#### Chapter 9 Objects in the Two-Dimensional Coordinate System

Section 9.1 Cartesian Coordinate System in the Plane

# 9.1.2 Points in Cartesian Coordinate Systems

Now, if we want to describe points in the plane by coordinates, we use variables. Typically, points are denoted by upper-case Latin letters $A,B,C,\dots$ or $P,Q,R,\dots$, and their coordinates are denoted by lower-case Latin letters $a,b,c,\dots$ or $x,y,\dots$. First, we will define what is meant by a point in the plane in which a coordinate system is given, and we will fix the notation that will be used in the rest of this course.
##### Info 9.1.1

With respect to a given coordinate system, a point in the plane is described by $P=\left(a;b\right)$, where $P$ is the variable denoting the point and $a$ and $b$ are its coordinates. Its abscissa or $x$-coordinate is $a$, and its ordinate or $y$-coordinate is $b$ as shown in the figure below.

For points, there is some variation in notation. In schools, $P\left(a|b\right)$ or $P\left(a,b\right)$ is often written instead of $P=\left(a;b\right)$. Throughout this course, the notation $P=\left(a;b\right)$ will be used. Since points are uniquely determined by their coordinates, we will not distinguish between a point $P$ and its coordinates $\left(a;b\right)$ in the following but we will consider both as the same object. For every coordinate system, the origin (the point with the coordinates $\left(0;0\right)$) is a special point. Generally, it is denoted by the variable $O$: $O=\left(0;0\right)$.
##### Example 9.1.2
The figure below shows the three points $P=\left(2;4\right)$, $Q=\left(-1;1\right)$, and $R=\left(0;-1\right)$. The point $Q$, for example, has the $x$-coordinate $-1$ (one unit length to the left on the axis of abscissas) and the $y$-coordinate $1$ (one unit length upwards on the axis of ordinates).

##### Exercise 9.1.3
Specify the coordinates of the points drawn in the following coordinate system.
1. $P$$=$ .
2. $Q$$=$ .
3. $R$$=$ .
Enter the points as $\left(x;y\right)$. For example, enter (12;-4) for a point with $x$-coordinate $12$ and $y$-coordinate $-4$.

In the following sections we will describe further geometrical objects, such as lines and circles, by coordinates. For this purpose, we first have to understand that points in the plane (described by their coordinates with respect to a given coordinate system) can be collected into so-called sets of points. This is illustrated by the example below.
##### Example 9.1.4
In the figure below, three points are plotted.
This set of points can be described by the following specification:

$\left\{\left(-0.5;-0.5\right);\left(1;1\right);\left(2;2\right)\right\}=\left\{\left(a;a\right) : a\in \left\{-0.5;1;2\right\}\right\}$

##### Exercise 9.1.5
Draw the following sets of points in a Cartesian coordinate system.
• $\left\{\left(i;i+1\right) : i\in \left\{-2,-1,0,1,2\right\}\right\}$
• $\left\{\left(\frac{1}{n};0\right) : n=1\vee n=2\vee n=4\right\}\cup \left\{\left(-1;-2\right)\right\}$
• The set of all points in the first quadrant with integer abscissa smaller than $5$ and ordinate $1$

As we know from Module 5, lines and circles are sets of an infinite number of points. It will be the subject of the following sections to describe their coordinates by means of sets of points and appropriate equations. A special infinite set of points is the collection of all points in a coordinate system in the plane. For this set, a specific notion exists.
##### Info 9.1.6

The set of all points in the plane described by the pairs of coordinates in a given Cartesian coordinate system is denoted by

$ℝ{}^{2}:=\left\{\left(x;y\right) : x\in ℝ\wedge y\in ℝ\right\} .$

The symbol $ℝ{}^{2}$ reads as "$ℝ$ two", "$ℝ$ to the power of two", or "$ℝ$ squared". This indicates that every point can be described by a pair of coordinates (also denoted as ordered pair) that consists of two real numbers.