote 36 amount of light boy wrote negative one. Since then, we have a Your old is a multiplication. Off them will be inside this one multiplied. And the inside of this one which will give us negative six The square root off road 36 assembly plus or minus six months of light Boy route Negative one. So route negative one. As we said before it is some signature does not exist. It is an imaginary number. So is there a road negative one? It is represented, boy off again. So it will be Quinto plus or minus six. So now we learned in this video About what? The imaginary number, the imaginary number assembly something which does not exist. It is when we are getting is a square road off a number like negative sort six or road negative four. We're all with negative one roll with negative 1.2 and so on. If would like the square root off negative number, then we start. We cannot find it. So we start using something which called the I always represent. Is there imaginary number or the root off? Negative one. So again, negative four. Assemble equal toe root for multiplied boy route and make the one which will give us postive or negative. Do so now we learn it. Is this video about the imaginary number in the next Yvette, you were going toe Discuss. Is the properties off the imaginary numbers? 3. Properties Of Imaginary Numbers: in this video are going toe understand more about the imaginary numbers. So would like to find their imagine a number or a boat or zero I, for one, are you power toe? I bore stream and I bought for how we can get these venues. So simply toe understands them instead off just memorizing them. My power. Zero it assembly. You know that I is a road negative one. But what zero. Okay, so it can be negative. One both off, multiplied by zero, which is equal to if we have hot multiplied by zero, which is equal to negative 140 which would be equal to zeal or what one? Okay, exactly. Damn. So I po or zero? It is the same as 140 to 403 Power zero and so on. All of them. Anything. The bar zero big toe one. And you understanding now why I bore zero is equal to one I power on. It is of course. Oy. There's no simplifications is I would like I square now. I square assembly. I multiplied by I And you know that I'm a rabbi. Is road negative one of the blood by wrote Negative one. Therefore, it will be equal to senses. They are the same. It will be a negative one. Both love plus off. Since this is a square root and another square root multiplied by each other, the same number it will give us half plus half, which is equal to Uh huh will give us what? Okay, negative. 141 which is and negative one. So I square is equal toe negative one I or zero assembly one and would like to get like you . OK, so I cube can be divided in tow. I Monta blood by our square and I square, as we found here, is equal to negative one. So I want the blood boy negative one which will give us a negative. Oy. So now we have This cube is a negative part. I square is negative. One Bord zero is equal to one now or bar for we can simplify it in tow. My square martyr Blood by always square. So I square by square. What does equal negative one, Which is this one? Uh huh. Multiplied by another negative one, which will give us one. So there was This is some simple operations on their imaginary numbers. Now, like to get an example about this. Now, let's have an example. If we would like to find the power six So how we can get this? Okay. Assembly, as we did before We know the maximum thing which we memorized is I power for? Okay, so I power six can be divided into I pulled for not the black boy I square. So that would be equal. High power. For as we said before, it is equal to one. I square is equal to what? Yes, Negative one. So our answer would be negative. One that's ahead on those are one If we would like to get oil born mine. Okay, I power lying can be divided into Uh huh. Yes, I give up O que oh cube or any singles, as you would like. So I cube. What does equality is equal to? Yes, negative. I negative. Always negative boy. So the multiplication off them assembly. Negative. I Q And this will give us what negative are. Yes. So we have here a negative one. Want a blood by negative. So our final answer will be all. So now we're in this video. We learn it some operations on our imaginary numbers in the next event. Do we are going to discuss what is the meaning off that complex numbers? 4. What is A Complex Number ?: in this when you are goingto discusses that complex number. So in the previous video we talked about with the imaginary number, their properties off the imaginary numbers and the nail in this video are going toe, discuss the meaning off complex number. So what is their complex number? A complex number. It is any number. Okay, which have is that form off a blow us being when a it's cold there. Rail board off the complex number and to be oil is called the imaginary bolt off their complex number. So the complex number consistence off towards the rail. BART, which is 1234 as we know, and a public support which represent is the imaginary report, as we discussed before, which contains our or the imaginary number which is brought negative one. So if we have X equal a plus B boy, then that rail board off X or X rail borne off X is equal toe and the Missionary Board X is equal toe. Be so the imaginary parties is this part, and the rial, or BART off X is a Now let's have an example off where the complex number can come from. We have here this equation X square plus four X plus five is equal to zero. And I would like to find is a solution off equation. Okay, so this is an automatic function off their second degree, okay? Or a function off a second degree. Okay, so how we can solve this equation? OK, we know that A and B and seeing so a is a number here beside X squared, which is one and the B is a number beside X, which is fourth, and C is the last number which A has X power zero. Okay, now I would like to find the solution. X will be what may get there be most of or negative root b squared minus for E c over toe A So let's substitute with that values Negative four plus or minus root Be square 16 minus four a C four multiplied by A and Monta led by C which is 20 over to air, which is one. This was one. Be equal to what? Okay, we have negative 4/2, which will give us and negative too plus or minus 16 minus 20. What does it equal? Equal Wrote negative for Okay, so now we have here negative for now. We will need to use that complex a phone off course over to because we have here at home Negative door, bless or minus wrote negative for what does it equal, As we said before, it will be called to. But that's one minus toe I. So I will say, though I be sense that we have here plus or minus over to So finally will have negative two plus or minus Ali. So we'll see that here We had a number or the solution off X is a complex number consists off negative two, which is the real bored. And I, which is that imaginary board. If we don't have I imagine meaning or imaginary part, we will not be able to solve this equation. So this is how our complex, the number came from. Okay, Now in Zannex, Steven, you won't still discussed some operations and properties off their complex numbers. 5. Operation and Properties of Complex Numbers: Now let's discuss some properties off their complex numbers Is that we're number one If we have X one, bless all you want equals X it'll plus I. Why? If this complex a number equal tothis complex number, then is aerial port here will be toe very important. Here it's one will be quite to exit door and the imaginary board by one is in Quito the imaginary board or you want to. Therefore, white one is a little. Why our second rule? If I said what x one bliss all one and that little extra Plus I y two and would like toa ad Is this two complex numbers? So did the one plus Zeta Tau will be went toe that really boards the addition off this to rail ports X one plus extort and the addition off the to imaginary parts one and white to all white one plus. Why do our search tool If Zet he will toe X plus i y and would like toe get that additive inverse off sent the additive inverse off. That is a photo negatives and so assembly negatives that it is negative X minus or what we want of light both sides of our negative one. So these are the 1st 3 runs now the rule number four If we have the one x one plus all you want and then we have to be quite X it'll plus I worry, too, and would like to get the multiplication off, said one. But the black boy said to so how we can get this. Okay, simply x one plus always more. You want Mark the black boy exit door. Plus why Why do this will be a quick how we come out of light Albrecht's okay, assembly the first want about by the first X one ex It'll and the second want the blood by the second. So we have here plus on square white one. Why do And that means And the blows extremes plus Oh, why you want extreme sport? Plus I wait o x one boy, Why do x Juan? So this would be equal to x one. Extol jeans the same I square It is a negative y one boy too. Okay, now we have the imaginary more. Plus why one extort plus white ex Juan. So it is very simple as it seems we have years of multiplication assembly. First tomato, Abbas a 1st 2nd prevents a second means extremes and you can get it. Assembly the imaginary port ends. Aerial bought. So now we learned how to most of Lloyd toe imaginary or complex numbers. Now, before we get toe that division off too complex a number, we have to know the multiplication tive inverse off a complex number. So we have a complex numbers it which are big toe X plus or your wife and don't like to get them off the implicated in verse Zamel complicated emphasis is 1%. So how we can get one of our city Okay, one over that is equal to one over its plus Roy boy and like toe get their imaginary board . And really? So you have to get to remember something. If we have a complex a number in the denim a NATO, what do we don't were simply Mark tabloid boy, Something which is called Is that contract it off the conviction. So they contradict assembly X all you want and the difference between the number complex and it's a contract. It it is a negative one. So this part is called the contract it off the complex and x A plus I y the contrary it off it is X minus one where the negative is with the imaginary boat. And here X minus. You will see that we did not do anything. If we remove this, it is the same. This cancer does this. Okay, So multiplying by each other. Uh huh. X square oy minus only way square. So it will give us why square and between them want negative or boasted most. Why I and Ali is I square, which is a negative one. And do we have here and get another negative year? So it will be plus y squared all you Why, it's minus r Y six so they will cancel each other So we'll see here we have here now only really part So x minus Roy Roy. So that will give us X over X plus y square minus only. Why X squared plus y squared. So we have here. This is a really bored and this part is the imaginary Bart. So now we will be We were able to get there. Multiplication embers offset boy about applying buys a conjugal. So why didn't we multiply by X plus r e y and instead off the conjugal because if you multiplies this, we will have again a complex number in the day in a minute. So we didn't solve the problem. We multiplied by the country in order to cancel the complex Bharti here you'll see that most of them the multiplication off them will give us a pure mom A pure rail, Mom. So that's why we multiplied by that country. Now we like to get their division off a two complex numbers. Then the one over said toe So that the one x one plus boy right? One exit all plus I Why so with the same idea off the contract it or the usage off consequent to get them out, implicated in verse in the break ins property, we will not deploy most of them boys a conjugate off system in order to cancel the complex in the denominator. So x, it'll minus. Are you doing ex? It'll minus oy. Why don't so we have similarly we can multiply boots off There we can x toe square plus y to square and the other board x one plus only 11 exito minus all worried too and similar. We can multiply, exists and get their complex alone end. Is that a rail port alone? So now we learn that have toe devoid of two numbers Now for our final property, which is power, or they're confident. So that is equal torches, a complex number again X plus y. We would like to find Zedd bar zit bored, bored, which means that we need that contra git Zet. So said the power It will be one x minus or warrant. As we said before, if we have a two complex numbers into one and said toe and would like to get is that congregate off their some mission? It is the same as we get the country get off. The fairest plus is that once you get off the second, if we would like to get their consequent off the multiplication off a complex a number similarly that the one bar multiplied by, said the two bar. Where's that? You want example X one plus one and bart off it. It will be excellent minus R Y. One similarly here and doing a lot of like moment. So this video we learned a very important rules and properties about the complex number 6. Solved Examples on Complex Numbers: in this video are room, toe, solve some examples on that complex. So first we have here, for example, is in this video we we need toe finds the very off X and y in this equations. So something Number one we have explicit or boy is he worked before minus three. Oy. So if you remember from the property number one, if we have a whole complex number equal to each other therefore the rial board would be good toe the real Bart. The imaginary part will be called to the imaginary about their four X equals four. And why will be going toe Negative city. Okay, so that is the first example. Our second example is door City X minus two, or why is equal to five minus I all the square. So we need to expand his birth in orderto equate the complex a port and the airport and the imaginary part off the complex numbers. So see ex minus story Roy equal to the expansion off. This is a room, square toe wind five the square off all square, negative one and negative to all. You want to let my wife is negative. 10 boy again five square is doing to fight and all squared is negative one. And do the blood by fire in the blood by negative, which is negative then are now This will be what 24 minus Didn't oy and C X minus toe already worried. So the rail port will be called to the real part. The imaginary board would be called to the imaginary board. Therefore X will be going toe 24. What city? Which will be called toe eight. Why Negative. Why negative? Why is equal to negative then? So why will be well, toe Fine. Okay, so now that is our second example. Now our served the example X minus Roy plus X minus. City oy is equal toe one. So now we have a complex A number equal toe aerial number. So what we do now? I remember. This one is considered as a complex, a number with a zero imaginary bart. So we can say that X minus y plus X minus. City boy is equal to one plus zero. All right, so this birth is again a complex a member with a zero imaginary value. So we would equate the aerial port with very in part and the imaginary bored with the imaginary boat So X minus war one and X minus three equal to zero So x will be equal to three. And by substituting whose X equal sitting in this equation is therefore city minus Roy equal to one. Therefore we will have What? Yes, why Equal toe toe. So that is our served example Now our force example is X plus Oh, Roy, over one Bless oy plus two x plus You, Roy one minus always equal to mine Over to so before you see is a solution of this equation X and y try so living exists Boy Stoppings is video right now and throwing toe solves this example So how weekend so for something like this? Okay, we have two methods we come out otherwise This boy it's a congregate here and here And this one by it's a congregate here and here here one minus one minus always one plus All right, number or weekend combines this together and see if we have a complex. So we're not obliged by contract it so it takes second mission. So we have here. If we Brian this together one Of course, it is a multiplication of this boat. One plus only one minus. Oy. Okay, Ex Iblis Oy. Why one minus Oy. The publication of his boat. This model lot boys This. And this month the blood boys is two x plus only. Why? Okay, Womble a soy. All of this will be equal to mine over to So we need to continue with this equation. Now we will continue with getting some space. So this part Okay, we'll continue it. One plus Ali multiplied by one minus oil. So a number is multiplied by its contract which will give us knowing over to K one month. Blood by one is one Bless away and minus only. I thought about my oy is all square, which is a negative one. And they make that they want the blood by. It would give us a postive one. Okay. Okay. This sport we would expand it. So let's take some space here. X, The blood by one is X. Oh, you want my blood by negative. All he might The blood by oy is how you square which will give us a negative one. Negative one month A blood buying negative is Blust one and we have y therefore be plus war . Oh, I want minus I X. So sees a boy minus six. Okay, this sport plus two x and all you want but by all is negative. Well, why and the means and extremes therefore all. And he had records. Why Look Door X Okay, so now we have here a simple equation and you'll see here one plus one, which is two. And we have here toe, so we can cancel this part with this part. Therefore, we have X plus war despots, which is a rare boat. Let's see if we have another play. A part plus door x minus, Roy. Now we don't have any rail port the imaginary, which is oy. Why? And oil is why minus six and plus two x socially of us plus six equals nine. Now, we would equate that airport with area boards and imaginary with imaginary. Since so we have here a pure lay rail number. It will be plus zero. Okay, so we will have X lost. Why? Plus two x Okay. So we can say instead off going like this, it will be three x Okay. Since y with Constant was why we will have a three x on. We will equipped it with night. Since it is the only rare part. Therefore, X will be toe three. Now let's sees the imaginary board toe. Why plus X equal to zero Zippel wallet will be able to native It's over. It will be negative sitting over. So now we solve it Some examples on our, um, complex A number this one. Now, in the next lecture, we will learn more about complex numbers. 7. Graphical Representation and Polar Form of Complex Numbers: in this video are going toe discusses the graphical representation off the complex numbers and the bowler form off the complex numbers. So the graphical representation off the complex number if we have, for example, zit equals four Los Cering. So how we can represent is this in there X y axes or in the rail on the complex plane? Okay, so first we will draw, Huh? This will be X. And this would be why where X is the area board. And why is that imaginary? Both. Okay, so for is really bored, which is X and serene is imaginary board, which is why. Okay, So our first step, we will take here on the x axis. We will take a distance off four. Okay. And when the y axis will take a distance off city. Okay. Therefore, we will have our point Zet. Okay, so this is the graphical representation off our complex number. Now we need to find the polar form off the complexity. So how we can write this complex a number in the form off a polar form or value, which contends is the magnitude off this and is the angle off the complex and So if we see here that if we just to connect it like this and we say is the sport school, the all so off will be equal to from buy cigarettes era. You would see that, or is equal toe square root or four square plus three square. So it will give us fine. So this is called the magnitude off the complex number for square Blas City square. So this is called the magnitude or the amplitude or magnitude. Oh, complex. No. Okay, now we need to find the angle. Okay, So the anger is here called Sita or Ceta or whatever it is according to your country or language. Okay, So Seita will be a quantum from the roll off that 10. You will see that, then see, that is equal to 10. Sita is equal tow opposite over there, Jason. Or was it which is serene over the adjacent which is for okay. Therefore, Sita will be quick toe Dan, minus one serene over four. So now we have our magnitude and we have our magnitude and we have our anger. Now we need to write this in a form off a polar form for the complex number. So now we have our magnitude, which is R equals five, and Seita or Sita, is equal to 10 minus one 3/4, which is equal to 36.8 all to be more precise. 6.86 Okay, so now we have magnitude. And so the polar form off the complex number is in this war is it would be quick to our because I am, say, that plus or science eater. And all of this is gold or or the imaginary part and you'll see here that represent is this porter absence X and this part it was in this war since. Okay, if you would like to know where this where this come from, X this port from their design rule. Okay, here go, Zion. Say it is equal to what? The adjacent over the high bottoms. So that Justin, which is for over the Hypo Tennis, which is our or instead off saying, for to make a general X so x will be Quito. Our design set force a wife. Okay, we can say sighing. Satar sign a quick tour The opposite opposite. Over the hypothesis opposite. Which is why over the botanist switches off, so why equal toe are sighing? State? So what's so? That's why our complex number has this for So now we can write this also in the form off what we can say that is equal toe the magnitude or and angle state. So this is the same as this form. And this escort is a polar for off complex number. Okay, so now we need to do have some examples right now, in order to understand how to write their poll or four. Before we get into some examples, we have to learn something new. We knew that that equals X plus r y. We could write it that equal are because Ryan satar plus oy science satar. Okay, so now we can see that or for the magnitude offset. Okay, is oh the magnitude or the amplitude off the complex, the number is and the angle or instead, off saying anger, we can say argument. Then it is said all seater. So argument that means that we are talking about the angle off the complex number. So now let's have some rules about the argument said if we know that argument sent is equal toe satar. So what is what about the argument off that both. So what is argument off center bar. Okay, so that the power is equal toe. Was I in Sita minus all science? It okay? And you know that from this graphical presentation that the science is negative. We're okay. So we know that our roll call drastic. OK, A s T. Okay. See? So this means all of their sign and because I intend all the pragmatic functions are boasted. Okay. Here it means that that sign is the only one which is posted here. Means that 10 is only is a Boston seen means that that the design only that was give here. Okay, so we have here and they get them. Always wins either. And we have a whole step. Was I in Seattle? So we can write it in the form off. Well, Zion, so 216 minus eater blow us boy. Sign 116 minus. Either your house, That is that the angle here must to be equal. Okay. And they must have here about Steph sign. And this must be a cause. I and this must be a sign. Okay, so we have your design, we have signed. Okay, That's the first thing. The second thing we have here a negative and would like it to make it a boasted value. So in order to make this almost that we need to change the angle here and here. And they must be equal to each other. We need the cosigned toe, people step and society would be negative. So cause I in both step and signed negative here was I am postive and all of the other documented functions are negative. So it will be here 316 minus said so on 16 minus said. Okay, so this will give us the argument off that more is equal to what? Yes, it will be one toe surrounded 16 minus seat or negative state. Because we can instead off doing this, we can see because I am Satan. Negative satar. Plus I sine negative state senses at Zion. Eats is anecdotal value. Okay, so was I need Sater is same escort science. It sign Negative. Sater is negative. I science seat. So that is a forest scene. Now let's define the argument off one of ours. It Okay, so we need to find the angle of one of ours that we have one overs it. If we have that X plus all you want Therefore one of ours It is one of our X plus y, but the black boy X minus y over X minus oil, Roy, to remove the complex apart in the denominator so it will give us X square. Plus why square x minus All ey. So this is as X over X squared plus y square minus y over X squared plus y squared. So I find is that we have here a magnitude on Donna's and magnitude of the year. This is called off aim, for example, and to be okay And do I have here a negative sign? So finally, one of present is a minus being. This case is the same as Zed ball where we said x minus Ali one. So they have the same anger, Okay, since we the white by the same magnitude here. So finally the angle is either or argument off. One of our scent is equal toe Seita equal to the angle off. It, of course, is equal to negative sit same as before. Since they have a negative sign here. And the magnitude off both of them is divided by the same value we can do. Of course. Something else which is dividing Dan, Win this one being over. Or why over X. Why have our X here? Uh huh. Or there be over. If we talk about is one of ours, it okay? I'm sorry for being too slow, but I would like toa give you prime to understand everything in details. That's why I am taking. I am talking and I am explaining very slowly in order for you to understand every single step in the complex, not in the next seven you are going to solve for some examples. 8. Solved Examples on Trignometric and Algebric Form: Now, let's have some examples. Owns the rig metric function and the algebraic form off the complex numbers. So we have here for examples in this video, our first example. We have set the magnitude off that is equal to five. The argument off that is a 1 11 boy over three. And I would like to write each off these functions in the trip. No metric form and algebraic form. So our fittest one, the magnitude off that is equal to five. The argument off that is equal to 11. Boy over three. Okay, we need to write this in the Trigana metric for so we know that that is equal toe all because I am said that. Plus, I sine say it. Okay, so our is the magnitude off the function five cause I in status quo Zion 11 boy over three plus. Oh, silly in 11 boy over three. Now we need toe soul visits and get the value required off design. 11. Boy over serie so X is equal to So this is that big metric for the algebraic form is that equals X plus Oh boy, where X is equal to 50 Zion 11 by over three. And why is equal to five sighing? 11 borrowing over three. So now we solve it their first example. We just to take the magnitude here or and see that is equal to hear that. See the hair. So this is that signal Metric four. And this is a deliberate for where X is five cosigning Ativan, biodiversity. And why is five signed? 11 bawling over three now, for example, Number two. That in quite negative Syrian Because I'm 16 Must I sign 60? So it will be acquired city minus cause I'm 60 plus or minus. Here, minus oy. Sighing 60 Now we need to find Zach. Recommit reform off this. Okay, so that rig metric for is that equal are goes I satar blows all I saw in cedar Okay, so we have here Are that make me don't hear boasted magnitude, which is city Okay, Is there cosigned and sign Must have a Boston very and the first of value or the X must be cause I and why must be signed. So what you see here cosigned and sign. Ok, so that is great. But we have here negative and negative. So how we can go something like this. So before we get through this, we need to understand something here. If we would like to. It changes our science off that design and so on. You will find here, here a all of that trick. Pneumatic functions are posted. Here Are the sign is supposed to then is only posted in the sub quadrant, and force a quadrant is the cosigned is the only most value. So if would if we would like you will see is that we have here cause I in and sign Most of them are negative, So yeah, are both of them are boasted So it is not acceptable. We have here sign Boston, but here, Negative. So it will not be acceptable here, Zach. Assignments posted, which is up here is negative. So not use this, but here is a sign. Negative and go, Zain. Negative. Okay, so we need it. Changes our science. Using is asserted quadrant. We have here the relation in order that changes assign only motel sign and function. So they're saying only for the second quadrant we will use 180 minus satar, or sita 180 plus eater 360 minus seat. So are using that sort of the quadrant were because I negative and sign need 180 plus seat . So I would say that it would be quite toe three because I am 180. Below us is the angle 60 plus sign 180 plus 60. So it will give us sitting. Was I 240 boss, Sign 240. And of course we have here, boy, and to have here. Okay, so now we wrote is a function here in the form off that signal. Metric function four. Is there a jailbreak phone? Explicitly. Boy, his ex would be Syrian Because I am 240. And why Siri signed 240 now for our third example, which is so you You can't do that. Minus. Uh huh. Plus, I go, Zion. So the 100 search. Okay, so now we need to find this. Okay, So that we have your negative six. So that will be equal to six. Negative science or 130 minus I cause I insert 100 30. Okay, so we have here toe things we need here, cause I and slight boasted and another was the value So we here have signed and we need because I and we need I will still value here. Negative. We need doubles the value and do here we need sign. Ok, so it's years that we have here negative and donated and the most of them are negative in that serve the quadrant K. Which is what Here. Okay, where is that then? So I negative and design made So what is what about what? They're the techno metric function. We need sign as a design and design as a science. So what are we going to do? Okay, you have to know that this room is used only for changing signs. If we would like to change the signs and the function, we will use 19 minus stater 90 blasts theatre 270 minus eater 1 70 Loss seat. So we have here this change sign blessed function. So we are here talking about to serve the Quadra so we'll use 270 minus seat So six So I am because I 270 minus 130. Okay. And we have here plus boy saw him 270 minus 700. Certain Okay. Therefore we will have six. I was lying What 30 and 30 and has 30. What will be 60 negative sacristy plus oy science. Negative. 60. So this is the drug No metric function wing Need it? And the fourth a Jobbik fall. It will be six because I am 60 which is X and six negative. 16. So I am negative 16 Which is that wall now for our lost examine that equal it. Why? Minus root Serene. We need the public form and that rig Metric phone. So if we have here something like this and you see here we have a complex number here. So you mark the blind boys that contract one plus quote three oy, one plus road serene. It will give us one. We have the negative road City of the blind by Rote City we have here. It's really three will give us city negative. Oy and oy, I square negative square, which is one. Therefore it will give us plus c and this part it plus eight Grocery. So this will give us four. So it will be it over four. Plus it roots 3/4. I 8/4 is what Yes, it will give us toe plus two. What? Serene. Really? Okay, so now we have the algebraic form required for that Trig metric. For we will yours he have. Now we have now. Is it a question? Oh, plus toe Roman city. So are, or the magnitude is route. Don't square. Plus, don't roll. Three. What a square. Okay. And what have the magnitude off our forsee angle, Dan? Minus one y over x. And we would have our anger Ford's them trigonometry for that equal or which is found from here cause I am Seita less oy sighing Sit. Which is from here. So now in this video, we solve with some examples when have toe get that signal metric for and help to get there at public form off the function. 9. Operations on Complex Numbers in Trigonometric Form: in this video are going to discuss Is the pregnant like for off a tomb complex and numbers off course. Is that'll complex numbers that can be a road off them or the division off them because we learn it before about with their addition and subtraction. So if you have one equals toe, are one Was I in seat A one plus always it on. And they do are to because I insist at all. Plus, I sine c two we would love to get their multiplication off that one more blood boys it Okay , so that one Montebrun by citadel is equal toe are one or two Zain, See the one love seat must always sighing See the one plus cedar to So is that if you must have, like two complex numbers in where you want to live Their magnitudes are one or two and we add the two anglers goes I sit on Oh, I see. They want to see that you can no more Isis okay for yourself The proof did one I want a blood boys into two is equal to or one or two Now if we were not the rises built brackets we would have go Zain. See that one? Go Zain. See that? The first month of blood by their first and second by second minus sowing seed The one saying See that all Plus boy that means and extremes. Oh, of course are here. So I don't see the one I was lying. See that all endorse. Plus because I am Cito one sign Sita one. Okay, so I don't want our toe will be equal The same one are to equal off the light bulb. Was I in Tito? One was I insert a tow Minor science Itaewon Science it What does it mean? It means that we are getting their design. That's a mission off Sita one plus sea turtle, right? Because I didn't see the one Prosciutto is cosigned. The first multiplied by Faris. Second minus sign fittest signs a second and this part represent Is there that sign Sita One plus. See that, which is the same as he. So now we learned how to get. Is that some mission or summat? Application off. Two complex numbers. Now let's get the division off a too complex and we have one. Or was it? Don't. Hey, quit toe Arwa over or two. Design Sita one minus citadel plus Alright. Sign See the one minus c. So that division off a toe complex A number is the division off the amplitude or the magnitude? Or the model s upset? Or one over R two. The difference between the two angles. See the one minus seated, as you see. Okay, so we need the proof of this equation for you. Of course. Abroad. That 1/2 equals R one over r two this divided by This goes I in seat two. Plus I sine see that design. See the one Los Oy sign Cito one. Now, in order to get the complex, the oil and the real and the imaginary separation off them we don't want the blood buys the quantity. Get off! Is that the new natto? So when Multiplied boy, Because I see that all minus oh, I saw and see the door over goes I see that Oh, minus always So you see the tool in orderto cancel their complex Bharti So if we continue our equation that one over there equals r one over r Okay, so this month about by this will give us schools. I see that square plus sign. See that square is was on square Science Square and plus or minus oil will give us one. Okay, The other bar. Go Zain. See the one? It was science for one. And because I see that to minus always science eater Don't Okay, so now is this part which is design Sita to square sign. See, that'll square is equal to one therefore will have our one over r two. Uh well, not the light goes off them Cosigned Sita one, cause I see that And this month the blood by exists will get us plus So I see the one sign See? That'll plus oy sawing Sita one See the Zion C do this multiple by exists will give us negative because I see the one sighing See the door. So this will give us what one of our design. The difference. See the one minus See that. And this will give us what? Plus all so I sita one minus See that which is the same as we got here in this equation. So the implication the addition and multiplication models for the division. The division off the amplitude and the sub direction off the anger. Now we would like to get sent for and and said one over K so that bar and assembly toe all was eyeing Sita and Sita, plus our design, then seat. Okay, so what did we get this? Okay, simply before have set more to which is equal to set off the blood by set. Therefore, it will have Zamata application off a phone number Is multiplication off the amplitude and that's a mission off the angles. So it will give us our square because I I don't see that plus oy science I don't see so generally told toe and the toe and in. And so you would like that more and it will be our foreign cosigned Insead If we would like that power for it will be our bar for design force it. And so now I would like to get that power came So to get that barber won over key Is it our work? A similarly as it more or our one of our key Go Zain Seita Over here Sign See it them over . Okay However, we will add an extra term which is doing boy boy off. Why is this? You will find out in their solutions off their solve. It examines in the next TV. But now you have to move that in order to get there. One of our key. It's the same else. Square root. Okay, If we say that Barnhart, then it means that we are getting the square root off then. Okay. That, uh, bar cube we are getting is that over one of our cities who are getting a cubic root offset ? So is that next event you will find out why we have toe by off and all is equal to zero. One door are 10-K minus one. So this route is really important and you will find out Why is this all with exhausts? So for this video will learn that some roads and in the next video, we will get Goto solve it examines 10. Solved Examples on Operations of Trigonometric Form: now let's asshole for some examples about what Sam econometric for we need in this examples we need that one on the blood boys in tow. So it is the first example we have that one equals do on one design. Plus i sine to see them and they don't quinto are too design Siri Sita Plus I sine saying so. Said the one Marta black boys in tow is equal to for one. What about my arto? Does that make me Jones application and some mission off their anger? So Sita and Sita and Sita plus my sign Why see, very simple exalted number two We have said one going right designed by varsity. Alright, signed by You were saying Exacto because I'm boy for six must always I'm boy over six. So said one. That too would be acquittal five like by toe exam, A publication office amplitude which is then there Some mission off the angle. So we have here for you for three. And the boy over six. We can say by sorcery. Is your boy over six? Okay, so the anger because I sitting boy over six plus boy sawing sitting boy over six. So These are two simple examples on that one. And no, let's have an example, which is more difficult. We need to solve this. Equations h off this. Examples. Three examples we need to find the year is the value of X is the value of present, the value of X and y. So let's take each of these exams Number one execute plus one equals e so excuse equals minus one. So if we think off this example in real life, we need answering X multiplied by each other will give us a negative one. So we need X. We could give it quote off a minus one, which will be called what, Of course, minus warm. Minus one. What about my minus one about by minus one will give us minus one here. Okay, so according toa our brave years knowledge before learning about the complex number, we know that X Q equals negative One is therefore X would be equal minus one. However, using the complex number, we can find more values off eggs. As you know that we hear have exports sitting. It means that we have a three values. Would she can give us negative one another example X wit quite for therefore X will be equal to don't or X equal minus storm. So we have here to values what you can give us for Okay in this equation. So this equation will have a serene so your sense So how we can get their city ocean. So we have here execute one minus one So minus one can be written in the form off a complex afford which is was I boy. Plus I sine Bali was I am bored or design 180 is equal to negative one and sign boy is equal toe zero So that's a mission off this complex, the number is negative one. So we just transform me from this four toe Technimetrics For now, let's get X was I boy plus boy, Sign following you'll one of our spring game. So now we need to get security route OK from them. Brave Yes, video. We said that it's about one of our king is our board one over King. And because I in Sita plus by our over Okay plus I sine sita less to buy own over k Therefore, by using this equation, we Carides this in this for their four x will be going toe one since one board cube is one was I am by over three Was I boy bluster by our que which is a can plus on site boy the last two by all or were sitting where oh, will be well, toe zero one. And as we said before that are is from 01 toasty and so on until k minus one work a Here is three therefore, K minus one is toe. So using this equation or these ours, we can find the three solutions those dysfunction You understand now why we use, though by our two by our is used in order to get a different city oceans. So let's see now finds the city ocean. If we substituted with our 10 therefore, x will be toe design Bali over three. Plus I sine boy over sitting right zero while you were three boy over serene. So what does it equal? Biodiversity is anger 60 Therefore, it will get us all. Plus what? Yes. Oh, right or wrong? Let's think about it, cause I 16 will be equal to half and sign 60 would make one. What would see over to not harm brought. Sing over to Okay, The only signature give us here Half is assigned 30. Okay, so that is the first solution at our one X would be equal to boy Blust Oh boy will give us sitting boy so close I'd boy boy must buy over sitting city Bali over our city which is boy plus oily sign Boy, it is the same as the first dissolution, right? Vacuity quote And what on I minus what I would give us negative one. This is a solution or the rail solution which we always know before learning about the complex And, um okay Oh, what X would be equal to if we adhere to martyr blood by Do is for four boy plus five boy, it will give us cosigned by boy over serene plus all signed for a boy over serene So now Well, only that you have three solutions here According toe the value on and we have articles. You are equal one and record. So this will give us what huh? Minus road three over toe. So we have here we have now city solution This and this and this and that we need execute, which means three solutions. So you think is the complex. Remember, we has 123 solutions. Now let's get door Another example Now, Now let's get toe another example, right? Something's this example before seeing is a solution in this video. So that said you're off. This will be we have wrote City plus I all power to over. So we cannot solve this without transforming in tow. Complex of all right. Yes. So let's transforms this in tow. Trig no metric for So this would be we will get Phyllis is a magnitude are which will be ableto walked Zaria board square plus the imaginary bought square What you're give us walked It would give us I don't know. No, it will give us his value. Okay, so All right, we'll get us toe wrote City square, this city and the plus one is four and four is equal toe. Okay now is angle Sita, then minus one. Is that rail board y over X, the imaginary over the area One overrode sitting So this will give us angle 60. Okay, now we have their amplitude and we have their anger. Now let's get to another scene when you don't write it in the form off the rig, No metric form so and do cause I m 16 plus I sine 60. Okay, well, over five. No, we also them okay, Don't power over Fight, okay? And we have a close eye in 60 and always find 60. Okay? So let this magnitude outside. Was I in 16 and we have to over five. Okay, we multiply too multiplied by it and I will tell you why now Do not applied by 60 is 120. Plus I sine 120. Okay, now we have one over. Fine. So we need five solution for this function. That's why we separated both off them toe Our interest is that two multiplied by here. As if I'm getting is a square off this value and faves toe get five values for our function . So faves will give us what to both over fight again. Design 100 went plus to buy or over five plus sign. Oh, I saw him 120 must buy or over five. Okay. Where our own will be called tau 0123 and 412345 toe have five solutions for this equation . So assembly boy substituting in this once a by zero. Second boy, one boy, two boys, three And before we can get for solutions offset in this function. So now we need willing it in this video about is those examines? Now we have one last example in this video. I know it is a long video, but I am talking slowly again in order, toe, toe, get and understand every single step in this complex number missiles I will lost Example Now x plus on Roy, it will do one plus four c. Ali one minus four road city. But what a ball we need to find the very off X and world price workings This before seen solution and game. So first step we will take this, Bart and they get that geometric function off this So you multiply it by the conjugate. Okay, now one this multiplied by this negative and I will give us Blust one four heart about my four road serene it will give us 16 multiplied by sitting which will give us 48. Okay, now is this part the first amount of blood by the 1st 2nd 2nd ends, means and extremes. It will give us one passport. Route three are all square. And I will tell you why I did not calculate this. Now we have 49 one blast four Road city Oy. Okay, You'll see that is this part is equivalent to one plus four Broad Siri and seven square. Okay, this divide the boys is this is square for tonight. This square is one plus forward three I always square So this is equivalent to all this now except bliss all you Why is equal tothis our home? So what happened here is that to cancel the Wiz this whole So we'll have finally 1/7 plus 4/7 Rolling city So x people toe one of our seven And why will be put forward? 3/7. So this complex sing Tornatore We have very example sing like this. So when you are trying to solve the equations in the complex number Troy solve exam slowly and step by step. The complex number is entertaining and really easy thing toe So I hope you benefit from this video and let's to go toe some signal in that make us video 11. Exponential Form of Complex Numbers: Now let's discuss how toe obtain Zack's financial for off the complex moms. So in the biggest videos we're learning about the jailbreak form, the trigger metric for and now we're going to study the exponential for So if we have is it equals X plus all wine. So this a public four can be written in the form off exponential function that the complex value equal r E O r oy seed is a magnitude off them complex. The number the same as the from Trig, No metric function or equal Route X squared plus y square area board, Square Blossom, Missionary board square and Sita. The angle istan minus one. Why, over X or off course, was I minus one X off or or Zion minus one? Why over all? So this is called their exponential form off complex. And now let's have some examples in l on the complex number 12. Solved Examples on Exponential Form: Now let's have some exam bits on there. Excellent food. So the first exam, it is that big 11 Minds wrote three oy and would like to obtain their exponential form. Where is that is equal toe are e born Sita. So the first thing winning it are, and we need to see that So are is equal toe brute X squared plus y squared. So X squared is one squared plus y squared, minus root three or square. So this will give us what it will give us too. Okay, Now for the angle, cedar is equal to Dan minus one y over X roi over picks, which is one which would give us an angle off several 100. The green. Okay, if you'll see here. Is that that Dan is negative in Zach, second Quadrant or their force order. But since that we have here negative wrote city and one so that you can understand the wares Angry came from where I feel negative root city. Okay, we have here Roy and ex, we have X as opposed to value, which is one and we have a negative root city. Naked city said for years is our that. Okay, so that now in the fourth quadrant. And it has a negative value. So what city will get us? What, Dan minus one. Good sitting is equal to 60 degree. But since you are dealing here with the force of quadrant, therefore the Sita will be called to see 116 minus safest, which is 700. Okay, so again, where that that's 300 came from 10 minus 1 may be the full 3/1. So we have here 10 off a negative value. So by using here and the representation off their dead function in X and y axes or rail and imaginary access, you will find here we have a boasted value and in a very and a little boy and almost of x will give us that in the fourth quadrant. And as you remember, we have here surround the 60 minus eater. And you have here on 180 Blas Sita on the 118 minus seat. Okay. As we said before in the previous videos. So now we have the angle Sita, and we have our therefore that will be quite toe. Don't e or negative. Several 100 or and for you. You have to understand that we shouldn't write it like this. Why? Since you are dealing with an exponential Therefore the angle here must be in the form off , boy. Okay, so this 300 will be removed. Okay. What? It moves that set 100 here. Okay. And we wouldn't right. 300 over 118 boy toy. Okay, so we must write it in the form off Radiant, which is boy over 180 in order to convert it into the radiant mood. So this now is correct. 700 is wrong. Okay, so that is the clearest example. So our first exam, it is okay now, I would like to get the job reform for this functions that quarto don't e our boy over three and you will see is that we're using voices who are dealing with the radiant index of Financial four. Therefore, it will be quite this is called the Are and this is the cedar. So what are we going to do? Assembly s or cosigned? Sita, Plus I sine seat, it will be equal toe. I was lying. Biodiversity. It is 180 over saying, which is 16 plus Oy sign 60. It will be toe month about design 60 which will be half plus Oy sign 16 which will give us what yes will give us wrote sitting over to So we have one plus oy And of course we forgot something and you have toe realize your own mistakes. Toe design, safety and the tomb always sign 16. So we have here. Therefore our final answer is one plus route stream on This is the algebraic form required now example number three. So we solved this. Now we need this example We need to write this index of financial for yet equals four design by over city minus all sign boy over sitting with the same miss would we? And did information off the rig Elementary form we will does the same Met steps here again . So our first step is that here we have our already It is almost every okay we have here cosigned and sign. OK, but the problem here is that we have here most value, which is ok, but we have here a negative value. So we have a negative sign and design Boston. Okay, so here Hey s de scene So this is not valid since we have a negative sign. Sigh. In Boston, it is not very to Dan. We have a most of design, so it's not valid. Therefore, we are in the fourth quadrant. Therefore, when he was around 60 minus see the or by minus see that as a radiant since way makes a linen shirt. For we use the radio message equals four design. Oh, boy. Minus boy Adversity and plus cosigned. So I am Oh, boy. Minus by sorcery. Okay, I don't forget that imaginary number. Okay, Now we can write it as that equal the amplitude and Xanga here. Tow us, boy over three. Okay, boy. Minus boy. We're sitting all as Ali. Now to our lost example. We have here that one equal Abe or 30 and we should not write it like this, So remove it. Yep. It is a mistake, which I did on purpose. So as you again understand and remember your own mistakes. So it is, boy over six and boy over three. Okay, so now we have here. Is that one equal to over five or six? And that, too, is equal to four evil biodiversity. Now I would like to have pain. The multiplication off that one's a little And the division off that one said OK, so very assembled equation. Does that door multiplied? Boys, they don't. Okay, so we have here at the exponential most about by each other, so don't not run by four. And it we have to explanation the multiplication off to exponential. What does it will give us? It will give us the same mission with exponential. Therefore, by over six of all plus boy over city boy. So this will give us for 98 boy for six end boy over Siri, whatever it will give us, boy over to only. So that's how you can solve the equation. Really easy. That don't send itto is a multiplication or dormant application off the toe. Amplitude is and submission off things. It is the same as if you remember from that rig, no metric functions. So we said Sita one loss seated toe and so on for the division off them, that one over there too. So we have to over four, which will give us half and to exponential divided by each other, will give us by over six oy minus boy over city. All so this will give us Oh, e About what? Okay. Minus this'll can be transformed into a boy over six, so it will give us minus boy over six. Okay, so this is the same as a metric function. We said that the division off the complex numbers is a division off. The attitudes here ends. That's up direction over there. Angles by over six. Oy minus boy over serene. 13. Cubic Roots of Unity: now in this video are going to discuss their comic roads off unity. And we will discuss this properties. And we will broke it in this video and then exhibit You will have some exams. Okay, So the first thing, What is that? Cube roads or cubic roads off unity. Okay. Assembly from its name. Cubic root. Cubic root off. Unity means that we needs a cubic wrote off one. So what is the cubic wrote off one. Okay, so one in that signal. Metric form. Is it toe design zero plus I sine zero. Okay, so this is equivalent to one. Okay, So what is a cubic wrote off? 11 our one of our city a quick was I zero plus I signing zero or were sitting. So we find here. Set, This can be written as equal Go Zion zero plus by our over city. Plus, I saw in Dubai are over three. Okay, since zero bluster by over, over city and the euro Plus to buy our over city from the role we discussed in their rivers renews, which is one over king. Okay, so now we would like toe get Zach cubic roots off one or will be called to 01 and two until came minus one. We came here is three. So now let's get r equals zero. At article zero, we will have a Zion zero and sign zero, which will give us one. Okay, that is number one. Their first disillusion, The second city Ocean are 11 will give us Was I in boy over sitting loss. All sighing, boy, or over three. And where r is equal to one now design by over three plus always silent toe or cities A quinto what equal to minus off. Plus wrote city over. All right. Okay, so now you'll see that this if we take the cubic group off, this is there. And sitting your more This was a power three executive wrote off with every day this 43 it will give us one. And this to the varsity will give us one now for our equal to. So we will see that we have now solutions. Okay, let's see. Are equal to at or equal dough Zin. We will have design four boy over sitting. Okay. Plus I sine four boy over city since we want the bloody boy toe. This will give us negative. Off. Minus brought three over. Toll on. Okay, so now we have this ring roads off that one. Okay, so this is a power three will give us one. This was a poor city. Will give us one. And this was the varsity. Will give us one. Okay, Now we will say that this name the S Omega. Okay. And this name the as Omega Square. So the first thing that the cubicles off unity or one Omega and Omega Square, where one is the same as one off course head omega is negative. Half plus roads. Three over Omega Square is negative. Love. Minus ropes. Three over to on. So why is this called Omega? We named this as omega and wisest. Make a square. Okay, So omega is equal to negative half plus three over to our therefore, Omega Square will be negative. Plus what city over toe I If you get that square off this, you'll finally get don't exist. Value, which is negative. Minus root city over to That's why is this called Omega Square And this court Omega. If we take the square off this value, it will give us this. Okay, So now it was the first property. Which is that is that city solutions are the following day now. So now we have the sea solutions. One Omega watches negative. Hopeless. Would Seaver toe oy and Omega square Negative. Half minus. Wrote 3/2. So let's suppose a second property one plus omega omega squared is equal to zero. So if we want and take the submission off this negative wolf blasting it evolve is negative one. And this most This will cancel each other. 10 This will give us minus what, and one will give us zero. So this is the first property. One plus omega Omega squared is equal to zero to the second property. Or the served a priority one multiplied by Omega by Omega Square, which is Omega. You will give us one. So let's see now. One Omega Oh, man. Got square. Is this Okay? So we have a number a complex Humber, and it's conjugated. So the multiplication off them will give us one negative urban. The bottom will give us 1/4 plus 70 over to our I minus rotary over. So I will give us plus 3/4. So this will give us one, which is the same as a certain property. Okay, Now let's find Zam. This property and this one Omega minus Omega square. Equal roots three on So omega minus omega square. Okay, this mine is this. We cancel it. Charles, on this one. Is this Well, give us sort city over door. I minus rule city over two, boy. And there is another negative year. Okay, negative. Rootsy over toe and minus. And the 1st 1 will you have us wrote sitting on, which is the same as this four Omega Square, minus omega equals wrote three minus with three oy. So omega square, this one minus omega. Okay, so negative off. Minor suits me over to R minus. Negative off. Blast rolled. Three over toe are Will give us this would be cancelled with this one. And this minus this one. It will give us, uh minus wrote city, which is the same as this property. Okay. Now, to our lost a priority one over Omegas, Omega Square and one of our Omega Square is Omega. So let's get this 11 over. Omega is the oil toe one over *** the world plus rewards me over toe. We don't have to apply, boys a contract it negative. Off minus Would city over toe ar minus off minus 13 over door. So this part will give us a really value off. What? Of course it will. Give us one. Okay. Which is this is an Omega square. Uh huh. And this is an omega some of publication of Omega Square and Omega. Give us one. Okay, so we have here one, and we were just to have this one negative off minus root city over door on this sport, which is negative, have mine. That's what city over to all is the same as Omega Square. So one over omega is equal to omega squared. Now, door, we lost one of our omega squared minus off minus root city over door. Similarly, we multiply by the contra git negative half, plus Route three over to all minors off, plus bored sitting over toe. So this one is omega. And this one is Omega Square Omega, multiplied by omega squared is omega que, which is one. So we have here what minus off lust would sitting over toe. This is one of our omega squared. OK, so minus off. Plus receiver to Ali is the same as Omega. So finally, we proved all of these equations. Okay, So instead, next, if you were going toe, have some examines on secular big coats off unity. 14. Solved Examples on Cubic Roots of Unity: Now, let's a have this store. Examples first broke that one minus omega. But the blind body one minus forming a square is equal to three and one plus Omega square. All our 15 is equal to minus one. So the first example one minus omega one minus omega squared, Which is the lift. The hand site. Okay, okay. Let's find this first. Okay, we went. Just expands this brackets. So the fairest tomato, but by the fairest called one second amount of lead by second is equal to plus may cure the means and extremes. Negative. Omega negative. Omega Square. So we may get you from here Is equal to one. So one plus one. Okay, finding us Omega minus omega square, we can write it. And negative Omega plus Omega squared. Right. Make the plus Or make us Where from here, is it? What? What? Negative one. So this negative one So negative and Donna's a negative year. Give us plus one. It was a submission off. This will give us sitting, which is the required to do. Okay. Try to solve this before continuing the video that was a solution is one plus Omega square or 15 which is also the left hand side. Okay. What? Blust omega squared. What does it equal? What? Plus Omega Square is a negative omega. So we can write this negative. Omega Power 15. Okay, so, Omega we can write it for We could write you, Nathan. We can take one of these powers make thank you, but one. Fine. Okay, so this is similar as this one. Okay, Zack, you'll Omega and the Cube off. Negative one is negative. One. It's up. Okay, so omega Q is it? But what one. So it will be called to mind us one, But what a fine which will give us what negative one, which is the same ass. That right hand side, this one. Okay, very simple toe examples. Now let's have another toe. Examines the Here we have here a bracket one minus door, Omega plus omega square. All multiply, boy. One plus omega minus omega squared. Equal six. Okay, so we need to prove that then, if downside equal terms. All right hand side. So the first step, we write the equation. Okay, so USIA number one one minus stole. Omega Blust, Omega square one plus omega minus omega squared which is that left hand side again. Okay, Now, before I saw this, try solving in for yourself. So that's to stop the video and try solving it. Okay, so in animals that situation one minus two omega Omega square and one plus omega minus, forming a square. So what we can do? OK, we know that one plus Omega square is equal to what is equal to negative Minga Square. So it'll write it like this. Negative. Omega one plus omega squared is equal to negative Omega. So we have negative omega narrative toe Omega. Okay. And this 11 Los Omega one plus Omega is negative on me. That square. Yeah. So this one will give us negative sitting omega. And do we have here negative door omega squared. So if you multiplies is it will give us sits Maiga, do you? And Omega Fuel is equal to what? What? We're called to six. So this is the same as required now, for it's a second example we need. Is there value off one Boston Omega blast in Omega Square over one minus three. Omega minus three. Omega squared. If you see something like this, you'll see I think that this is a very complex A problem. But actually, it's very easy. Just think off this rules. And how can I apply ice is the front rolls into this. Okay, so we have here one lost eight omega blasting Omega square. So we can hear take then as a common factor and see what would happen. One plus 10 Omega must mean that square and the year one My enough city Omega plus Omega Square. So now it looks is something familiar to me. What is it? Omega plus Mega Square is equal to negative one. So it will be went toe wrote one plus 10 Monta blood by narrative one. And this one is looking familiar. Toe what? Omega. Also negative. 11 minus three. And then you get the one so it will be equal to one. Minus 10 is equal toe minus my right. And this 11 minus city. Negative one. This will give us plus four blustery. Therefore, the Dalton is plus four rooter negative value. Again. We'll get back those a complex of game so negative outside his eye and wrote 9/4 mind Over four is a tow plus or minus Syrian over to boy. So the complex is hitting Game turned to be a very simple, complex month. Now let's I have is our last toe examples here. Okay, so I would like to prove that this sport is equal to 1/3 and this party Quito one. So let's assault. Sold each off them on Troy solving it by yourself or by stopping the video again. So number one omega over one plus Tau Omega Square, All square plus omega square over one plus Tau Omega or the square. Okay, so the first step again is the same as we did before. What? Plus, to make a square. There is something familiar here we can take one must make a square as negative omega so we can write. It will exist Negative Omega. And we have here a remaining on me in the square, so this can be made plus omega squared plus omega squared and two plus one on one plus omega squared plus mega square. One plus omega squared is negative, Megan. And so make a square is the same as it's a okay. And do we have here Omega Square this part and, uh this part is Omega. What a square. Okay, plus this part Omega over four. And down here, this comparison as Omega plus Omega one plus Omega is the same as negative Omega Square. So we corrected negative on me and that's where and we have here. Plus will make Okay, now this all the square. Okay, so now we have a tool. Numbers now would like Toa have a common between them. What is the common between them? You will see your negative omega plus Omega squared and negative will make a squared. Plus, for me, the difference between them is a negative sign. However, we have here to square and that we have here square. So the negative sign between them it does not matter. So we can take one of them and collect them together. So he hadn't had example. We will take this as a common between them, or make a square minus omega all the square. Okay, we will add Sisto Village, Omega Square plus Omega by +14 So this will give us make a square and don't make up for made up. All four is what omega que, which is one and omega. So Omega About four. Is Omega. Okay? Omega Q is one and the model, right. But our own Miguel will give us Omega. Okay, so I only get plus Omega Square Omega, los Omega squared will give us negative one. So we have here. Negative one. And do we have here? What? Omega square. Minus omega. All the square. So omega squared, minus omega. From here is wrote City good three or all the square. So negative one. This square will give us negative sitting therefore will have 1/3, as required heat. So it seems complex. But just by thinking how toe applies these rules, you can solve any problem like this. Now, Darwin lost. Exam it one. Okay. Over four. Plus three for me again. Plus tau Omega Square plus 1/5 plus three. Omega plus four Omega squared. Okay, we need tobe roses. Equal one. So the first step, we will just yours. The same rules. Okay. We can hear, though, something we can devise this in tow. Two plus two. And this in tow. Mega plus omega. Okay. And this to Omega Square. I chose to Why two? According to the lowest coefficient here, that was toe coefficient e is to omega squared. So I would like to find to omega and to for this one with the least a coefficient is sitting. So we'll devices into three plus two sitting omega, and this one will be sitting omega squared. Plus Omega Square senses that almost a coefficient is city. No, we will take Don't and America. And to make a square sitting city Omega and city made the squid. This would make well what zero. Okay, so this will against them, which my exists. Okay, so we'll have one plus one over. Don't that remain in the door and the remaining his omega and here will be equal to door plus omega square. Okay, so now we have this simple seem and would like to get again one. So how we can do this. Okay, we have. And Mr Now, which is just combining this together plus with me again. Dole plus Omega Square, two plus Omega Square and two plus omega. Okay, Now, let's continue here so that you can see the solution forward plus Omega squared, plus or make right now for this boat, it will be four Mustoe Maiga plus or make a square must omega que Now let's simple for Is this again? Now, let's continue four plus omega squared plus omega. This one, we'll make a squared. Plus omega is minus one. Therefore four minus one. And we have here four. We'll make a Q is equal to what one? Okay. And do we have here toe omega and Amega Square? So we can write. And omega plus only Inga Square. So we have sorry for you, Blust do Omega. Blessed made a square me again, forming a plus Omega Square Omega plus omega squared is minus one. So we have here minus do since this is minus one and minus two. So finally we have 3/3, which is one as required here. So I hope you benefit from all of this about complex number. If you have any questions, you can ask me in there Discussion barred for this course. And we can message me directly. I will be available for 24 hours for your own questions. Thank you. And don't forget, don't reviews the scores if you enjoyed it. Okay? Not lets them for going to five stars if you like it. Thank you.