Coordinate Systems – 2D, part 1

How do you locate a point on a two-dimensional (2D) surface. Since we are now in two dimensions, it will take a minimum of 2 numbers to locate a point. As in the case for 1D, the 2D surface used can be flat (which this post talks about) or curved: for example the surface of the Earth where the most common system to locate a point is the Geographic Coordinate System using latitude and longitude (again, two numbers to locate a point).

Cartesian Coordinate System

The coordinate system most used by students of mathematics is the Cartesian Coordinate System. This was invented (and named after) René Descartes in the 17th century. This system is used in 3D as well as higher dimensions, but this post is limited to 2D. As most people best learn and retain mathematical concepts visually, this system of plotting was, and still is, indispensable in algebra, calculus, geometry, trigonometry, and many more subjects. So what is the Cartesian Coordinate System?

If you take two 1D number lines, one horizontal and the other vertical so that they are at 90° to one another and that their origins intersect, voilà, you have a Cartesian Coordinate System:

The system above also has a superimposed grid so that we can more easily located a point.

Conventionally, the horizontal line is called the x-axis, and the vertical one the y-axis. Note the negative numbers are to the left and down. A point on a plane which has this system of location, is said to have coordinates (x, y). Note that x is always first. So a general point (x, y) will have a position such that it is x units left or right of the y-axis and y units above or below the x-axis. Here are some examples:

Analysing points and shapes plotted on a Cartesian coordinate system is called Coordinate Geometry. The lengths and midpoints of plotted lines with defined endpoints can be calculated. But the much more interesting use of a 2D coordinate system is plotting all the points that satisfy a relation between x and y values. This is called plotting an equation.

Suppose you have a relationship (equation) x2 + y2 = 4. What are the values of x and y that satisfy this equation? There are an infinite number of (x, y) pairs that will solve this equation. For example, (0, 2) solves this equation because 02 + 22 = 4. Even though there are infinite solutions, we can draw a picture of all the points that do solve the equation:

As you can see, the set of all points that solve this equation plots as a circle of radius 2. Plots of other equation can look quite strange:

But it is important to remember that the (x, y) coordinates of any point on the graph of a relation, makes the equation true when you substitute those values into it.

The Cartesian coordinate system is not the only way to locate a point in 2D. I will talk about another popular 2D coordinate sytstem in my next post.

Coordinate Systems – 1D

Many of the posts I have written, had plots of functions or relations between two variables, usually x and y. Most of teaching algebra and calculus relies on graphs to illustrate concepts. These graphs are plots of all the points that satisfy an algebraic relation between the two (or more) variables. Behind these plots is the coordinate system used. This series of posts explores the different coordinate systems commonly used in maths. Let’s first look at a one dimension (1D) coordinate system.

1D means that one number is needed to locate a point. The most used 1D coordinate system is the number line:

Number lines can be vertical or even curvy, for example, to show distance along a path. Usually though, the number line is a straight horizontal line. But they all have some things on common. First, they have to have a reference point: a point from which all other points obtain their position. This point here and in all coordinate systems is called the origin. And second, there is a scale: the distance between the tick marks that allow us to place a point. In the example above, the scale is 1 unit between tick marks. For example, if we want to plot the variable x = 5, the plot would be

There are an infinite number of points on this line: an infinite number of tick marks and an infinite number of points between each tick mark. What are the kinds of numbers that can be plotted?

Any number on the number line is called a real number. This is an actual mathematical term to distinguish these from other types of numbers used in maths such as imaginary numbers (despite the name, imaginary numbers have a real meaning in science and engineering). The set of real numbers is represented by the symbol ℝ. There are several subsets of real numbers.

The first set of numbers you learned as a child were the natural numbers. These are the counting numbers 1, 2, 3, … but do not include 0. This set of numbers is given the symbol ℕ.

Then you learned about 0 and negative integers. Integers are whole numbers (no decimals or fraction parts) and include the natural numbers, 0, and the negative integers. This set of numbers is given the symbol ℤ. Why not 𝕀? Because 𝕀 is the symbol for imaginary numbers which are not real numbers and 𝕀 is also sometimes used to refer to irrational numbers which I will talk about soon. Notice that ℕ is a subset of ℤ which is a subset of ℝ.

The next type of real numbers is the set of rational numbers. These are numbers that can be put into the form p/q where p and q are integers. Any integer is a rational number like 2 since 2 can be written as 2/1. Any decimal number with a repeating pattern of decimals (even if that is a repeating 0) is a rational number. As ℝ is already used for real numbers, this set of numbers is given the symbol ℚ. This stands for quotient as p/q is a quotient (a maths term for division). All of the previous sets of numbers are subsets of ℝ.

That leaves the set of irrational numbers: the numbers that cannot be put into the form p/q. Numbers like 𝜋 or √2 are irrational and symbols like these are the only way to represent the exact values. They cannot be exactly represented as a decimal number as their decimal parts never repeat. There is no common symbol for these but ℙ or 𝕀 are sometimes used. There are few occasions where only irrational numbers are required, but a more common notation would be ℝ\ℚ which means “all real numbers except rational numbers”. Here is a nice picture of how all these types of real numbers are related:

It’s the irrational and some of the rational numbers that lie between the tick marks. So 𝜋 would be approximately

Plotting single points on the number line is rather boring. But it can also be used to indicate intervals of numbers like all the numbers between −6 and 2. This is shown as −6 < x < 2 where the endpoints are not included or −6 ≤ x ≤ 2 if both endpoints are included or a combination. When plotting these, an open circle means that the endpoint is not included and a filled in circle means that it is included. So −6 < x ≤ 2 would plotted

There’s not much else we can do when using the 1D number line, but we have a lot more options when expanding to 2D: to be continued.