 I'm Zor. Welcome to Unisor Education. This is problem number five about complex numbers, and the problem actually is it's kind of a arithmetic exercise if you wish. If you have two different complex numbers, A plus B, I, and C plus B, I were A, B, C, and D are real numbers. I would like to represent this result of this division as a complex number with X and Y real numbers. The question is, how can we do it? All right. Well, let's just do it together. I never tried quite frankly to do anything like this, but let's think about how can we do it? Well, obviously, the result of the division is supposed to satisfy the multiplication of denominator and the result. So X plus Y, times C plus B, I should be equal to A plus B, I. When we are saying that this should be equal, it means real part of this should be equal to A, and the coefficient in an imaginary part should be equal to B. Well, let's just open these parentheses and see what happens. First of all, the real part will be X times C, and Y i times D i would be Y times D times i square, and i square is minus one, so it's minus Y D, and that will be my real part of the result of this multiplication. Now, imaginary part will be coefficient with i will be C times i, sorry, C times Y, C times Y, and X times D, and that will be B equal to A plus B i. Okay, so we have a real number should be equal to real number, and this real number, the coefficient with i should be equal to B. So basically, we have a system of two linear equations with two different unknown variables X and Y. So let's just write it down as such, and we will see what happens. So X C minus Y D should be equal to i A, and X D plus Y C is equal to B. I have changed the order of these two variables just to have X and Y ranges one after another. Okay. How can we solve this equation in the simplest way? Well, with linear equations, it's quite simple. Let's say you multiply this by C and this by G, and add them up, what happens? So this would be multiplied by C, this would be multiplied by G, and then we will add them up together. Obviously, Y, D, and C will be Y, C, and G with opposite signs, so Y will disappear when we do this, and we have only X equation for X, and that's exactly how we will find out what the X is. So if we multiply it by C, it would be X times C square, and this multiplied by G would be X times D square, and that's equal to AC plus BG. AC BG. That's the result of this, from which we conclude that X is equal to AC plus BG over C square plus G square. What's interesting, by the way, obviously any kind of a fraction does not exist when the denominator is equal to zero, and this particular denominator is equal to zero when both C and D are equal to zero, which means in our original example, A plus BI divided by C plus DI, if C and D both are equal to zero, it means we divide by zero, which is no-no, right? So basically our formula is correct exactly when our division is correct. So if C square plus D square is not equal to zero, it means that both together are not at the same time equal to zero, and that means that the denominator is not zero. So that's quite fun. All right, so having done that, let's just change it slightly and get rid of X from the same two equations. How can we get rid of X? Well, the same way, actually. We multiply this by D and multiply this by C, so this will be XCG and this will be XGC, which is the same, and then we will subtract one from another. Let's say from the second, we subtract the first one. So X disappears and we will have here YC square. Now this is minus, but we subtract, so it will be YG square is equal to BC minus AD from which we conclude that Y is equal to BC minus AD over C square plus D square. And again, notice that in the denominator, we also have exactly the same thing, C square plus D square, which means it's correct whenever we divide by something which is not equal to zero. So that's the answer. So this thing is equal to, let me start from this, BC minus AD divided by C square plus D square I plus AC plus BD divided by C is equal to zero. C square plus E square. So that's the result of the division. So we have divided two complex numbers and we've got the result, a complex number, which has the real part calculated using this formula and the imaginary part calculated using this formula. That's the answer and again, what's very important is that both left and right part are valid when one of the C or G not equal to zero. If both are zero, this isn't invalid and all these are invalid. So we have the same kind of a domain where both formulas are defined with the same set of C and G. Thank you very much. That's it.