 This is the first video in my new course on more advanced topics in linear algebra. Very much a second course in linear algebra at college or university. Now to start this course we are going to consider complex vectors and complex matrices. So in this first video I talk about the imaginary unit and complex numbers. For some of you this might be brand new but for most I suspect it's just a bit of revision. I've created many resources for this course. Firstly there are these video lectures. You will see me taking to pencil and paper explaining the concepts and working through some example problems. Now there are also videos on how to do the calculations using simple Python code. The Python videos will be interspersed between the normal pencil and paper lectures. You don't need to know any Python to watch these videos. Using Python to do your math homework is very easy and if you wish to watch the Python videos then you will learn Python while doing math in no time at all. You also don't have to download and install Python. You can use your Google Drive just as you use Google Docs. In fact I have a video to show you just how easy it is to set up your Google Drive so that you can write Python code for mathematics. There's a link to the video in the description and I think it is worthwhile watching the Python videos. In real life we use computer code to do mathematics. It can actually help you understand the work better. It is also great at checking your homework. There's also a set of lecture notes which form a free open textbook. There's a link to these notes in the description. The notes read very much like a textbook and also contains code examples. Now the code in the notes that is written in MATLAB. Now MATLAB is still used in many colleges and universities, mathematics and science departments. You can still read the textbook and type your code in Python though. But back to this video. Let's explore the imaginary unit and complex numbers. Grab your pencil and some paper. There we have, let's get our trusty Nvidia ruler here. An imaginary number. So we know from the real numbers and we usually use a variable X for a real number. We're going to say that X is a member of this set of real numbers. We'll do that double struck R. Because we know if we square every real number it has going to be equal to or larger than zero. If we square a positive number three squares nine, if we square a negative number negative three squares nine, if we square zero with zero, so we're never going to get anything less than zero. This absolute lower bound is going to be zero. We have this little issue that we might want to take the square root of a negative number say the square root of negative one. Now if I square any number we've just said well as far as the real is the set of real numbers concerned that's not going to work for us. So we define this imaginary number in the following way. We say I now imaginary is a horrible name but it's stuck and we've got to deal with it. There's nothing imaginary about it. But anyway we have this number i, a lowercase i, and we say if we square that we get negative one. That's how we define this imaginary number, this imaginary unit. That's probably a better way to call it. And so this imaginary unit is defined in the following way we say it's i such that when we square it we get negative one. So what am I going to do? I'm going to put a little green arrow there for us. It is a definition and I want to you know sort of highlight where all our definitions are. This means if I take the square root of both sides I'm going to have this idea that this imaginary unit is going to equal the square root of negative one. But really not how I want to remember it. I want to remember this unit as a thing that when I square it I get negative one. And so that brings us now that we've defined this to the set of complex numbers. So there's the set of complex numbers. We have got some notation for these numbers. To set we had this double struct R. We're going to have some notation. So notation and the notation for the set is the set C and we also put a little double struct line through the so C. The set of all complex numbers. Now we want to define a complex number and for any element of this complex number we usually use symbols. This as we used to use X and Y etc for R. We have some symbols that we commonly use but you can obviously use any symbol you want. And by that symbol I mean an element of and we usually say this we use this symbol Z. We say Z is a member of the set of complex numbers. This is X as an element of the real numbers. You can use whatever you want. This is common to use. Other common ones that we do use as W etc. And now we've just got to define what this actually what what is this complex number. This complex number Z. How do we write it out? How do we think about it? A complex number and the variable that we use Z equals A plus B I. Now B I means B times I. And I've just got to be clear about this A and B. They are both elements of the set of real numbers. And it's B times B times I. I can also write I times B same thing. But I have these two parts of a complex number. So we have a complex number. We better put some some little green arrow there just to remind us that's a definition for us. We're always going to use that. And that means there's these two components to this number. By the way let's just do let's just say Z equals 3 plus 4 times I. Perfectly valid complex number. Perfectly valid complex number. Now there are two parts to this and this is these parts A and B. So I'm going to call A and B. And remember I can also use other terms. We'll have a look at that very shortly. We have A and B. And we're going to say A equals the real part. I'm going to call that the real part of the imaginary number Z. The real part. And this one I'm going to call the imaginary part. Nothing imaginary about it. And it is a real number. The imaginary part. So let's put it out there because it counts for the both of them. The real part and the imaginary part. And we very quickly can see what happens when let's let B the imaginary part equals zero. So if B was zero I might have Z equals A plus zero times I. Now that as I said that unit is it's just we can think of what is the number. It's multiplied by zero. So that's just going to equal A. And A is just a real number. So in this instance Z is A a real number which helps us to understand the fact that all real numbers this whole set of real numbers they are a proper subset of the complex numbers. So they are a smaller version of I'm going to call it smaller let's just keep it there. I don't want to be specific and pedantic about that definition smaller at the moment the cardinality of the sets etc. I'm just going to say it is a proper subset of C of C in as much as I can always just let the imaginary part be zero that gives me a real number. But for every real number you know there's an uncountably infinite number of values I can put for B because the set of real numbers are uncountably infinite. And so you can see how I've expanded my world by introducing this imaginary part. With these two parts obviously another nice thing that we can think about is to plot these and I'm going to talk about the argand plane argand plane. Now we know one plane already and that's the cartesian plane for real numbers both axes are real numbers and here we have sort of the similar idea of an argand plane other than the fact we're going to call this one the real axis and we're going to call this one the meant this orthogonal axis we're going to call that the imaginary axis imaginary axis. So once we've done that we can see a slightly different notation we used to this being an x and this being a y and that both x and y are real numbers which means there's another way that I can write these I can say z equals x plus now for some reason we swap we commute because it's this multiplication we can commute when you use x and y for some reason most of us we just write x plus i y instead of x plus y i doesn't matter but because you know we used to this being x and that being y and both sets are real numbers I can think of let's think keep on thinking of them as a and b doesn't really matter I can think of this being a vector if I have a vector from the origin to this point here a comma b that a comma b on this argand plane is another way to think about an imaginary number so instead of writing it out x plus i y or a plus bi I write it as a as this ordered pair for a stereo part then the imaginary part inside of a set of parentheses separated by a comma and that is as far as the argand plane is concerned at least that is an imaginary number and because I can visualize this as a vector it does mean I can think of an angle and I can think of a length I can definitely think of an angle and I can think of a length so let's start with the length and we call that as far as complex numbers are concerned we call that the norm the norm or just the length of this and we can make use of good old the good old Pythagorean theorem because what we do have here is mutually perpendicular axes and there we go I can think of the Pythagorean theorem in as much as the length of this component is a the length of that component is b this is a right angle I have a right angle triangle and I can think of the norm and the way that we would usually write that is we would say there's our imaginary number z it's a variable at the moment and we're going to put these little bars in the front and in the back there we go a z with these double bars that's the norm of z and what would that be well it's going to be the square root of a squared plus b squared just according to the Pythagorean theorem so that's just going to be and it's always going to be positive for us it's a positive length and if z was so let's just define z here z was a plus bi that means this is going to be a squared plus b squared so we can see here that the norm of a complex number is always going to be a real number because I have a real part squared the imaginary part both of them are real numbers and so this is always going to be an element of real numbers but it's going to be very special because it's always going to be positive it can be zero of course if the imaginary number that I'm talking about is zero comma zero so that's the norm and then we have to talk about this angle and we call that angle the argument so there's the argument so by the way let's put that there for our norm right there now we're going to talk about the argument this is the angle counterclockwise that this vector makes with the positive real axis and that we're going to call the argument and if we think about this let's make this theta at the moment theta we can use once again our right angle triangle and obviously we can call out the tangent here so if I call out the tangent of theta at the moment as it stands that's opposite divided by adjacent that's going to be b over a and that means that this angle this argument is going to be the arc tangent let's use arc tangent of b divided by a if I take the arc tangents of both sides those are inverses of each other that's what I have so if I have some way of calculating the arc tangent of the imaginary part divided by the real part I'm going to get this argument and we go from 0 all the way to 2 pi and of course I can just go around again but on this interval from 0 to 2 pi I'm going to actually call that later on I'll call that the principal argument so I think we know enough of the complex numbers now that we can go on in the next video and we're going to talk about some arithmetic with the complex numbers