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) *x*^{2} + *y*^{2} = 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 0^{2} + 2^{2} = 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.