Complex Number || 11th Math || Chapter 1 lec 1 #complexnumbers @a2zpractical991

Power of IOTA I means imaginary number How do we write "I"? It looks like an electronic piano. What happens in an electronic piano? When you turn the volume knob to a specific frequency The numbers represented on the Y-axis are called imaginary axes The numbers represented on the X-axis are called real axes Hello friends, how are you? I hope you're all doing well. My name is Rahul Singh I am a B.Tech. Computer Science student. I love studying and teaching mathematics. That's why I'm going to help you on your mathematics journey We are starting. our Maharashtra Board's First Chapter Complex Numbers, which is Mathematics Part 2 So we are starting Mathematics Part 2 From its first chapter, Complex Numbers Here you can see The definition of Complex Numbers But before reading this definition We will know its history. We will know how the complex numbers system works. was developed How people used their input to develop this number system How this number system was developed When will we get its insight? then we will read this definition Then you will understand this definition. very clearly So before reading this definition We will try to understand. its history So In 1st AD There was a person. His name was Heron of Alexandria These are the same herons. With the help of whom we got the formula of Area of Triangle You must have read in Class 8 9th or 10th Heron's formula With the help of whom we used to get Area of Triangle This is the same person His name is Heron of Alexandria. In 1st AD He has given some insight about Imaginary Number System It is very difficult to explain. Who was the person? Complex numbers were developed by several mathematicians over time, but they are most commonly associated with the work of Italian mathematician Gerolamo Cardano in the 16th century, who first used them in solving cubic equations. Later, the foundational work was expanded by Rafael Bombelli, who provided systematic rules for manipulating complex numbers. In the 18th and 19th centuries, mathematicians such as Leonhard Euler, Carl Friedrich Gauss, and William Rowan Hamilton further formalized and popularized the concept, leading to the modern understanding of complex numbers. So It is very difficult to explain. But with the help of that document I got the information. I have obtained the information. with the help of Wikipedia With the help of different platforms So from that document The name of the person Is Heron of Alexandria In 1st AD In his book Stereometria He introduced the concept of imaginary numbers. He raised a question. We will understand that question. But what do we know If we take the square root of Square root of 16 It's plus or minus 4. If we take the square root of Square root of 25 It is plus or minus 5 So he said What will happen if we take the square root of Square root of minus 16 So he raised the question. Let's see We will try to find the square root of We know that If we take the square root of We know that If we take the square root of Square root of 16 It's plus or minus 4. If we take the square root of Square root of minus 16 It's plus or minus 4. If I can gain 16, then see, I can gain how if I Even if I do 4 × 4, I'll get the square root of 16, which is 4. If I do minus 4 × Even if you do minus 4, will the answer inside the root be 16? If I can, I'll try to see whether it's possible, maybe it won't be possible if I If you make it 4 x 4, then the result will be the square root of 16, right? Is that correct? The square root of 16 cannot be greater than the square root of minus 16. Yes, we've already studied up to the end of Class 7. Now you might be wondering how something like this could happen. Minus x minus becomes plus x, plus x plus becomes plus x, and plus x minus minus becomes We'll also cover this in another lecture—let's create a 10-15 minute lecture where I'll give you inside insight into why plus x minus minus equals plus. Who gets it? Up to now, the equation we have is plus x minus minus. Minus x minus plus becomes The one you earned during class 7 and 8, Janakadhi About his lyrical works, we are advancing today. We know that if a human being is born, then what do we know about humans? What happens, plus what happens So, where is the answer to this? Root Ob 16, come alive. Attidi, Bandalanga Bonagan evaporate Tagnanagantavaya Kani's half milestone Oh, the beautiful sky, the moon, the stars That place, I saw it clearly, it was imposed. What we do not exist in reality, but exist in imagination. So, in their time, they tried every possible means to convince such stubborn people. People did not take their words as seriously, nor did they pay them much attention, but after a long time, three Metameticians arrived whose names were Cadono, Lusa Pascoli, and Scorpianeder Fedo. These three What they faced and how they overcame it, exactly. So, their equation, which is x cubed plus x squared plus x, the matter is clear, the indicator of your life, solve this equation, those who attempt to solve it, these people, among whom Vaccines required to address problems in this manner So this is the XQ Plus x squared Plus x plus one equals two Zero And we people have already Class Eleven Have read We have codetician Class Aquacian x Q x equals equation Class Aquacian X Plus One Co. Lying down, zero Why the hell And how do those who have fallen see us, who are still standing? What happened? that when we know this x square, here this plus one is present, What will happen if I take him along with me? It will go into minus, right? And when I take the square root of this together, what will I get? Roots of minus one do not even exist on our real number line. The real number line I had created, which was this number line. Now, if this kind of root minus one, then what will come? They said that whenever the square root of minus one appears, okay, we will represent it as being dependent on i, okay. So they named this square root of minus one as plus or minus i, and this i is referred to as an imaginary number, okay. What's written here is taken from the lowercase letter of our English alphabet; they didn't write it using the letter "I," they wrote it using "iotā." So you said to represent it with I; what if that happens? If root up minus 16 came, root up minus 25 came, So shall we also represent them there, J, K, I, EL, in this manner using alphabetical symbols? What did Kādon do, that whenever root minus one appears, it should represent i, it should represent i, which is called iota Even on the 25th, see how it can be written, if it's perceber, who am I to whom I can break it down this way: I can write it as 25 minus one, and then I can I can write, "Teach me, belief minus one," and what I said was, whenever We have to wake him up from the depression caused by her, right? Now you'll say why get depressed from I, when there are so many other things, L? Yes, one can also be depressed by other things, but we have chosen this number system, If Imaginary Number was speaking, then the ones that are in our alphabet, It starts with the letter 'I' in Imaginary, so we have depressed it from 'I', okay. Let's move forward. As we solve more problems, you'll gradually get the idea on your own, and once you become comfortable with it, you won't feel overwhelmed anymore. That's right. So, give it a little time, keep moving forward, and try to understand things. When we solve the problem, gradually, as you solve problems, you'll get used to these things, and they'll become second nature to you. These are very easy chapters; we'll easily solve them. And you will also have full confidence that yes, any problem of this kind Yes, we can easily handle the confidence number. You must be seeing the tree-like graph of the Peda language. which is a tree ball graph. Any graph of this kind exists Such as these Like this, huh Like this Whenever graphs like this appear Such a graph is called a tree-like graph. You don't need to worry about how these graphs are made. When we move forward into Class 11 So we will read one chapter; the paragraph said, "that fool." So how do we create this graph there? Now we understand what a graph is and how. Using this graph, we can understand the ketogenic diet. For now, I'm wiping it out. When you are in class 10, you would have seen an equation The one who is drawing the quadratic graph And when the aquifer of provision annat, when you make a graph Thus, a parabolic graph is formed. And all the ones you have so far By solving the quadratic equation, they obtained the solution. Whatever solutions he came up with were real roots, okay. Let me take an equation, quadratic equation and I solve it, okay, and whatever we have You know, there were three methods to solve a quadratic equation. The first factorization method was, the second method was, completing the square method, And what was the third method? The formula method, okay. So, let's go over the formula method. will solve the problem and see what comes up, but before that, we Let's take a question and solve it by factorization. by the method, and see what solution it yields, and its How graphs are created, whenever those solutions appear, quadratic equation, we know that a quadratic equation has two solutions And that sogolution, if they were real solutions, So what will they do, they'll go along the x-axis If you type the line like this, then put the code here Two lines on the x-axis intersect at the real solution An equation solver that solves equations and provides a graph, meaning a tool. I bring the desmos graph there and try to make an equation happy using that tool. x^4 + 3x - 6 equals zero. What can I do here? Can I factor out 2x? I factor out 2x, and what will come out? x - 2 If I take 3 as common from here So what will come? 3x - 2 equal to 0 Alright How can I write this then? 2x plus 3 How can I write this? x - 2 equal to 0 How can I write this then? 2x plus 3 equal to 0 x - 2 equal to 0 x equals 2 2x plus 3 equals 0 Here it will come How can I write this then? -3 Two solusorns How much They exist on our real number system. So, whatever the two solutions to this are We see this as a tool. By plotting the Desmos graph and going there, we have these two equations Wear the tool And what else is there? That's right, look, whenever a line intersects the x-axis, the y-coordinate at that point is zero. We've already learned this in class when we were studying graphs, especially when dealing with concepts like the Cartesian product. If some kind of line product is zero, then its y-axis value is zero. So, these y-axis values are zero, zero, correct? We must pay attention to this point. What is our solution? Our solution was, my... Look at the solutions he got, and see at which two points on the x-axis they intersect. So it was intersecting at minus 3 over 2, and at 2, right? So, for all quadratic equations, how are their graphs formed? But when their solutions come, they intersect the x-axis if real solutions appear; if imaginary solutions appear, then They intersect the x-axis and also the y-axis, right? So this is an equation, an equation, and we solve this equation. And let's see when the solution to this arrives? Even then, the equation was x squared plus x plus one equals zero. In class, we studied the formula method in the chapter on determinants, where we also learned about the discriminant. Using that, we... They will appear very much, they will see a lot. The equation's roots—here, the roots are not roots, but roots were—first, we will determine the values of a, b, and c, then compare them. What will we compare? The question is, the general form is used when you are dealing with this equation. And what is this called? Immersed number Alright, if she becomes a barmaid. So, are they emergency numbers, right? And what kind of book is this, exactly? A, B, C, D, this formula in Take care of B, C's sand, keep it safe. And what comes out when you extract the x from the sand? So, how much is the mine's B sand? One is. Because I know that for any quadratic equation, the solutions are always two. There will be two solutions for this: the first one is minus one plus root minus three over two, and the second one will be that two solutions come out, but these two solutions involve the imaginary unit, because whenever there is a minus sign inside the square root, it becomes an imaginary number. These two solutions, when you plot them on a graph, will intersect the Y-axis. Right on So we go back to Desmos and write down this equation to see what its graph looks like. We write down the equation and see what comes out. If the roots of a certain equation are real, then if you graph that equation, it intersects the x-axis, right? So, we've understood this using the help of two equations. I understood it with the help of these two equations.