 In the second video, in my course on more advanced topics in linear algebra, I talk about arithmetic with complex numbers. It is very simple to do and is very much like working with real numbers. When we work with complex vectors and complex matrices, we will use the simple arithmetic that is covered in this lecture. So grab your pencil and paper and let's do some complex arithmetic. So what we are going to do now is just do some complex arithmetic. And the first thing we are going to do is scalar complex number multiplication. And so let's start with this scalar complex number, but we are going to just constrain ourselves a little bit. And we are going to say let this scalar and let's make this scalar k, let that be an element for now only of the real numbers. Later on we will look at what happens when we have a complex number so that we multiply two complex numbers with each other. So let's have this number z, this complex number z, that is going to be a plus b i. And we now know real number, real number, imaginary unit i and multiply b by i, this is our complex number. And we are going to define this multiplication k times z, that is going to be k times a plus b times i. And all we are going to have here, we are just going to have the normal distribution that we learned at school. So nothing complicated there, so we are going to have k times a and we are going to have k times b times i. And so let's have a little example of that, well let's first of all define that. So that is going to be our definition, we multiply a real number by a complex number. And so there is our definition, so let's just have a little example. And we are just going to say let's do three times four, let's do that five times i. Pretty simple, three times four and three times five, that is what we are going to have. So that is three times four plus here we have three times five times i. And that is going to be twelve plus fifteen i. So really nothing difficult there whatsoever. Let's go to the addition of complex numbers, let's look at the addition of complex numbers. And now we are going to have two complex numbers, let's make this one z. We are going to make this be a plus b times i. And let's make a second one w is another common variable that we have when we do talk about complex numbers. And let's make this c plus di. And now do please remember the fact that c and d, they are both real numbers. So if we are going to define this addition z plus w, that is going to be a plus b times i plus c plus d plus i. d times i I should say. And now I am just going to group all my real parts and my imaginary parts. So the a and the c, they are both the real parts. So I am going to put them together, that is my real part. And I am going to take out i as a common factor there so that we have our imaginary part b plus d times i. And there is our definition for complex number addition. And you can see very easily there it is just adding the two real parts and adding the two imaginary parts. There is really nothing to it. Shall we do one of these? Let's say 3 plus 4i. And I am going to add to that 7 plus 3i for instance. And that is just going to equal these two real parts, go together. 3 plus 7 is 10. Plus it is 4i plus 3i. So remember it is 4i plus 3i, that is what I am talking about there. And it is the same as you would say 4x plus 3x. We know how to do that or we can take out this common factor i. And then we just left with 4 plus 3. In other words that is a 7i. So plus 7i, very simple. Which reminds me of something because we have always been dealing just with these plus and plus and plus and plus. It is always a positive. But what if we want to do negative? And so one little other thing that we can add here. Let's do let k equal negative 1. And so if I have negative 1 times, let's do 3 plus 4i. And I am just going to get negative 3 negative 4i. So certainly we can have negatives in there. That is not an issue for us at all. Because what that allows us is to do the subtraction of complex numbers. So let's do the subtraction of complex numbers. Let's have this definition again. We are going to have z equals a plus bi. Now remember that b is a real number. So it might be negative 3. So I can have a plus negative 3i. That would be negative 3i. I think you get what I am talking about. Not particularly difficult. So let's have c plus di. And all I want to calculate now is z minus w. I am subtracting w from z. So I am going to have a plus bi. And I am going to subtract from that the following. c plus di. And really nothing difficult here. You can clearly see where this is going. 3 plus b times i. I am going to distribute this negative 1 throughout. So that is going to become negative c. And that is going to become negative d times i. And all I am going to do is group together the real and imaginary parts. So I am going to have a minus c. That is going to be my new real part. And if I add to that, I am going to have the b minus d. b minus d times i. So let's say that we have 3 plus 4i. And I am going to subtract from that 7 plus 7i. What am I going to get here? I am going to get 3 plus 4 times i. Minus 7 minus 7 times i. Just group these together. It is going to be 3 minus 7. And I add to that 4 minus 7i. 4 minus 7 times i. And I am going to get 3 minus 7 is negative 4. And 4 minus 7 is negative 3. Negative 3i. As simple as that. And what we have to find here is the subtraction of complex numbers. So where shall we put it? Let's put it right there. We have to find the subtraction of complex numbers. So let's do something that is slightly more fun. Although we do get to add complex numbers. That is a real thing. And when we are going to talk about vectors and matrices. That is of course what we are going to do. So let's talk about the product of complex numbers. Here we go to the product of complex numbers. So I am going to have two complex numbers. Z and W. Let's make this A plus B i. Let's make this one C plus D i. Where all these things A and B and C and D. Remember they are all members of this set of real numbers. Real part, imaginary part, real part, imaginary part. Two complex variables. And I want to multiply them. I want to say Z times W. And it is going to be nothing other than just the expansion of what you already know. So I am going to have A plus B i. And I am going to multiply that by C plus D i. And guess what? We are just going to do this same distribution. So I am going to have A times C and then A times D i. And I am going to have B i times C and B i times D i. Simple, simple stuff. A times C is A C. A times D i is positive A D i. I am going to have... I am just going to put these things in order because they do commute B i C. So I am going to say that is B times C times i. And now I just have to be careful. I have got B times D. So that is B times D. This is my B times D. But I am also doing i times i. And remember what i times i is? Well that is i squared. And remember that is how we defined our complex, our imaginary number. I should say that is negative one. So it is B times D i times i is negative one. So that becomes negative B D. And now I am just going to group together the real and imaginary parts. So that is going to be A C minus B D. Plus I am just going to group those two. Take at i as a common factor. B D plus B C times i. So you have just got to watch out for that i times i. Which is going to give you a negative one. So that you end up with your negative B D. So let us write that down there. That is an important definition for us. The product of complex numbers. Let us do a little bit of an example. Let us just have 3 plus 4 i. And I am going to be slightly more difficult. Let us multiply that by 3 minus 3 i. We are going to have 3 times 3 that equals 9. I am going to have 3 negative 3 that gives me negative 9 i. I am going to have 4 i times 3 that is positive 12 i. And now I have got to be careful. 4 times negative 3 is negative 12. But i times i is negative one. So that gives me a positive 12. So I am going to end up with 9 plus 12 for my real part. And I can add to that. Let us do, it is a negative 9 and a 12. So let us make 12 minus 9 times i. 9 and 12 is 21. And 12 minus 9 that is going to be 3. So plus 3 i. So just check the arithmetic there. I am famous for making arithmetical errors. And that is why we have computer languages and calculators. Because who wants to do arithmetic? So check on that one for me. But I think you can see clearly what we do when we do the product of complex numbers. So let us move on. And I want to talk to you about something very special. And that is called the complex conjugate. And the way that we define that, if I have a complex number z, there is my variable z. And that is a plus bi, a real part, an imaginary part. I define the complex conjugate. Let us put it there as z with a little bar on top. So just a little bar. That is going to be a plus. And I am just going to multiply this b part by negative 1. So that would just mean z bar in this instance is going to be a minus bi. So in practice what we usually just do is just swap the sign of that imaginary part. So a plus 3 i, a plus bi I should say it is complex conjugate is a negative bi. And so that is a very important thing. So let us do a little example. Let us have that z equals 3 plus 3 i. And that means z bar, this complex conjugate. Well I am going to multiply that positive 3 by negative 3 and that gives me 3 minus 3 i. That is great. Let us do another one. Let us have w equals 4 minus 4 i. So what is w conjugate going to be? Well I have got to just multiply negative 4 by negative 1. That gives me a positive. So that is 4 plus 4 i. Can you see that we always just swap that sign in practice? That is all we are going to do. So if we think about the argn plane. Let us just put a little sticker there. All these things that are so important that I want you to remember. That is how we define the complex conjugate. Right there, complex conjugate. And so if we think about the argn plane. It just really means if I display or think about a complex number as a vector. And so if I have my vector there. Remember that is going to be my a part component. That is my b component. It just literally means we are just reflecting this along this real axis line. So I am just going to still have the positive a. But now I have the negative a positive a. But from the positive b I am going to go to the negative b. Or from negative b I go to the positive b. So the fact is if this was my complex number z. This one right here. That is going to be z bar. So that is kind of cool. Complex conjugate. Where can I use the complex conjugate? Well let us go back to something we have already done. And we looked at the norm of a complex number. So let us just remind ourselves very quickly what the norm of the complex number was. So we had this idea if z was equal to a plus b i. I had this norm of z. And remember that was the two bars. And that was just this real number. That I take the positive square root of the real part squared. Plus the imaginary part squared. Now let us think about something. Let us do something very special here. I am just going to write it out. So you can see where this is going. I am going to say let us do the norm of z. But we are going to square the norm. Let us square the norm. Well if I square the norm what am I going to get? a squared plus b squared isn't it? I am squaring the square root. So I am going to get a squared plus b squared. Now let us do something crazy. Let us do something crazy. Let us multiply z by z bar. That is crazy isn't it? z times z bar. So what is this going to be? a plus b i times what is its complex conjugate. Well it is going to be a minus b i. And if I do that multiplication what do I get? a times a is a squared. And I have a times negative b i. That is negative a b i. Positive b i times a. That is plus positive b i. And negative a b i. Positive a b i. Those two can's allow. And now I am left with b times negative b. That is negative b squared. But I have i times i which is another negative one. So that becomes positive b squared. So just do that so that you are comfortable with that. And hey look at that. We have the fact that. Let us get a little geniart. We have this nice little thing there. Let us put it down there. Because z bar is nothing other than, let us put it out here, that is the norm of this complex number squared. The norm of the complex number squared is equal to the complex number times its complex conjugate. These things commute. So I can have them the other way around. It doesn't really matter. And then if I take the square root of this. So if I take the square root of the complex number times its complex conjugate, then I am just going to get the norm, the length. If I want to think of it as a vector, I am just going to get the length. So this is kind of an important and quite interesting if you think about it. And so that would be one use of us thinking about this idea of a complex conjugate is because I can do these wonderful things. Instead of calculating this Pythagorean theorem as far as the norm of a complex number is concerned, thinking about the length there of that vector from the origin. For any point, and remember any point is a point, a comma b, any point on this plane defines for me the whole set of complex numbers. But I can think of the square, at least of the norm, as being the product of a complex number and its complex conjugate. As I say, all the other way around. Now that we've revisited the norm, we can also revisit the argument. Remember the principle argument, this angle that we make? Now there's a slightly different way to think about it. I'm not going to discuss it here, but if you do want to watch the extra videos on using Matlab to do these calculations, we'll just look at another way of thinking about this argument where we're going to have a positive argument on this side. So from zero to pi, we'll call that a positive angle. But as soon as we go beyond pi, as far as the radiance, this angle is concerned, we're going to rather go clockwise. And so on this open interval from zero to pi, and we're going to call that a negative angle. And that's it. I think you know enough of complex numbers so that we can start thinking about complex vectors, which is not this. Remember this is a complex number turned into a vector. I'm talking about a whole vector of complex numbers and even better, a matrix full of complex numbers. That'll be coming up next in this lecture series.