 Hi, I'm Zor. Welcome to a new Zor education. I would like to say a few words about a very important tool which mathematicians are using in many, many different places. It's called coordinates or coordinate systems. There are many different coordinate systems, and as I will be using these coordinate systems throughout many different lectures in algebra, geometry, or somewhere else, trigonometry. I probably would suggest you to refer back to these lectures devoted to coordinates just to refresh your understanding of what this actually is all about. So, what kind of coordinate systems exist in mathematics and used, and what they are basically all about? There are actually quite a few. So, I will start with something which is by far the most frequently used system of coordinates. It's called Cartesian after the French mathematician Descartes, and basically here it is. How to construct this particular coordinate system? First of all, there are different coordinate systems in different spaces, if you wish. I will use the word space very carefully, because I would like to really specify what I'm talking about. Space is a different concept, and basically there are many different aspects of this word, but I will concentrate on three major incarnations, if you wish, of the word space in mathematics. I'm not talking about real life or anything like this. So, in mathematics, we are talking about these three main types of spaces, and I'm talking about elementary mathematics. One is called a line, this is a straight line. Another is a plane. This is, for instance, the surface of this whiteboard is a plane, and the third one is the space we live in. We usually call it a three-dimensional space, but I will explain what that means in mathematics terms. So, let me start with coordinate system. In the line. Now, what is actually the coordinate system? The coordinate system is the way to numerically characterize a geometric object. A geometric object on the line is something like a point, or a segment or something, many segments or something like this. So, how can we numerically characterize these things? Well, the first thing we should start is numerically characterize the point. So, let me start with this. What is a numerical characteristic of a point on the line? How can it be constructed? Actually, it's very simple. What you do is, number one, you choose one particular point on the line, which you call origin, or the beginning of coordinate. This is also sometimes called a zero point. Now, line by this point is divided into two pieces. On one hand, it's called the positive direction, and usually it's signified by an error or something like this. On another side, it's a negative direction. Now, we also have to have a unit of measurement. Some kind of a segment doesn't really matter what it is, which signifies a segment of a unit length. Length equals one in whatever numbers. I mean, it's just one. That's it. Now, what we can do is, every point can be characterized by this distance, and distance can be measured in these units. Not necessarily in integer number units. It can be a rational or irrational number, whatever the number is. And we were talking about different numbers, and we will probably discuss all these different numbers, what they are about, et cetera. And by the way, I will use certain concepts here, like for instance, irrational numbers, which you might learn a little further down the line, in which case you can always return back to the coordinate lectures which are using these concepts. So in this case, we are saying that this particular segment from O to A, O is origin and A is any point, can be measured in these units. And if A is on the positive side of the line, we will characterize the position of the A as this particular length with a plus sign, with positive sign. And if some point B is on a different, on a negative side of the line, we will characterize this length with a minus sign. So, lengths with the proper sign is a sufficient information to identify the position of the point on a line relative to specific origin, relative to specifically chosen positive direction of the line, and a unit of measurement. Well, that's it basically. This is a coordinate system on the line. That's absolutely enough, because now what we can do for instance, how can I characterize the position of a segment on the line? Well, what is a segment? Segment basically is characterized by two points. The beginning can be the end of the segment. Each point has its own characteristic as a coordinate, numerical characteristic, which means the segment can be characterized as a pair of numbers. One number being one end of the segment and another number being another coordinate of another end of the segment. So, in this case, segment is a set of two coordinates. Now, if I have more than one segment, obviously it can be expanded. So, that's it. This is a Cartesian coordinate system, which I have introduced on the line and using this origin, the positive direction and the unit of measurement, I can numerically characterize position of any point. Let's move on. Two-dimensional is a plane. By the way, I didn't mention it. This line is called one-dimensional space because it's enough to have one number to characterize the position of the point. Now, let's talk about two-dimensions. Now, in two-dimensions, and I'll use this whiteboard as a surface, the Cartesian coordinate is built the following way. First, we have two perpendicular lines. Now, the intersection is called origin. Now, we also have unit of measurement. Now, in mathematics, usually, unit of measurement is the same for this line as well as for this line. So, we are introducing a Cartesian system on each line. On this line, we have origin, positive direction, and the unit of measurement. And on this line, we also have positive direction from the same origin, by the way, and usually the same unit of measurement. In practice, the unit of measurement can be really different on different axis. So, this is called x-axis, and this is called y-axis. Now, traditionally, if you are drawing something on the paper, the x-axis is horizontal and directed, positively directed towards the right hand, and y-axis is traditionally vertical, and the positive direction is upward. Now, that's good. We still didn't define the coordinate system. So, coordinate system is supposed to numerically identify or characterize any point, right? So, what can we do? Well, here is the explanation, which is partially geometrical, which means you have to know geometry to really understand that this is the right definition. But in any case, I will just use the geometry which is presented in this course as given. So, what we are doing here, so we are using the concept of perpendicularity of these lines, which is a geometric concept. And then we are using another concept of dropping the perpendicular. Sometimes it's called a projection of the point A. This is projection on the x-axis, and this is projection on the y-axis. Now, in geometrical terms, this is a rectangle. Which means this is parallel to this. This is parallel to this. This is the right angle. This is the right angle to this and this. Now, how can I characterize this particular point? Well, this point is uniquely characterized by its two projections and two x's. Now, each projection is uniquely characterized because this is a grisa-cartesian system of coordinate on this line. So I can measure this distance, all x, in these units with a proper sign, whether positive or negative. Now, this is exactly the same thing. So I have two coordinates. One is called x-coordinate, which is this. And another is called y-coordinate, which is this. Which, by the way, are equal to this y and this x, because it's a rectangle. Now, in this particular case, my both x and y are positive because projection on the x-axis is on the positive side and projection on the y-axis also on the positive side. But if I will have the point somewhere here, for instance, my projection on the x-axis would be negative and projections on the y-axis would be positive. So in this case, x is negative, y is positive. Now, these are four parts of the plane. These two lines are divided into. And they're called quadrant. This is the first quadrant. I'll use the Roman numerals. This is the second. This is the third. This is the fourth. So whenever we're talking about, let's say, a point is in the fourth quadrant. It means it's here. It means it's x-coordinate is positive and y-coordinate is negative. Well, that's it. We have defined a system of co-ordinate on the plane. Now, it's two-dimensional because we need two numbers and it's time to go to three dimensions. So let's go to space. Now, space is something where we live and how can we identify a position of a point somewhere in the space using something similar to the systems which we were talking before. So we are introducing right now a Cartesian system in the space. Here's how we can do it. Now, I will try to draw something three-dimensional on the two-dimensional board. And here is how. Well, let's imagine that we also have somewhere a position which we call origin, a point of origin. Now, we will have a plane which goes through this point, which I will call x-y-plane. And on this plane, I will introduce a Cartesian co-ordinate, two-dimensional one, which I have just explained before. So let's assume that on this plane, this plane, and these are x's. We do have this Cartesian system, two-dimensional Cartesian system. Now, from the origin perpendicular to the plane. So let's say if my black, if my whiteboard actually is a plane, then this is a perpendicular to the plane. If my floor is the plane, then this is a perpendicular to the plane. So in this case, if this x or y is the plane which I was talking about, then this is called z-axis, which is originated from the origin, and perpendicular to the plane, which in particular means it's perpendicular to each line on the plane. By the way, let me just demonstrate it very quickly. If my plane is the whiteboard, and this is a perpendicular, then every line which is going through this point would be perpendicular to my perpendicular to the plane. So that's how we start. Now, we obviously introduce positive directions on all three axis, x-axis, y-axis, and z-axis. Now let's talk about a point. Now let's say we have somewhere a point here. It's hanging in the air. I mean, it's on my whiteboard, but again, I'm trying to convey a three-dimensional picture on the two-dimensional board. So let's drop the perpendicular to this plane from this point. All right? And from this, I'll drop the perpendicular to x-axis and y-axis. So this is a, and also perpendicular to z. This is ax, this is ay, and this is az. So basically, we are projecting this point to each of the three mutually perpendicular axes. On each of them, we have Cartesian system of coordinates. And therefore, these three numbers, ax, ay, and az, characterize position in space. That's it. Now, how can I obtain az? Well, I just dropped the perpendicular from a to the z-axis. How can I obtain the point of ax? Well, I can either directly drop the perpendicular or to this axis, or I can drop the perpendicular to the plane, xy plane. And then from this perpendicular within the plane, same thing with this one. You just have to imagine this in the three-dimensional also. This is not maybe the right best picture. But it does represent basically the way how we are dealing with any point in space. So how can it be more graphically? OK, let's consider this is my vertical z-axis. This is my x-axis. And the perpendicular to the x-axis is my y-axis. So if point is somewhere here, I can drop a perpendicular to this, this, and this. And these three points have three coordinates. This triplet is the coordinate of the point. Now, we have actually, now, in two-dimensional case on the plane, we have quadrant. In this case, we have octant, octants. This plane divides the space in two. And on this plane, I have four quadrants because this is a two-dimensional Cartesian system. So all these four quadrants define on the positive side, where the z is positive, the corresponding z-positive octants on the space. And whatever is below this plane would be, again, four quadrants, four octants, which are negative. So eight altogether. That's why it's octants. Quadrants, four octants, eight. Well, that's it, basically. This is a Cartesian system in three-dimensional. And it needs three, as I was saying, coordinate systems. Now, mathematics in a purely abstract fashion can consider not only pairs or triplets of coordinates. It can consider actually something like this. A point in an n-dimensional space. We cannot have a visual representation. Even with three-dimensional representation, we have a problem representing on a flat whiteboard. Now, we can actually have an abstract point in n-dimensional space. And the higher mathematics are actually studying the properties of n-dimensional spaces. All right, next. What's next? Next is polar coordinates. Polar coordinates are on the plane only. So that's very easy. How can I find a point on a plane using the polar coordinates? Again, we have a more origin. We also have a ray from this origin, which is called polar axis. Now, any point can be characterized by two things. Let's draw a ray from the origin to our point. Now, we can start measuring this angle from the polar axis counterclockwise to the position of this particular vector. That's how we define the direction where A is located on this particular plane relative to origin and the polar axis. Now, once we found the direction, we can find the distance from the origin. So if it's this point, for instance, and this is my ray which goes from A to B, this is my angle. I have to turn my polar axis to get to this direction. And then this lens characterizes the position on this line. So usually, you're using r for a distance and 5 for the angle. Sometimes, instead of r, they're using Greek letter rho. Well, that's basically it. It's kind of simple, but less convenient, I would say. Cartesian system is, by far, the most used in mathematics. Now, what else can we say? Let's talk about system of coordinates in space. We were introducing the Cartesian system, but that's not the only one. We can have a combination of the polar and Cartesian system in the following way. Again, let's imagine that we have a plane and a z-axis which goes up from this plane perpendicular to the plane. So this was our Cartesian system. Now, instead of using x, y on the plane to identify the position of this point, I can use polar coordinates. So this would be my polar axis. Forget about this. This would be my polar axis. This would be my direction to the point A. So I need angle and the distance. So the point on the plane would be identified by the polar coordinate. Now, from the plane, I can go to any point in space vertically. So if my point in the space needs these cylindrical coordinates, then what I do is I project it to this basic plane, the fixed plane which I have. This point has polar coordinates, which means we can have r and phi. And then z coordinate, it's called altitude H. So these three characteristics. So from point, we drop down the projection onto this plane. In the plane, we have polar coordinates. And the height or altitude from this plane can be positive or negative depending on where it is relative to the positive direction of the z axis. This last component is added. So that's cylindrical coordinate system. Why? Because it's kind of rotary on the plane because it depends on the angle. And then it goes up like in a cylinder. Now, the last one which mathematicians sometimes use in space is a spherical coordinate. Now, the spherical coordinate is the following. Again, let me try to draw it this way. So you have a point in space. Again, drop the perpendicular to this plane. And we can identify this direction by this angle on the plane, almost like in a polar system. But we are not considering right now this length. Instead, we are considering the lengths from the origin directly to the point. And we also need another angle between this vector and the z axis. I think it's called azimuth. Yeah, this is called zines. The direction, the vertical direction is called zines. This angle, I think it's called azimuth. And I forgot how this one is called. It's a rarely used system. But it also allows you to define, based on the distance, and two angles, I think it's that way. And two angles, it allows you to identify the position. Rarely used. I will never probably use it in this course. But anyway, what's important is, as you have noticed, both Cartesian and polar systems on the plane are systems where you need two numbers. All coordinate systems in the space, you need three numbers. Now, is it coincidence? Actually not. Because there is a concept of dimensionality. Our space has three dimensions, which means no matter how we want to identify our point, we need some three numbers, three linear numbers, or one linear number and two angles, or two linear numbers and one angle. So whatever the combination is, we need three numbers. Because this property of our space to be three dimensional, it depends on a space not on the coordinate system. It's called dimensionality, property of the space. Same as the plane. The plane has a property to be two dimensional, no matter what kind of system of coordinate we are introducing. I would also like to very briefly talk about geographical system, longitude and latitude. Just for your information, again, not because we are going to use it. So we are, we live on something which is almost like a sphere, our planet. How can we identify the point on a sphere? Well, here is how they do it. First of all, we have two diametrically opposed poles. We call it North Pole and South Pole. That's number one. Now, if you draw the line, it goes through the center of the sphere, right? If you in the middle draw a plane which cuts the sphere in half, perpendicular to the axis, this is the main axis, North-South axis, and the plane is perpendicular. It's called equatorial plane. And this is equator, which is an intersection between this equatorial plane and the surface of the screen. So this is equator. Now, any plane which is parallel to this equatorial plane cuts a line on the surface of this sphere called a parallel. Now, on another hand, this is an axis, right? I can have a plane which is going through the axis, actually have a plane where this axis is a border. So it goes something like this. And its intersections are meridians. So all we need to do, actually, to identify a point on a plane is to find which parallel it belongs to and which meridian it belongs to. Very easy, right? So how the parallels are measured? Well, let's draw a line from the center to a point on the parallel and the opposite point to the equator. Now, this angle can be a characteristic of a parallel. So the equator itself is a parallel which has a 0 degree. It's measured in degrees, these samples. And the parallel which goes, basically, around this north pole will have 90 degrees. So the parallel can be characterized by a degree from 0 to 90. But we have to say whether it's towards north or towards south. So this is a latitude. So latitude is a number of degrees north or south, north latitude or south latitude of this angle from this point to this point where the parallel is. That's how we identify the parallel. Now, how can we identify the meridian? Well, one particular meridian which goes through London is called zero meridian. And then any other meridian can be actually measured again by the angle from this meridian. How can we turn the plane? How many degrees we have to turn the plane? This way, which is eastward, actually, or westward. Now we can do it up to 180 towards the east and 180 towards the west. So that's how the longitude is measured. So we have east-west longitude from 0 to 180. And we have north-west, north-south latitude. And each one is from 0 to 90 degrees. Well, that's how we measure our position. That's how we identify our position on our planet. In reality, the earth is not really a sphere. So it's slightly distorted. It's kind of an ellipsoid. But anyway, the system is slightly distorted. But the main sense of the whole system is exactly this. Well, that's all I wanted to say as an introduction to what kind of coordinate systems exist. Most likely, in most cases, I will use the Cartesian system. In some very rare cases, I will use the Polar system of coordinates. Probably never I will use spherical or spherical. So that's it. Thank you very much. I do suggest you to review this in the notes. Notes are quite detailed. Just to have a more or less good understanding of what we are talking about. Thank you very much.