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The Cartesian Plane: Definition & Explanation

Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

The Cartesian plane is like a map grid. Just like the map helps you find the shoe store or post office, the Cartesian plane helps you locate a pair of values.

Definition

There is a myth that the mathematician Rene Descartes was watching a fly on the ceiling when he was in bed. He figured out a way of tracking the fly by thinking of the ceiling as a grid system. This system is like a map, and we call it the Cartesian plane. Just like on a map, you can locate any point in the Cartesian plane if you know two things - how far the point is to the left/right and how far it is up/down.

Using a Map Grid

Let's start by looking at a map.

City map
City Map

In this city map, the dog is right at the center of town. As you travel east of the center of town, the streets are named 100 East, 200 East, 300 East, and so forth. You find similar patterns as you drive west, north, or south. Main Street is really 0 East AND 0 West simultaneously, so it is easiest to just give it a name. Similarly, Center Street is really 0 North and 0 South.

You can easily find any location on the map just by knowing how far it is east/west and north/south of the center of town. The car is at about 200 East and 200 North. The shoes are at about 200 West and 100 South.

Using Numbers Instead of Street Names

The Cartesian Plane is very much like a map grid. Instead of using the compass directions (north, south, east, west), you use positive and negative numbers. East and north are positive, while west and south are negative. The picture below is exactly the same as the last map, except for replacing the compass directions with positive and negative numbers (integers).

Simple Cartesian plane
Simple Cartesian Plane

Now the car is at the crossing of the red line '2' and the blue line '2.' You could say it is at 'red 2' and 'blue 2.' That approach works and keeps the vertical and horizontal straight, but it is a little cumbersome. Instead of relying on colors or compass directions, mathematicians use a different trick: they use order to describe which number is the horizontal distance from the center of town and which is vertical. The horizontal distance always comes first.

Here is an easy way to remember which direction comes first: an airplane has to go over before it can go up. To find the shoes, go over to -2 then up to -1. Strange as it sounds, going 'up' by -1 means going down!

Since you always go horizontally first and vertically second, you just need to list the two numbers (or coordinates) in order. The car is at (2,2). You always put parentheses around the two coordinates as a clue that the numbers represent a location on the Cartesian plane. Written this way, we call (2,2) an ordered pair.

As another example, the shoes are at (-2,-1).

We still don't have an accurate Cartesian plane, though. Let's make a few more changes.

The Cartesian Plane

Cartesian plane
Cartesian Plane

In the picture above, several things are different. First, we've put arrows at the ends of all the lines. This simply means that the grid goes on forever in all directions.

Next, do you see the word origin? You don't really have to write that in. It shows us that the middle point, the 'center of town,' has a special name. The point (0,0) is the origin, or the place that you start from when finding your way through the Cartesian plane.

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