Descartes and the Cartesian Plane

So far in this course, you have been introduced to the mathematical world and the mathematical objects that exist in this world. You've read a bit about the mathematical discipline of geometry, which involves lines, points and shapes. You've also read a lot about the mathematical discipline of algebra, which involves numbers, variables, expressions, and the practices of evaluation and deduction. In this part of the course, you'll see how the mathematician Descartes joined these two types of math together.

The reason this was an important development is that, as humans, we often like to think in pictures, rather than just words. However, so far, algebra has been all about numbers and symbols and words. This can make it much harder to deduce new facts and information and true statements about real world problems. Descartes realized that if he could join the part of the mathematical world that involved pictures (geometery) with the part of the mathematical world that involved equations (algebra) it would be much easier for people to understand and solve mathematical problems.

To do this, Descartes had to create a connection between shapes and equations. He did this by creating the Cartesian Plane.

The Cartesian Plane is something in the mathematical world, but to understand it, it is easier to compare it with something in the real world. So, start by thinking about a place where two skidoo paths cross at right angles. Let's say one of the paths has been made by Jeannie's skidoo, and the other by Joy's skidoo.

Now suppose that we wanted to give someone directions to get to a particular place. We could tell someone, "Start at the place where the two skidoo paths cross. Walk north down the skidoo path that runs north-south for 50 meters. Then turn right, and walk east along beside the other skidoo track for 30 meters. You'll find the spot I mean there."

We can match this real world situation up with objects in the mathematical world, just as the Greeks matched objects in the real world up with shapes in the mathematical world (see section 2.1). For instance, we could start by representing each skidoo path as a line, and giving each line a variable name- for example, JeanniesLine and JoysLine. We could draw the two lines so that they crossed each other at right angles. We could also give this crossing point a name- for example, CrossingPoint. We could then mark off points on each of the lines so that each point represented a space of a meter along one of the actual skidoo paths.
 

Now we have described the situation mathematically, in terms of objects that exist in the mathematical world.

If you use maps or GPSs, all of these ideas will not be new to you. You will probably also be familiar with the idea of giving someone a set of coordinates so that they can locate a particular point on the map. This idea will come in handy in the next section, section 4.2